Daily Papers

Transformers and large language models~(LLMs) have seen rapid adoption in all domains. Their sizes have exploded to hundreds of billions of parameters and keep increasing. Under these circumstances, the training of transformers is very expensive and often hits a ``memory wall'', i.e., even when using 3D parallelism (pipeline, tensor, data) and aggregating the memory of many GPUs, it is still not enough to hold the necessary data structures (model parameters, optimizer state, gradients, activations) in GPU memory. To compensate, state-of-the-art approaches offload the optimizer state, at least partially, to the host memory and perform hybrid CPU-GPU computations. However, the management of the combined host-GPU memory is often suboptimal and results in poor overlapping between data movements and computations. This leads to missed opportunities to simultaneously leverage the interconnect bandwidth and computational capabilities of CPUs and GPUs. In this paper, we leverage a key observation that the interleaving of the forward, backward and update phases generate fluctuations in the GPU memory utilization, which can be exploited to dynamically move a part of the optimizer state between the host and the GPU memory at each iteration. To this end, we design and implement \proj, a novel technique to split the LLM into subgroups, whose update phase is scheduled on either the CPU or the GPU based on our proposed performance model that addresses the trade-off between data movement cost, acceleration on the GPUs vs the CPUs, and competition for shared resources. We integrate our approach with DeepSpeed and demonstrate 2.5times faster iterations over state-of-the-art approaches using extensive experiments.

FP8 training has emerged as a promising method for improving training efficiency. Existing frameworks accelerate training by applying FP8 computation to linear layers while leaving optimizer states and activations in higher precision, which fails to fully optimize memory usage. This paper introduces COAT (Compressing Optimizer States and Activations for FP8 Training), a novel FP8 training framework designed to significantly reduce memory footprint when training large models. COAT addresses current limitations through two key innovations: (1) Dynamic Range Expansion, which aligns optimizer state distributions more closely with the FP8 representation range, thereby reducing quantization error, and (2) Mixed-Granularity Activation Quantization, which optimizes activation memory using a combination of per-tensor and per-group quantization strategies. Experiments demonstrate that COAT effectively reduces end-to-end training memory footprint by 1.54x compared to BF16 while achieving nearly lossless performance across various tasks, such as Large Language Model pretraining and fine-tuning and Vision Language Model training. COAT also achieves a 1.43x end-to-end training speedup compared to BF16, performing on par with or surpassing TransformerEngine's speedup. COAT enables efficient full-parameter training of large models on fewer GPUs, and facilitates doubling the batch size in distributed training settings, providing a practical solution for scaling large-scale model training. The code is available at https://github.com/NVlabs/COAT.

The Muon optimizer, based on matrix orthogonalization, has recently shown faster convergence and up to 2x computational efficiency over AdamW in LLM pretraining. Like AdamW, Muon is stateful, requiring storage of both model weights and accumulated gradients. While 8-bit AdamW variants mitigate this overhead using blockwise quantization, they are typically stable only under dynamic quantization - which improves stability on linear quantization for extreme values. In this paper, we introduce the 8-bit Muon optimizer using blockwise quantization, supporting both linear and dynamic schemes. We demonstrate that 8-bit Muon maintains stability under both, while delivering sim74\% reduction in memory footprint compared to full-precision Muon. In extensive experiments, 8-bit Muon closely matches the performance of Muon while outperforming AdamW and 8-bit AdamW in pre-training a 1.6B model on 4B FineWeb tokens. It also shows competitive results when fine-tuning the Llama 3.2 3B model on post-training data. We also provide a theoretical perspective to help explain this robustness under quantization.

Optimizer states are a major source of memory consumption for training neural networks, limiting the maximum trainable model within given memory budget. Compressing the optimizer states from 32-bit floating points to lower bitwidth is promising to reduce the training memory footprint, while the current lowest achievable bitwidth is 8-bit. In this work, we push optimizer states bitwidth down to 4-bit through a detailed empirical analysis of first and second moments. Specifically, we find that moments have complicated outlier patterns, that current block-wise quantization cannot accurately approximate. We use a smaller block size and propose to utilize both row-wise and column-wise information for better quantization. We further identify a zero point problem of quantizing the second moment, and solve this problem with a linear quantizer that excludes the zero point. Our 4-bit optimizers are evaluated on a wide variety of benchmarks including natural language understanding, machine translation, image classification, and instruction tuning. On all the tasks our optimizers can achieve comparable accuracy with their full-precision counterparts, while enjoying better memory efficiency.

With the increase in the number of parameters in large language models, the process of pre-training and fine-tuning increasingly demands larger volumes of GPU memory. A significant portion of this memory is typically consumed by the optimizer state. To overcome this challenge, recent approaches such as low-rank adaptation (LoRA (Hu et al., 2021)), low-rank gradient projection (GaLore (Zhao et al., 2024)), and blockwise optimization (BAdam (Luo et al., 2024)) have been proposed. However, in all these algorithms, the effective rank of the weight updates remains low-rank, which can lead to a substantial loss of information from the gradient. This loss can be critically important, especially during the pre-training stage. In this paper, we introduce FRUGAL (Full-Rank Updates with GrAdient spLitting), a new memory-efficient optimization framework. FRUGAL leverages gradient splitting to perform low-dimensional updates using advanced algorithms (such as Adam), while updates along the remaining directions are executed via state-free methods like SGD or signSGD (Bernstein et al., 2018). Our framework can be integrated with various low-rank update selection techniques, including GaLore and BAdam. We provide theoretical convergence guarantees for our framework when using SGDM for low-dimensional updates and SGD for state-free updates. Additionally, our method consistently outperforms concurrent approaches across various fixed memory budgets, achieving state-of-the-art results in pre-training and fine-tuning tasks while balancing memory efficiency and performance metrics.

Large Language Models (LLMs) have demonstrated impressive performance across various downstream tasks. When training these models, there is a growing inclination to process more tokens on larger training scales but with relatively smaller model sizes. Zero Redundancy Optimizer (ZeRO), although effective in conventional training environments, grapples with scaling challenges when confronted with this emerging paradigm. To this end, we propose a novel LLM training framework AMSP, which undertakes a granular partitioning of model states, encompassing parameters (P), gradient (G), and optimizer states (OS). Specifically, AMSP(1) builds a unified partitioning space, enabling independent partitioning strategies for P, G, and OS; (2) incorporates a scale-aware partitioner to autonomously search for optimal partitioning strategies: (3) designs a dedicated communication optimizer to ensure proficient management of data placement discrepancies arising from diverse partitioning strategies. Our evaluations show that AMSP achieves up to 90.3% scaling efficiency across 1024 GPUs.

Mixture-of-Experts (MoE) has emerged as a promising sparse paradigm for scaling up pre-trained models (PTMs) with remarkable cost-effectiveness. However, the dynamic nature of MoE leads to rapid fluctuations and imbalances in expert loads during training, resulting in significant straggler effects that hinder training performance when using expert parallelism (EP). Existing MoE training systems attempt to mitigate these effects through expert rearrangement strategies, but they face challenges in terms of memory efficiency and timeliness of rearrangement. This paper proposes Fully Sharded Sparse Data Parallelism (FSSDP), an innovative approach that tackles the parallelization of MoE layers and potential straggler effects caused by imbalanced expert loads from a new perspective. FSSDP fully shards the parameters and optimizer states of MoE layers across devices and sparsely materializes MoE parameters from scratch in each iteration with two sparse collectives SparseAllGather and SparseReduceScatter. We build Hecate, a high-performance MoE training system that incorporates FSSDP to fully unlock its potential. Hecate introduces heterogeneous sharding, sparse materialization, and re-materialization techniques to construct flexible and efficient expert placements with low memory and communication overhead. Our evaluation reveals that Hecate achieves up to 3.54x speedup compared over state-of-the-art MoE training systems and consistently demonstrates improvements across model architectures and hardware environments.

Training large neural networks typically requires sharing gradients between accelerators through specialized high-speed interconnects. Drawing from the signal processing principles of frequency decomposition and energy compaction, we demonstrate that synchronizing full optimizer states and model parameters during training is unnecessary. By decoupling momentum updates and allowing controlled divergence in optimizer states across accelerators, we achieve improved convergence compared to state-of-the-art optimizers. We introduce {De}coupled {Mo}mentum (DeMo), a fused optimizer and data parallel algorithm that reduces inter-accelerator communication requirements by several orders of magnitude. This enables training of large neural networks even with limited network bandwidth and heterogeneous hardware. Our method is topology-agnostic and architecture-independent and supports scalable clock-synchronous distributed training with negligible compute and memory overhead. Empirical results show that models trained with DeMo match or exceed the performance of equivalent models trained with AdamW, while eliminating the need for high-speed interconnects when pre-training large scale foundation models. An open source reference PyTorch implementation is published on GitHub at https://github.com/bloc97/DeMo

Large language models (LLMs) require immense resources for training and inference. Quantization, a technique that reduces the precision of model parameters, offers a promising solution for improving LLM efficiency and sustainability. While post-training quantization methods typically achieve 4-8 bits per parameter, recent research suggests that training LLMs with 1.58 bits per weight parameter from scratch can maintain model accuracy while greatly reducing memory requirements and energy consumption at inference time. Here, we investigate a training strategy for quantization-aware pre-training, where the models are first trained with 16-bit precision and then transition into 1.58-bit quantization-aware training. Our results on 11 downstream tasks show that this 16-to-1.58-bit training strategy is preferable over full 1.58-bit training and leaves models closer to those which have undergone 16-bit training. We further investigate the effects of retaining the optimizer state at the transition point and gradually phasing in quantization strength -- finding that both techniques alleviate the magnitude of loss spikes, but also that these effects can be compensated through further training.

Full-parameter fine-tuning has become the go-to choice for adapting language models (LMs) to downstream tasks due to its excellent performance. As LMs grow in size, fine-tuning the full parameters of LMs requires a prohibitively large amount of GPU memory. Existing approaches utilize zeroth-order optimizer to conserve GPU memory, which can potentially compromise the performance of LMs as non-zero order optimizers tend to converge more readily on most downstream tasks. In this paper, we propose a novel optimizer-independent end-to-end hierarchical fine-tuning strategy, HiFT, which only updates a subset of parameters at each training step. HiFT can significantly reduce the amount of gradients and optimizer state parameters residing in GPU memory at the same time, thereby reducing GPU memory usage. Our results demonstrate that: (1) HiFT achieves comparable performance to parameter-efficient fine-tuning and standard full parameter fine-tuning. (2) HiFT supports various optimizers including AdamW, AdaGrad, SGD, etc. (3) HiFT can save more than 60\% GPU memory compared with standard full-parameter fine-tuning for 7B model. (4) HiFT enables full-parameter fine-tuning of a 7B model on single 48G A6000 with a precision of 32 using the AdamW optimizer, without using any memory saving techniques.

Training Large Language Models (LLMs) presents significant memory challenges, predominantly due to the growing size of weights and optimizer states. Common memory-reduction approaches, such as low-rank adaptation (LoRA), add a trainable low-rank matrix to the frozen pre-trained weight in each layer, reducing trainable parameters and optimizer states. However, such approaches typically underperform training with full-rank weights in both pre-training and fine-tuning stages since they limit the parameter search to a low-rank subspace and alter the training dynamics, and further, may require full-rank warm start. In this work, we propose Gradient Low-Rank Projection (GaLore), a training strategy that allows full-parameter learning but is more memory-efficient than common low-rank adaptation methods such as LoRA. Our approach reduces memory usage by up to 65.5% in optimizer states while maintaining both efficiency and performance for pre-training on LLaMA 1B and 7B architectures with C4 dataset with up to 19.7B tokens, and on fine-tuning RoBERTa on GLUE tasks. Our 8-bit GaLore further reduces optimizer memory by up to 82.5% and total training memory by 63.3%, compared to a BF16 baseline. Notably, we demonstrate, for the first time, the feasibility of pre-training a 7B model on consumer GPUs with 24GB memory (e.g., NVIDIA RTX 4090) without model parallel, checkpointing, or offloading strategies.

Recently, a wide range of memory-efficient LLM training algorithms have gained substantial popularity. These methods leverage the low-rank structure of gradients to project optimizer states into a subspace using projection matrix found by singular value decomposition (SVD). However, convergence of these algorithms is highly dependent on the update rules of their projection matrix. In this work, we provide the first convergence guarantee for arbitrary update rules of projection matrix. This guarantee is generally applicable to optimizers that can be analyzed with Hamiltonian Descent, including most common ones, such as LION, Adam. Inspired by our theoretical understanding, we propose Online Subspace Descent, a new family of subspace descent optimizer without SVD. Instead of updating the projection matrix with eigenvectors, Online Subspace Descent updates the projection matrix with online PCA. Online Subspace Descent is flexible and introduces only minimum overhead to training. We show that for the task of pretraining LLaMA models ranging from 60M to 7B parameters on the C4 dataset, Online Subspace Descent achieves lower perplexity and better downstream tasks performance than state-of-the-art low-rank training methods across different settings and narrows the gap with full-rank baselines.

As the size of large language models grows exponentially, GPU memory has become a bottleneck for adapting these models to downstream tasks. In this paper, we aim to push the limits of memory-efficient training by minimizing memory usage on model weights, gradients, and optimizer states, within a unified framework. Our idea is to eliminate both gradients and optimizer states using zeroth-order optimization, which approximates gradients by perturbing weights during forward passes to identify gradient directions. To minimize memory usage on weights, we employ model quantization, e.g., converting from bfloat16 to int4. However, directly applying zeroth-order optimization to quantized weights is infeasible due to the precision gap between discrete weights and continuous gradients, which would otherwise require de-quantization and re-quantization. To overcome this challenge, we propose Quantized Zeroth-order Optimization (QZO), a novel approach that perturbs the continuous quantization scale for gradient estimation and uses a directional derivative clipping method to stabilize training. QZO is orthogonal to both scalar-based and codebook-based post-training quantization methods. Compared to full-parameter fine-tuning in bfloat16, QZO can reduce the total memory cost by more than 18times for 4-bit LLMs, and enables fine-tuning Llama-2-13B and Stable Diffusion 3.5 Large within a single 24GB GPU.

Large language model (LLM) training and finetuning are often bottlenecked by limited GPU memory. While existing projection-based optimization methods address this by projecting gradients into a lower-dimensional subspace to reduce optimizer state memory, they typically rely on dense projection matrices, which can introduce computational and memory overheads. In this work, we propose Grass (GRAdient Stuctured Sparsification), a novel approach that leverages sparse projections to transform gradients into structured sparse updates. This design not only significantly reduces memory usage for optimizer states but also minimizes gradient memory footprint, computation, and communication costs, leading to substantial throughput improvements. Extensive experiments on pretraining and finetuning tasks demonstrate that Grass achieves competitive performance to full-rank training and existing projection-based methods. Notably, Grass enables half-precision pretraining of a 13B parameter LLaMA model on a single 40GB A100 GPU--a feat infeasible for previous methods--and yields up to a 2times throughput improvement on an 8-GPU system. Code can be found at https://github.com/aashiqmuhamed/GRASS .

Recent work has shown that orthonormal matrix updates speed up neural network optimization, improve training stability, and offer better hyperparameter transfer across model sizes. Applying these updates efficiently when model weights and optimizer states are sharded across a large-scale distributed LLM training system remains a major challenge. We introduce Dion (DIstributed OrthoNormalization), a scalable and communication-efficient orthonormalizing optimizer. Dion leverages low-rank approximation and decoupled momentum buffers, eliminating the need for full gradient synchronization while producing numerically equivalent results. It is compatible with simultaneous DDP, FSDP, and TP parallelism, and it computes an orthonormalized update without unsharding a full parameter matrix on any single device. We evaluate Dion on language models from 120M to 3B parameters and find that its benefits improve with increasing model size and batch size.

This paper is dedicated to an efficient compression of weights and optimizer states (called checkpoints) obtained at different stages during a neural network training process. First, we propose a prediction-based compression approach, where values from the previously saved checkpoint are used for context modeling in arithmetic coding. Second, in order to enhance the compression performance, we also propose to apply pruning and quantization of the checkpoint values. Experimental results show that our approach achieves substantial bit size reduction, while enabling near-lossless training recovery from restored checkpoints, preserving the model's performance and making it suitable for storage-limited environments.

The growing prevalence of Large Language Models (LLMs) and their substantial memory requirements have prompted renewed interest in CPU offloading as a method to compensate for limited GPU memory. In particular, when CPU memory is leveraged to temporarily store intermediate states of LLMs, CPU memory becomes a new bottleneck and soon reaches the capacity limitation of commodity CPUs. In this work, we investigate the effectiveness of Compute Express Link (CXL) add-in card (AIC) memory as an extension to CPU memory, enabling larger model sizes and longer context lengths during fine-tuning. Through extensive benchmarking, this study quantifies the performance overhead introduced by transferring data between CXL memory, CPU, and GPUs, focusing on how concurrency and data volume influence bandwidth utilization and latency. This study also compares CPUbased optimizer steps when model parameters, gradients, and optimizer states reside in local memory versus CXL memory, revealing that naive adoption of CXL often degrades performance during the optimizer phase. To overcome these challenges, this study proposes a CXL-aware allocation to strategically partition CPU offloading workloads across both local and CXL memory. This study further demonstrates that employing multiple AICs significantly reduces bandwidth contention, thus improving scalability. Experimental results show that these optimizations enable efficient long-context LLM fine-tuning, underscoring CXL as a promising avenue for unlocking the full potential of CPU offloading in long-context LLM fine-tuning.

We propose a new variant of the Adam optimizer [Kingma and Ba, 2014] called MICROADAM that specifically minimizes memory overheads, while maintaining theoretical convergence guarantees. We achieve this by compressing the gradient information before it is fed into the optimizer state, thereby reducing its memory footprint significantly. We control the resulting compression error via a novel instance of the classical error feedback mechanism from distributed optimization [Seide et al., 2014, Alistarh et al., 2018, Karimireddy et al., 2019] in which the error correction information is itself compressed to allow for practical memory gains. We prove that the resulting approach maintains theoretical convergence guarantees competitive to those of AMSGrad, while providing good practical performance. Specifically, we show that MICROADAM can be implemented efficiently on GPUs: on both million-scale (BERT) and billion-scale (LLaMA) models, MicroAdam provides practical convergence competitive to that of the uncompressed Adam baseline, with lower memory usage and similar running time. Our code is available at https://github.com/IST-DASLab/MicroAdam.

In this paper, we explore FP8 low-bit data formats for efficient training of large language models (LLMs). Our key insight is that most variables, such as gradients and optimizer states, in LLM training can employ low-precision data formats without compromising model accuracy and requiring no changes to hyper-parameters. Specifically, we propose a new FP8 automatic mixed-precision framework for training LLMs. This framework offers three levels of FP8 utilization to streamline mixed-precision and distributed parallel training for LLMs. It gradually incorporates 8-bit gradients, optimizer states, and distributed learning in an incremental manner. Experiment results show that, during the training of GPT-175B model on H100 GPU platform, our FP8 mixed-precision training framework not only achieved a remarkable 42% reduction in real memory usage but also ran 64% faster than the widely adopted BF16 framework (i.e., Megatron-LM), surpassing the speed of Nvidia Transformer Engine by 17%. This largely reduces the training costs for large foundation models. Furthermore, our FP8 mixed-precision training methodology is generic. It can be seamlessly applied to other tasks such as LLM instruction tuning and reinforcement learning with human feedback, offering savings in fine-tuning expenses. Our FP8 low-precision training framework is open-sourced at {https://github.com/Azure/MS-AMP}{aka.ms/MS.AMP}.

Second-order optimizers, maintaining a matrix termed a preconditioner, are superior to first-order optimizers in both theory and practice. The states forming the preconditioner and its inverse root restrict the maximum size of models trained by second-order optimizers. To address this, compressing 32-bit optimizer states to lower bitwidths has shown promise in reducing memory usage. However, current approaches only pertain to first-order optimizers. In this paper, we propose the first 4-bit second-order optimizers, exemplified by 4-bit Shampoo, maintaining performance similar to that of 32-bit ones. We show that quantizing the eigenvector matrix of the preconditioner in 4-bit Shampoo is remarkably better than quantizing the preconditioner itself both theoretically and experimentally. By rectifying the orthogonality of the quantized eigenvector matrix, we enhance the approximation of the preconditioner's eigenvector matrix, which also benefits the computation of its inverse 4-th root. Besides, we find that linear square quantization slightly outperforms dynamic tree quantization when quantizing second-order optimizer states. Evaluation on various networks for image classification demonstrates that our 4-bit Shampoo achieves comparable test accuracy to its 32-bit counterpart while being more memory-efficient. The source code will be made available.

Scaling foundation model training with Distributed Data Parallel (DDP) methods is bandwidth-limited. Existing infrequent communication methods like Local SGD were designed to synchronize only model parameters and cannot be trivially applied to adaptive optimizers due to additional optimizer states. Current approaches extending Local SGD either lack convergence guarantees or require synchronizing all optimizer states, tripling communication costs. We propose Desynced Low Communication Adaptive Optimizers (DES-LOC), a family of optimizers assigning independent synchronization periods to parameters and momenta, enabling lower communication costs while preserving convergence. Through extensive experiments on language models of up to 1.7B, we show that DES-LOC can communicate 170x less than DDP and 2x less than the previous state-of-the-art Local ADAM. Furthermore, unlike previous heuristic approaches, DES-LOC is suited for practical training scenarios prone to system failures. DES-LOC offers a scalable, bandwidth-efficient, and fault-tolerant solution for foundation model training.

Large language models (LLMs) face the challenges in fine-tuning and deployment due to their high memory demands and computational costs. While parameter-efficient fine-tuning (PEFT) methods aim to reduce the memory usage of the optimizer state during fine-tuning, the inherent size of pre-trained LLM weights continues to be a pressing concern. Even though quantization techniques are widely proposed to ease memory demands and accelerate LLM inference, most of these techniques are geared towards the deployment phase. To bridge this gap, this paper presents Parameter-Efficient and Quantization-aware Adaptation (PEQA) - a simple yet effective method that combines the advantages of PEFT with quantized LLMs. By updating solely the quantization scales, PEQA can be directly applied to quantized LLMs, ensuring seamless task transitions. Parallel to existing PEFT methods, PEQA significantly reduces the memory overhead associated with the optimizer state. Furthermore, it leverages the advantages of quantization to substantially reduce model sizes. Even after fine-tuning, the quantization structure of a PEQA-tuned LLM remains intact, allowing for accelerated inference on the deployment stage. We employ PEQA-tuning for task-specific adaptation on LLMs with up to 65 billion parameters. To assess the logical reasoning and language comprehension of PEQA-tuned LLMs, we fine-tune low-bit quantized LLMs using a instruction dataset. Our results show that even when LLMs are quantized to below 4-bit precision, their capabilities in language modeling, few-shot in-context learning, and comprehension can be resiliently restored to (or even improved over) their full-precision original performances with PEQA.

Low-precision formats such as float8 have been introduced in machine learning accelerated hardware to improve computational efficiency for large language models training and inference. Nevertheless, adoption by the ML community has been slowed down by the complex, and sometimes brittle, techniques required to match higher precision training accuracy. In this work, we present Scalify, a end-to-end scale propagation paradigm for computational graphs, generalizing and formalizing existing tensor scaling methods. Experiment results show that Scalify supports out-of-the-box float8 matrix multiplication and gradients representation, as well as float16 optimizer state storage. Our JAX implementation of Scalify is open-sourced at https://github.com/graphcore-research/jax-scalify

In this work, we question the necessity of adaptive gradient methods for training deep neural networks. SGD-SaI is a simple yet effective enhancement to stochastic gradient descent with momentum (SGDM). SGD-SaI performs learning rate Scaling at Initialization (SaI) to distinct parameter groups, guided by their respective gradient signal-to-noise ratios (g-SNR). By adjusting learning rates without relying on adaptive second-order momentum, SGD-SaI helps prevent training imbalances from the very first iteration and cuts the optimizer's memory usage by half compared to AdamW. Despite its simplicity and efficiency, SGD-SaI consistently matches or outperforms AdamW in training a variety of Transformer-based tasks, effectively overcoming a long-standing challenge of using SGD for training Transformers. SGD-SaI excels in ImageNet-1K classification with Vision Transformers(ViT) and GPT-2 pretraining for large language models (LLMs, transformer decoder-only), demonstrating robustness to hyperparameter variations and practicality for diverse applications. We further tested its robustness on tasks like LoRA fine-tuning for LLMs and diffusion models, where it consistently outperforms state-of-the-art optimizers. From a memory efficiency perspective, SGD-SaI achieves substantial memory savings for optimizer states, reducing memory usage by 5.93 GB for GPT-2 (1.5B parameters) and 25.15 GB for Llama2-7B compared to AdamW in full-precision training settings.

Training large language models (LLMs) is often bottlenecked by extreme memory demands, with optimizer states dominating the footprint. Recent works mitigates this cost by projecting gradients into low-dimensional subspaces using sophisticated update strategies. In this paper, we analyze the dynamics of gradient space and its underlying subspaces. We find that while a small subspace captures most gradient energy, a significant portion still resides in the residual bulk; moreover, the influence of the core subspace diminishes over time and in deeper layers. We also observe that the gradient space exhibits near-flat curvature, calling for algorithms that explicitly account for this geometry. Motivated by these insights, we introduce a suite of randomized algorithms, GrassWalk and GrassJump, which exploit subspace and achieve state-of-the-art memory savings while improving performance on LLaMA-1B and LLaMA-7B pretraining.

Pretraining large language models is a complex endeavor influenced by multiple factors, including model architecture, data quality, training continuity, and hardware constraints. In this paper, we share insights gained from the experience of training DMaS-LLaMa-Lite, a fully open source, 1.7-billion-parameter, LLaMa-based model, on approximately 20 billion tokens of carefully curated data. We chronicle the full training trajectory, documenting how evolving validation loss levels and downstream benchmarks reflect transitions from incoherent text to fluent, contextually grounded output. Beyond standard quantitative metrics, we highlight practical considerations such as the importance of restoring optimizer states when resuming from checkpoints, and the impact of hardware changes on training stability and throughput. While qualitative evaluation provides an intuitive understanding of model improvements, our analysis extends to various performance benchmarks, demonstrating how high-quality data and thoughtful scaling enable competitive results with significantly fewer training tokens. By detailing these experiences and offering training logs, checkpoints, and sample outputs, we aim to guide future researchers and practitioners in refining their pretraining strategies. The training script is available on Github at https://github.com/McGill-DMaS/DMaS-LLaMa-Lite-Training-Code. The model checkpoints are available on Huggingface at https://huggingface.co/collections/McGill-DMaS/dmas-llama-lite-6761d97ba903f82341954ceb.

Low-rank adaptation (LoRA) has become the default approach to fine-tune large language models (LLMs) due to its significant reduction in trainable parameters. However, trainable parameter demand for LoRA increases with increasing model embedding dimensions, leading to high compute costs. Additionally, its backward updates require storing high-dimensional intermediate activations and optimizer states, demanding high peak GPU memory. In this paper, we introduce large model fine-tuning via spectrally decomposed low-dimensional adaptation (LaMDA), a novel approach to fine-tuning large language models, which leverages low-dimensional adaptation to achieve significant reductions in trainable parameters and peak GPU memory footprint. LaMDA freezes a first projection matrix (PMA) in the adaptation path while introducing a low-dimensional trainable square matrix, resulting in substantial reductions in trainable parameters and peak GPU memory usage. LaMDA gradually freezes a second projection matrix (PMB) during the early fine-tuning stages, reducing the compute cost associated with weight updates to enhance parameter efficiency further. We also present an enhancement, LaMDA++, incorporating a ``lite-weight" adaptive rank allocation for the LoRA path via normalized spectrum analysis of pre-trained model weights. We evaluate LaMDA/LaMDA++ across various tasks, including natural language understanding with the GLUE benchmark, text summarization, natural language generation, and complex reasoning on different LLMs. Results show that LaMDA matches or surpasses the performance of existing alternatives while requiring up to 17.7x fewer parameter updates and up to 1.32x lower peak GPU memory usage during fine-tuning. Code will be publicly available.

Large language models (LLMs) are notoriously memory-intensive during training, particularly with the popular AdamW optimizer. This memory burden necessitates using more or higher-end GPUs or reducing batch sizes, limiting training scalability and throughput. To address this, various memory-efficient optimizers have been proposed to reduce optimizer memory usage. However, they face critical challenges: (i) reliance on costly SVD operations; (ii) significant performance trade-offs compared to AdamW; and (iii) still substantial optimizer memory overhead to maintain competitive performance. In this work, we identify that AdamW's learning rate adaptation rule can be effectively coarsened as a structured learning rate update. Based on this insight, we propose Approximated Gradient Scaling for Memory-Efficient LLM Optimization (APOLLO), which approximates learning rate scaling using an auxiliary low-rank optimizer state based on pure random projection. This structured learning rate update rule makes APOLLO highly tolerant to further memory reductions while delivering comparable pre-training performance. Even its rank-1 variant, APOLLO-Mini, achieves superior pre-training performance compared to AdamW with SGD-level memory costs. Extensive experiments demonstrate that the APOLLO series performs on-par with or better than AdamW, while achieving greater memory savings by nearly eliminating the optimization states of AdamW. These savings provide significant system-level benefits: (1) Enhanced Throughput: 3x throughput on an 8xA100-80GB setup compared to AdamW by supporting 4x larger batch sizes. (2) Improved Model Scalability: Pre-training LLaMA-13B with naive DDP on A100-80GB GPUs without system-level optimizations. (3) Low-End GPU Friendly Pre-training: Pre-training LLaMA-7B on a single GPU using less than 12 GB of memory with weight quantization.

LLMs have seen rapid adoption in all domains. They need to be trained on high-end high-performance computing (HPC) infrastructures and ingest massive amounts of input data. Unsurprisingly, at such a large scale, unexpected events (e.g., failures of components, instability of the software, undesirable learning patterns, etc.), are frequent and typically impact the training in a negative fashion. Thus, LLMs need to be checkpointed frequently so that they can be rolled back to a stable state and subsequently fine-tuned. However, given the large sizes of LLMs, a straightforward checkpointing solution that directly writes the model parameters and optimizer state to persistent storage (e.g., a parallel file system), incurs significant I/O overheads. To address this challenge, in this paper we study how to reduce the I/O overheads for enabling fast and scalable checkpointing for LLMs that can be applied at high frequency (up to the granularity of individual iterations) without significant impact on the training process. Specifically, we introduce a lazy asynchronous multi-level approach that takes advantage of the fact that the tensors making up the model and optimizer state shards remain immutable for extended periods of time, which makes it possible to copy their content in the background with minimal interference during the training process. We evaluate our approach at scales of up to 180 GPUs using different model sizes, parallelism settings, and checkpointing frequencies. The results show up to 48times faster checkpointing and 2.2times faster end-to-end training runtime compared with the state-of-art checkpointing approaches.

Recent advances in large language models have brought immense value to the world, with their superior capabilities stemming from the massive number of parameters they utilize. However, even the GPUs with the highest memory capacities, currently peaking at 80GB, are far from sufficient to accommodate these vast parameters and their associated optimizer states when conducting stochastic gradient descent-based optimization. One approach to hosting such huge models is to aggregate device memory from many GPUs. However, this approach introduces prohibitive costs for most academic researchers, who always have a limited budget for many high-end GPU servers. In this paper, we focus on huge model fine-tuning on a single, even low-end, GPU in a commodity server, which is accessible to most AI researchers. In such a scenario, the state-of-the-art work ZeRO-Infinity suffers from two severe issues when running in a commodity server: 1) low GPU utilization due to inefficient swapping, and 2) limited trainable model size due to CPU memory capacity. The underlying reason is that ZeRO-Infinity is optimized for running on high-end GPU servers. To this end, we present Fuyou, a low-cost training framework that enables efficient 100B huge model fine-tuning on a low-end server with a low-end GPU and limited CPU memory capacity. The key idea is to add the SSD-CPU communication as an optimization dimension and thus carefully co-optimize computation and data swapping from a systematic approach to maximize GPU utilization. The experimental results show that 1) Fuyou is able to fine-tune 175B GPT-3 on a consumer GPU RTX 4090 with high GPU utilization, while ZeRO-Infinity fails to fine-tune; and 2) when training a small GPT-3 13B model, Fuyou achieves 156 TFLOPS on an RTX 4090 GPU while ZeRO-Infinity only achieves 45 TFLOPS.

Large language models have achieved remarkable success, but their extensive parameter size necessitates substantial memory for training, thereby setting a high threshold. While the recently proposed low-memory optimization (LOMO) reduces memory footprint, its optimization technique, akin to stochastic gradient descent, is sensitive to hyper-parameters and exhibits suboptimal convergence, failing to match the performance of the prevailing optimizer for large language models, AdamW. Through empirical analysis of the Adam optimizer, we found that, compared to momentum, the adaptive learning rate is more critical for bridging the gap. Building on this insight, we introduce the low-memory optimization with adaptive learning rate (AdaLomo), which offers an adaptive learning rate for each parameter. To maintain memory efficiency, we employ non-negative matrix factorization for the second-order moment estimation in the optimizer state. Additionally, we suggest the use of a grouped update normalization to stabilize convergence. Our experiments with instruction-tuning and further pre-training demonstrate that AdaLomo achieves results on par with AdamW, while significantly reducing memory requirements, thereby lowering the hardware barrier to training large language models.

Stateful optimizers maintain gradient statistics over time, e.g., the exponentially smoothed sum (SGD with momentum) or squared sum (Adam) of past gradient values. This state can be used to accelerate optimization compared to plain stochastic gradient descent but uses memory that might otherwise be allocated to model parameters, thereby limiting the maximum size of models trained in practice. In this paper, we develop the first optimizers that use 8-bit statistics while maintaining the performance levels of using 32-bit optimizer states. To overcome the resulting computational, quantization, and stability challenges, we develop block-wise dynamic quantization. Block-wise quantization divides input tensors into smaller blocks that are independently quantized. Each block is processed in parallel across cores, yielding faster optimization and high precision quantization. To maintain stability and performance, we combine block-wise quantization with two additional changes: (1) dynamic quantization, a form of non-linear optimization that is precise for both large and small magnitude values, and (2) a stable embedding layer to reduce gradient variance that comes from the highly non-uniform distribution of input tokens in language models. As a result, our 8-bit optimizers maintain 32-bit performance with a small fraction of the memory footprint on a range of tasks, including 1.5B parameter language modeling, GLUE finetuning, ImageNet classification, WMT'14 machine translation, MoCo v2 contrastive ImageNet pretraining+finetuning, and RoBERTa pretraining, without changes to the original optimizer hyperparameters. We open-source our 8-bit optimizers as a drop-in replacement that only requires a two-line code change.

Extensive system scales (i.e. thousands of GPU/TPUs) and prolonged training periods (i.e. months of pretraining) significantly escalate the probability of failures when training large language models (LLMs). Thus, efficient and reliable fault-tolerance methods are in urgent need. Checkpointing is the primary fault-tolerance method to periodically save parameter snapshots from GPU memory to disks via CPU memory. In this paper, we identify the frequency of existing checkpoint-based fault-tolerance being significantly limited by the storage I/O overheads, which results in hefty re-training costs on restarting from the nearest checkpoint. In response to this gap, we introduce an in-memory fault-tolerance framework for large-scale LLM pretraining. The framework boosts the efficiency and reliability of fault tolerance from three aspects: (1) Reduced Data Transfer and I/O: By asynchronously caching parameters, i.e., sharded model parameters, optimizer states, and RNG states, to CPU volatile memory, Our framework significantly reduces communication costs and bypasses checkpoint I/O. (2) Enhanced System Reliability: Our framework enhances parameter protection with a two-layer hierarchy: snapshot management processes (SMPs) safeguard against software failures, together with Erasure Coding (EC) protecting against node failures. This double-layered protection greatly improves the survival probability of the parameters compared to existing checkpointing methods. (3) Improved Snapshotting Frequency: Our framework achieves more frequent snapshotting compared with asynchronous checkpointing optimizations under the same saving time budget, which improves the fault tolerance efficiency. Empirical results demonstrate that Our framework minimizes the overhead of fault tolerance of LLM pretraining by effectively leveraging redundant CPU resources.

Large language models (LLMs) have shown impressive performance across a range of natural language processing tasks. However, their vast number of parameters introduces significant memory challenges during training, particularly when using memory-intensive optimizers like Adam. Existing memory-efficient algorithms often rely on techniques such as singular value decomposition projection or weight freezing. While these approaches help alleviate memory constraints, they generally produce suboptimal results compared to full-rank updates. In this paper, we investigate the memory-efficient method beyond low-rank training, proposing a novel solution called Gradient Wavelet Transform (GWT), which applies wavelet transforms to gradients in order to significantly reduce the memory requirements for maintaining optimizer states. We demonstrate that GWT can be seamlessly integrated with memory-intensive optimizers, enabling efficient training without sacrificing performance. Through extensive experiments on both pre-training and fine-tuning tasks, we show that GWT achieves state-of-the-art performance compared with advanced memory-efficient optimizers and full-rank approaches in terms of both memory usage and training performance.

Training large-scale neural networks in vision, and multimodal domains demands substantial memory resources, primarily due to the storage of optimizer states. While LoRA, a popular parameter-efficient method, reduces memory usage, it often suffers from suboptimal performance due to the constraints of low-rank updates. Low-rank gradient projection methods (e.g., GaLore, Flora) reduce optimizer memory by projecting gradients and moment estimates into low-rank spaces via singular value decomposition or random projection. However, they fail to account for inter-projection correlation, causing performance degradation, and their projection strategies often incur high computational costs. In this paper, we present COAP (Correlation-Aware Gradient Projection), a memory-efficient method that minimizes computational overhead while maintaining training performance. Evaluated across various vision, language, and multimodal tasks, COAP outperforms existing methods in both training speed and model performance. For LLaMA-1B, it reduces optimizer memory by 61% with only 2% additional time cost, achieving the same PPL as AdamW. With 8-bit quantization, COAP cuts optimizer memory by 81% and achieves 4x speedup over GaLore for LLaVA-v1.5-7B fine-tuning, while delivering higher accuracy.

We introduce two complementary techniques for efficient adaptive optimization that reduce memory requirements while accelerating training of large-scale neural networks. The first technique, Subset-Norm adaptive step size, generalizes AdaGrad-Norm and AdaGrad(-Coordinate) by reducing the second moment term's memory footprint from O(d) to O(d) through step-size sharing, where d is the model size. For non-convex smooth objectives under coordinate-wise sub-gaussian gradient noise, we prove a noise-adapted high-probability convergence guarantee showing improved dimensional dependence over existing methods. Our second technique, Subspace-Momentum, reduces the momentum state's memory footprint by operating in a low-dimensional subspace while applying standard SGD in the orthogonal complement. We establish high-probability convergence rates under similar relaxed assumptions. Empirical evaluation on LLaMA models from 60M to 1B parameters demonstrates the effectiveness of our methods, where combining subset-norm with subspace-momentum achieves Adam's validation perplexity in approximately half the training tokens (6.8B vs 13.1B) while using only 20% of the Adam's optimizer-states memory footprint and requiring minimal additional hyperparameter tuning.

Training LLMs presents significant memory challenges due to growing size of data, weights, and optimizer states. Techniques such as data and model parallelism, gradient checkpointing, and offloading strategies address this issue but are often infeasible due to hardware constraints. To mitigate memory usage, alternative methods like Parameter-Efficient-Fine-Tuning (PEFT) and GaLore approximate weights or optimizer states. PEFT methods, such as LoRA, have gained popularity for fine-tuning LLMs, though they require a full-rank warm start. In contrast, GaLore allows full-parameter learning while being more memory-efficient. This work introduces Natural GaLore, a simple drop in replacement for AdamW, which efficiently applies the inverse Empirical Fisher Information Matrix to low-rank gradients using Woodbury's Identity. We demonstrate that incorporating second-order information speeds up optimization significantly, especially when the iteration budget is limited. Empirical pretraining on 60M, 130M, 350M, and 1.1B parameter Llama models on C4 data demonstrate significantly lower perplexity over GaLore without additional memory overhead. By fine-tuning RoBERTa on the GLUE benchmark using Natural GaLore, we demonstrate significant reduction in gap 86.05% vs 86.28% for full-finetuning. Furthermore, fine-tuning the TinyLlama 1.1B model for function calling using the TinyAgent framework shows that Natural GaLore achieving 83.09% accuracy on the TinyAgent dataset, significantly outperforms 16-bit LoRA at 80.06% and even surpasses GPT4-Turbo by 4%, all while using 30% less memory. All code to reproduce the results are available at: https://github.com/selfsupervised-ai/Natural-GaLore.git

Finetuning large language models (LLMs) has been empirically effective on a variety of downstream tasks. Existing approaches to finetuning an LLM either focus on parameter-efficient finetuning, which only updates a small number of trainable parameters, or attempt to reduce the memory footprint during the training phase of the finetuning. Typically, the memory footprint during finetuning stems from three contributors: model weights, optimizer states, and intermediate activations. However, existing works still require considerable memory and none can simultaneously mitigate memory footprint for all three sources. In this paper, we present Quantized Side Tuing (QST), which enables memory-efficient and fast finetuning of LLMs by operating through a dual-stage process. First, QST quantizes an LLM's model weights into 4-bit to reduce the memory footprint of the LLM's original weights; QST also introduces a side network separated from the LLM, which utilizes the hidden states of the LLM to make task-specific predictions. Using a separate side network avoids performing backpropagation through the LLM, thus reducing the memory requirement of the intermediate activations. Furthermore, QST leverages several low-rank adaptors and gradient-free downsample modules to significantly reduce the trainable parameters, so as to save the memory footprint of the optimizer states. Experiments show that QST can reduce the total memory footprint by up to 2.3 times and speed up the finetuning process by up to 3 times while achieving competent performance compared with the state-of-the-art. When it comes to full finetuning, QST can reduce the total memory footprint up to 7 times.

Conventional wisdom dictates that small batch sizes make language model pretraining and fine-tuning unstable, motivating gradient accumulation, which trades off the number of optimizer steps for a proportional increase in batch size. While it is common to decrease the learning rate for smaller batch sizes, other hyperparameters are often held fixed. In this work, we revisit small batch sizes all the way down to batch size one, and we propose a rule for scaling Adam hyperparameters to small batch sizes. We find that small batch sizes (1) train stably, (2) are consistently more robust to hyperparameter choices, (3) achieve equal or better per-FLOP performance than larger batch sizes, and (4) notably enable stable language model training with vanilla SGD, even without momentum, despite storing no optimizer state. Building on these results, we provide practical recommendations for selecting a batch size and setting optimizer hyperparameters. We further recommend against gradient accumulation unless training on multiple devices with multiple model replicas, bottlenecked by inter-device bandwidth.

Heuristic design with large language models (LLMs) has emerged as a promising approach for tackling combinatorial optimization problems (COPs). However, existing approaches often rely on manually predefined evolutionary computation (EC) optimizers and single-task training schemes, which may constrain the exploration of diverse heuristic algorithms and hinder the generalization of the resulting heuristics. To address these issues, we propose Meta-Optimization of Heuristics (MoH), a novel framework that operates at the optimizer level, discovering effective optimizers through the principle of meta-learning. Specifically, MoH leverages LLMs to iteratively refine a meta-optimizer that autonomously constructs diverse optimizers through (self-)invocation, thereby eliminating the reliance on a predefined EC optimizer. These constructed optimizers subsequently evolve heuristics for downstream tasks, enabling broader heuristic exploration. Moreover, MoH employs a multi-task training scheme to promote its generalization capability. Experiments on classic COPs demonstrate that MoH constructs an effective and interpretable meta-optimizer, achieving state-of-the-art performance across various downstream tasks, particularly in cross-size settings.

Reinforcement Learning (RL) from Human Preference-based feedback is a popular paradigm for fine-tuning generative models, which has produced impressive models such as GPT-4 and Claude3 Opus. This framework often consists of two steps: learning a reward model from an offline preference dataset followed by running online RL to optimize the learned reward model. In this work, leveraging the idea of reset, we propose a new RLHF algorithm with provable guarantees. Motivated by the fact that offline preference dataset provides informative states (i.e., data that is preferred by the labelers), our new algorithm, Dataset Reset Policy Optimization (DR-PO), integrates the existing offline preference dataset into the online policy training procedure via dataset reset: it directly resets the policy optimizer to the states in the offline dataset, instead of always starting from the initial state distribution. In theory, we show that DR-PO learns to perform at least as good as any policy that is covered by the offline dataset under general function approximation with finite sample complexity. In experiments, we demonstrate that on both the TL;DR summarization and the Anthropic Helpful Harmful (HH) dataset, the generation from DR-PO is better than that from Proximal Policy Optimization (PPO) and Direction Preference Optimization (DPO), under the metric of GPT4 win-rate. Code for this work can be found at https://github.com/Cornell-RL/drpo.

Empirical scaling laws prescribe how to allocate parameters, data, and compute, while maximal-update parameterization (muP) enables learning-rate transfer across widths by equalizing early-time update magnitudes. However, in modern scale-invariant architectures, training quickly enters an optimizer-governed steady state where normalization layers create backward scale sensitivity and the effective learning rate becomes width dependent, degrading muP transfer. We address this by introducing a weight-decay scaling rule for AdamW that preserves sublayer gain across widths. Empirically, the singular-value spectrum of each matrix parameter scales in norm as eta/lambda with an approximately invariant shape; under width scaling d, we observe that the top singular value scales approximately as eta/lambdacdot d^{0.75}. Combining this observation with the muP learning-rate rule eta_2propto d^{-1} for matrix-like parameters implies an empirical weight-decay scaling rule lambda_2propto d that approximately keeps sublayer gains width invariant. Together with vector-like parameters trained at eta_1=Theta_d(1) and lambda_1=0, this yields zero-shot transfer of both learning rate and weight decay from proxy to target widths, removing per-width sweeps. We validate the rule on LLaMA-style Transformers and in a minimal synthetic setting, and we provide a simple diagnostic, matching top singular values, to check sublayer-gain invariance. Our results extend muP beyond the near-init regime by explicitly controlling steady-state scales set by the optimizer, offering a practical recipe for width-robust hyperparameter transfer under AdamW.

We introduce Kimi K2, a Mixture-of-Experts (MoE) large language model with 32 billion activated parameters and 1 trillion total parameters. We propose the MuonClip optimizer, which improves upon Muon with a novel QK-clip technique to address training instability while enjoying the advanced token efficiency of Muon. Based on MuonClip, K2 was pre-trained on 15.5 trillion tokens with zero loss spike. During post-training, K2 undergoes a multi-stage post-training process, highlighted by a large-scale agentic data synthesis pipeline and a joint reinforcement learning (RL) stage, where the model improves its capabilities through interactions with real and synthetic environments. Kimi K2 achieves state-of-the-art performance among open-source non-thinking models, with strengths in agentic capabilities. Notably, K2 obtains 66.1 on Tau2-Bench, 76.5 on ACEBench (En), 65.8 on SWE-Bench Verified, and 47.3 on SWE-Bench Multilingual -- surpassing most open and closed-sourced baselines in non-thinking settings. It also exhibits strong capabilities in coding, mathematics, and reasoning tasks, with a score of 53.7 on LiveCodeBench v6, 49.5 on AIME 2025, 75.1 on GPQA-Diamond, and 27.1 on OJBench, all without extended thinking. These results position Kimi K2 as one of the most capable open-source large language models to date, particularly in software engineering and agentic tasks. We release our base and post-trained model checkpoints to facilitate future research and applications of agentic intelligence.

While polyhedral compilers have shown success in implementing advanced code transformations, they still face challenges in selecting the ones that lead to the most profitable speedups. This has motivated the use of machine learning based cost models to guide the search for polyhedral optimizations. State-of-the-art polyhedral compilers have demonstrated a viable proof-of-concept of such an approach. While promising, this approach still faces significant limitations. State-of-the-art polyhedral compilers that use a deep learning cost model only support a small subset of affine transformations, limiting their ability to explore complex code transformations. Furthermore, their applicability does not scale beyond simple programs, thus excluding many program classes from their scope, such as those with non-rectangular iteration domains or multiple loop nests. These limitations significantly impact the generality of such compilers and autoschedulers and put into question the whole approach. In this paper, we introduce LOOPer, the first polyhedral autoscheduler that uses a deep learning based cost model and covers a large space of affine transformations and programs. LOOPer allows the optimization of an extensive set of programs while being effective at applying complex sequences of polyhedral transformations. We implement and evaluate LOOPer and show that it achieves competitive speedups over the state-of-the-art. On the PolyBench benchmarks, LOOPer achieves a geometric mean speedup of 1.84x over Tiramisu and 1.42x over Pluto, two state-of-the-art polyhedral autoschedulers.

The optimization algorithm and its hyperparameters can significantly affect the training speed and resulting model accuracy in machine learning applications. The wish list for an ideal optimizer includes fast and smooth convergence to low error, low computational demand, and general applicability. Our recently introduced continual resilient (CoRe) optimizer has shown superior performance compared to other state-of-the-art first-order gradient-based optimizers for training lifelong machine learning potentials. In this work we provide an extensive performance comparison of the CoRe optimizer and nine other optimization algorithms including the Adam optimizer and resilient backpropagation (RPROP) for diverse machine learning tasks. We analyze the influence of different hyperparameters and provide generally applicable values. The CoRe optimizer yields best or competitive performance in every investigated application, while only one hyperparameter needs to be changed depending on mini-batch or batch learning.

This work proposes a Momentum-Enabled Kronecker-Factor-Based Optimizer Using Rank-1 updates, called MKOR, that improves the training time and convergence properties of deep neural networks (DNNs). Second-order techniques, while enjoying higher convergence rates vs first-order counterparts, have cubic complexity with respect to either the model size and/or the training batch size. Hence they exhibit poor scalability and performance in transformer models, e.g. large language models (LLMs), because the batch sizes in these models scale by the attention mechanism sequence length, leading to large model size and batch sizes. MKOR's complexity is quadratic with respect to the model size, alleviating the computation bottlenecks in second-order methods. Because of their high computation complexity, state-of-the-art implementations of second-order methods can only afford to update the second order information infrequently, and thus do not fully exploit the promise of better convergence from these updates. By reducing the communication complexity of the second-order updates as well as achieving a linear communication complexity, MKOR increases the frequency of second order updates. We also propose a hybrid version of MKOR (called MKOR-H) that mid-training falls backs to a first order optimizer if the second order updates no longer accelerate convergence. Our experiments show that MKOR outperforms state -of-the-art first order methods, e.g. the LAMB optimizer, and best implementations of second-order methods, i.e. KAISA/KFAC, up to 2.57x and 1.85x respectively on BERT-Large-Uncased on 64 GPUs.

Given the massive cost of language model pre-training, a non-trivial improvement of the optimization algorithm would lead to a material reduction on the time and cost of training. Adam and its variants have been state-of-the-art for years, and more sophisticated second-order (Hessian-based) optimizers often incur too much per-step overhead. In this paper, we propose Sophia, Second-order Clipped Stochastic Optimization, a simple scalable second-order optimizer that uses a light-weight estimate of the diagonal Hessian as the pre-conditioner. The update is the moving average of the gradients divided by the moving average of the estimated Hessian, followed by element-wise clipping. The clipping controls the worst-case update size and tames the negative impact of non-convexity and rapid change of Hessian along the trajectory. Sophia only estimates the diagonal Hessian every handful of iterations, which has negligible average per-step time and memory overhead. On language modeling with GPT-2 models of sizes ranging from 125M to 770M, Sophia achieves a 2x speed-up compared with Adam in the number of steps, total compute, and wall-clock time. Theoretically, we show that Sophia adapts to the curvature in different components of the parameters, which can be highly heterogeneous for language modeling tasks. Our run-time bound does not depend on the condition number of the loss.

We introduce ADAHESSIAN, a second order stochastic optimization algorithm which dynamically incorporates the curvature of the loss function via ADAptive estimates of the HESSIAN. Second order algorithms are among the most powerful optimization algorithms with superior convergence properties as compared to first order methods such as SGD and Adam. The main disadvantage of traditional second order methods is their heavier per iteration computation and poor accuracy as compared to first order methods. To address these, we incorporate several novel approaches in ADAHESSIAN, including: (i) a fast Hutchinson based method to approximate the curvature matrix with low computational overhead; (ii) a root-mean-square exponential moving average to smooth out variations of the Hessian diagonal across different iterations; and (iii) a block diagonal averaging to reduce the variance of Hessian diagonal elements. We show that ADAHESSIAN achieves new state-of-the-art results by a large margin as compared to other adaptive optimization methods, including variants of Adam. In particular, we perform extensive tests on CV, NLP, and recommendation system tasks and find that ADAHESSIAN: (i) achieves 1.80%/1.45% higher accuracy on ResNets20/32 on Cifar10, and 5.55% higher accuracy on ImageNet as compared to Adam; (ii) outperforms AdamW for transformers by 0.13/0.33 BLEU score on IWSLT14/WMT14 and 2.7/1.0 PPL on PTB/Wikitext-103; (iii) outperforms AdamW for SqueezeBert by 0.41 points on GLUE; and (iv) achieves 0.032% better score than Adagrad for DLRM on the Criteo Ad Kaggle dataset. Importantly, we show that the cost per iteration of ADAHESSIAN is comparable to first order methods, and that it exhibits robustness towards its hyperparameters.

As optimizers are critical to the performances of neural networks, every year a large number of papers innovating on the subject are published. However, while most of these publications provide incremental improvements to existing algorithms, they tend to be presented as new optimizers rather than composable algorithms. Thus, many worthwhile improvements are rarely seen out of their initial publication. Taking advantage of this untapped potential, we introduce Ranger21, a new optimizer which combines AdamW with eight components, carefully selected after reviewing and testing ideas from the literature. We found that the resulting optimizer provides significantly improved validation accuracy and training speed, smoother training curves, and is even able to train a ResNet50 on ImageNet2012 without Batch Normalization layers. A problem on which AdamW stays systematically stuck in a bad initial state.

This paper establishes a mathematical foundation for the Adam optimizer, elucidating its connection to natural gradient descent through Riemannian and information geometry. We rigorously analyze the diagonal empirical Fisher information matrix (FIM) in Adam, clarifying all detailed approximations and advocating for the use of log probability functions as loss, which should be based on discrete distributions, due to the limitations of empirical FIM. Our analysis uncovers flaws in the original Adam algorithm, leading to proposed corrections such as enhanced momentum calculations, adjusted bias corrections, and gradient clipping. We refine the weight decay term based on our theoretical framework. Our modified algorithm, Fisher Adam (FAdam), demonstrates superior performance across diverse domains including LLM, ASR, and VQ-VAE, achieving state-of-the-art results in ASR.

Classical machine learning models such as deep neural networks are usually trained by using Stochastic Gradient Descent-based (SGD) algorithms. The classical SGD can be interpreted as a discretization of the stochastic gradient flow. In this paper we propose a novel, robust and accelerated stochastic optimizer that relies on two key elements: (1) an accelerated Nesterov-like Stochastic Differential Equation (SDE) and (2) its semi-implicit Gauss-Seidel type discretization. The convergence and stability of the obtained method, referred to as NAG-GS, are first studied extensively in the case of the minimization of a quadratic function. This analysis allows us to come up with an optimal learning rate in terms of the convergence rate while ensuring the stability of NAG-GS. This is achieved by the careful analysis of the spectral radius of the iteration matrix and the covariance matrix at stationarity with respect to all hyperparameters of our method. Further, we show that NAG- GS is competitive with state-of-the-art methods such as momentum SGD with weight decay and AdamW for the training of machine learning models such as the logistic regression model, the residual networks models on standard computer vision datasets, Transformers in the frame of the GLUE benchmark and the recent Vision Transformers.

Yes. SE data can have "smoother" boundaries between classes (compared to traditional AI data sets). To be more precise, the magnitude of the second derivative of the loss function found in SE data is typically much smaller. A new hyper-parameter optimizer, called SMOOTHIE, can exploit this idiosyncrasy of SE data. We compare SMOOTHIE and a state-of-the-art AI hyper-parameter optimizer on three tasks: (a) GitHub issue lifetime prediction (b) detecting static code warnings false alarm; (c) defect prediction. For completeness, we also show experiments on some standard AI datasets. SMOOTHIE runs faster and predicts better on the SE data--but ties on non-SE data with the AI tool. Hence we conclude that SE data can be different to other kinds of data; and those differences mean that we should use different kinds of algorithms for our data. To support open science and other researchers working in this area, all our scripts and datasets are available on-line at https://github.com/yrahul3910/smoothness-hpo/.

Neural networks are deployed widely in natural language processing tasks on the industrial scale, and perhaps the most often they are used as compounds of automatic machine translation systems. In this work, we present a simple approach to fool state-of-the-art machine translation tools in the task of translation from Russian to English and vice versa. Using a novel black-box gradient-free tensor-based optimizer, we show that many online translation tools, such as Google, DeepL, and Yandex, may both produce wrong or offensive translations for nonsensical adversarial input queries and refuse to translate seemingly benign input phrases. This vulnerability may interfere with understanding a new language and simply worsen the user's experience while using machine translation systems, and, hence, additional improvements of these tools are required to establish better translation.

Large Language Models (LLMs) have showcased remarkable impacts across a wide spectrum of natural language processing tasks. Fine-tuning these pre-trained models on downstream datasets provides further significant performance gains, but this process has been challenging due to its extraordinary resource requirements. To this end, existing efforts focus on parameter-efficient fine-tuning, which, unfortunately, fail to capitalize on the powerful potential of full-parameter fine-tuning. In this work, we propose QFT, a novel Quantized Full-parameter Tuning framework for LLMs that enables memory-efficient fine-tuning without harming performance. Our framework incorporates two novel ideas: (i) we adopt the efficient Lion optimizer, which only keeps track of the momentum and has consistent update magnitudes for each parameter, an inherent advantage for robust quantization; and (ii) we quantize all model states and store them as integer values, and present a gradient flow and parameter update scheme for the quantized weights. As a result, QFT reduces the model state memory to 21% of the standard solution while achieving comparable performance, e.g., tuning a LLaMA-7B model requires only <30GB of memory, satisfied by a single A6000 GPU.

Transformer-based large language models (LLMs) have demonstrated impressive capabilities in a variety of natural language processing (NLP) tasks. Nonetheless, it is challenging to deploy and fine-tune LLMs on mobile edge devices with limited computing, memory, and energy budgets. In this paper, we propose Confidant, a multi-backend collaborative training framework for customizing state-of-the-art LLMs on commodity mobile devices like smartphones. Confidant partitions an LLM into several sub-models so that each fits into a mobile device's memory. A pipeline parallel training mechanism is further developed to ensure fast and efficient distributed training. In addition, we propose a novel backend scheduler to allocate different attention heads to heterogeneous compute hardware, including mobile CPU and GPUs, to maximize the compute resource utilization on each edge device. Our preliminary experimental results show that Confidant achieves at most 45.3% memory reduction and 8.03x inference speedup in practical settings.

Recent advances in large vision-language models (LVLMs) have shown promise for embodied task planning, yet they struggle with fundamental challenges like dependency constraints and efficiency. Existing approaches either solely optimize action selection or leverage world models during inference, overlooking the benefits of learning to model the world as a way to enhance planning capabilities. We propose Dual Preference Optimization (D^2PO), a new learning framework that jointly optimizes state prediction and action selection through preference learning, enabling LVLMs to understand environment dynamics for better planning. To automatically collect trajectories and stepwise preference data without human annotation, we introduce a tree search mechanism for extensive exploration via trial-and-error. Extensive experiments on VoTa-Bench demonstrate that our D^2PO-based method significantly outperforms existing methods and GPT-4o when applied to Qwen2-VL (7B), LLaVA-1.6 (7B), and LLaMA-3.2 (11B), achieving superior task success rates with more efficient execution paths.

Pre-trained language models (LMs) have been shown to memorize a substantial amount of knowledge from the pre-training corpora; however, they are still limited in recalling factually correct knowledge given a certain context. Hence, they tend to suffer from counterfactual or hallucinatory generation when used in knowledge-intensive natural language generation (NLG) tasks. Recent remedies to this problem focus on modifying either the pre-training or task fine-tuning objectives to incorporate knowledge, which normally require additional costly training or architecture modification of LMs for practical applications. We present Knowledge Infused Decoding (KID) -- a novel decoding algorithm for generative LMs, which dynamically infuses external knowledge into each step of the LM decoding. Specifically, we maintain a local knowledge memory based on the current context, interacting with a dynamically created external knowledge trie, and continuously update the local memory as a knowledge-aware constraint to guide decoding via reinforcement learning. On six diverse knowledge-intensive NLG tasks, task-agnostic LMs (e.g., GPT-2 and BART) armed with KID outperform many task-optimized state-of-the-art models, and show particularly strong performance in few-shot scenarios over seven related knowledge-infusion techniques. Human evaluation confirms KID's ability to generate more relevant and factual language for the input context when compared with multiple baselines. Finally, KID also alleviates exposure bias and provides stable generation quality when generating longer sequences. Code for KID is available at https://github.com/microsoft/KID.

We study a class of optimization problems motivated by automating the design and update of AI systems like coding assistants, robots, and copilots. We propose an end-to-end optimization framework, Trace, which treats the computational workflow of an AI system as a graph akin to neural networks, based on a generalization of back-propagation. Optimization of computational workflows often involves rich feedback (e.g. console output or user's responses), heterogeneous parameters (e.g. prompts, hyper-parameters, codes), and intricate objectives (beyond maximizing a score). Moreover, its computation graph can change dynamically with the inputs and parameters. We frame a new mathematical setup of iterative optimization, Optimization with Trace Oracle (OPTO), to capture and abstract these properties so as to design optimizers that work across many domains. In OPTO, an optimizer receives an execution trace along with feedback on the computed output and updates parameters iteratively. Trace is the tool to implement OPTO in practice. Trace has a Python interface that efficiently converts a computational workflow into an OPTO instance using a PyTorch-like interface. Using Trace, we develop a general-purpose LLM-based optimizer called OptoPrime that can effectively solve OPTO problems. In empirical studies, we find that OptoPrime is capable of first-order numerical optimization, prompt optimization, hyper-parameter tuning, robot controller design, code debugging, etc., and is often competitive with specialized optimizers for each domain. We believe that Trace, OptoPrime and the OPTO framework will enable the next generation of interactive agents that automatically adapt using various kinds of feedback. Website: https://microsoft.github.io/Trace

Optimization plays a costly and crucial role in developing machine learning systems. In learned optimizers, the few hyperparameters of commonly used hand-designed optimizers, e.g. Adam or SGD, are replaced with flexible parametric functions. The parameters of these functions are then optimized so that the resulting learned optimizer minimizes a target loss on a chosen class of models. Learned optimizers can both reduce the number of required training steps and improve the final test loss. However, they can be expensive to train, and once trained can be expensive to use due to computational and memory overhead for the optimizer itself. In this work, we identify and quantify the design features governing the memory, compute, and performance trade-offs for many learned and hand-designed optimizers. We further leverage our analysis to construct a learned optimizer that is both faster and more memory efficient than previous work. Our model and training code are open source.

We study the learning dynamics of self-predictive learning for reinforcement learning, a family of algorithms that learn representations by minimizing the prediction error of their own future latent representations. Despite its recent empirical success, such algorithms have an apparent defect: trivial representations (such as constants) minimize the prediction error, yet it is obviously undesirable to converge to such solutions. Our central insight is that careful designs of the optimization dynamics are critical to learning meaningful representations. We identify that a faster paced optimization of the predictor and semi-gradient updates on the representation, are crucial to preventing the representation collapse. Then in an idealized setup, we show self-predictive learning dynamics carries out spectral decomposition on the state transition matrix, effectively capturing information of the transition dynamics. Building on the theoretical insights, we propose bidirectional self-predictive learning, a novel self-predictive algorithm that learns two representations simultaneously. We examine the robustness of our theoretical insights with a number of small-scale experiments and showcase the promise of the novel representation learning algorithm with large-scale experiments.

Tremendous progress has been made in reinforcement learning (RL) over the past decade. Most of these advancements came through the continual development of new algorithms, which were designed using a combination of mathematical derivations, intuitions, and experimentation. Such an approach of creating algorithms manually is limited by human understanding and ingenuity. In contrast, meta-learning provides a toolkit for automatic machine learning method optimisation, potentially addressing this flaw. However, black-box approaches which attempt to discover RL algorithms with minimal prior structure have thus far not outperformed existing hand-crafted algorithms. Mirror Learning, which includes RL algorithms, such as PPO, offers a potential middle-ground starting point: while every method in this framework comes with theoretical guarantees, components that differentiate them are subject to design. In this paper we explore the Mirror Learning space by meta-learning a "drift" function. We refer to the immediate result as Learnt Policy Optimisation (LPO). By analysing LPO we gain original insights into policy optimisation which we use to formulate a novel, closed-form RL algorithm, Discovered Policy Optimisation (DPO). Our experiments in Brax environments confirm state-of-the-art performance of LPO and DPO, as well as their transfer to unseen settings.

While deep learning models have replaced hand-designed features across many domains, these models are still trained with hand-designed optimizers. In this work, we leverage the same scaling approach behind the success of deep learning to learn versatile optimizers. We train an optimizer for deep learning which is itself a small neural network that ingests gradients and outputs parameter updates. Meta-trained with approximately four thousand TPU-months of compute on a wide variety of optimization tasks, our optimizer not only exhibits compelling performance, but optimizes in interesting and unexpected ways. It requires no hyperparameter tuning, instead automatically adapting to the specifics of the problem being optimized. We open source our learned optimizer, meta-training code, the associated train and test data, and an extensive optimizer benchmark suite with baselines at velo-code.github.io.

As new optimizers gain traction and model quantization becomes standard for efficient deployment, a key question arises: how does the choice of optimizer affect model performance in the presence of quantization? Despite progress in both areas, systematic evidence on optimizer-quantization interactions remains limited. To fill this gap, we study the impact of optimizer choice on model robustness under quantization, considering both post-training quantization (PTQ), and quantization-aware training (QAT). We first train full-precision models, ranging from 50M to 1.5B parameters, with six optimizers, to explore the hyperparameter landscape, and establish well-tuned baselines. We then apply PTQ to evaluate how model performance degrades when trained with different optimizers. We find that outlier-related metrics, such as the max-to-mean ratio (MMR) and Kurtosis, fail to predict the PTQ performance across different optimizers. We show analytically that this is due to the MMR capturing only isolated layer errors, while ignoring how quantization errors accumulate and propagate through the network. To study the QAT degradation, we train quantized models from scratch and compare them to our original-precision baselines. We find that optimizers performing well in the original pretraining setup may not remain optimal under QAT, and that models trained with Shampoo show the lowest accuracy degradation. Finally, we derive scaling laws for quantization-aware training under different optimizers, showing that Shampoo achieves the highest parameter efficiency of all tested optimizers.

We describe an iterative procedure for optimizing policies, with guaranteed monotonic improvement. By making several approximations to the theoretically-justified procedure, we develop a practical algorithm, called Trust Region Policy Optimization (TRPO). This algorithm is similar to natural policy gradient methods and is effective for optimizing large nonlinear policies such as neural networks. Our experiments demonstrate its robust performance on a wide variety of tasks: learning simulated robotic swimming, hopping, and walking gaits; and playing Atari games using images of the screen as input. Despite its approximations that deviate from the theory, TRPO tends to give monotonic improvement, with little tuning of hyperparameters.

Traditional theories of optimization cannot describe the dynamics of optimization in deep learning, even in the simple setting of deterministic training. The challenge is that optimizers typically operate in a complex, oscillatory regime called the "edge of stability." In this paper, we develop theory that can describe the dynamics of optimization in this regime. Our key insight is that while the *exact* trajectory of an oscillatory optimizer may be challenging to analyze, the *time-averaged* (i.e. smoothed) trajectory is often much more tractable. To analyze an optimizer, we derive a differential equation called a "central flow" that characterizes this time-averaged trajectory. We empirically show that these central flows can predict long-term optimization trajectories for generic neural networks with a high degree of numerical accuracy. By interpreting these central flows, we are able to understand how gradient descent makes progress even as the loss sometimes goes up; how adaptive optimizers "adapt" to the local loss landscape; and how adaptive optimizers implicitly navigate towards regions where they can take larger steps. Our results suggest that central flows can be a valuable theoretical tool for reasoning about optimization in deep learning.

We provide an optimization-based framework to perform counterfactual analysis in a dynamic model with hidden states. Our framework is grounded in the ``abduction, action, and prediction'' approach to answer counterfactual queries and handles two key challenges where (1) the states are hidden and (2) the model is dynamic. Recognizing the lack of knowledge on the underlying causal mechanism and the possibility of infinitely many such mechanisms, we optimize over this space and compute upper and lower bounds on the counterfactual quantity of interest. Our work brings together ideas from causality, state-space models, simulation, and optimization, and we apply it on a breast cancer case study. To the best of our knowledge, we are the first to compute lower and upper bounds on a counterfactual query in a dynamic latent-state model.

Due to the nonlinear nature of Deep Neural Networks (DNNs), one can not guarantee convergence to a unique global minimum of the loss when using optimizers relying only on local information, such as SGD. Indeed, this was a primary source of skepticism regarding the feasibility of DNNs in the early days of the field. The past decades of progress in deep learning have revealed this skepticism to be misplaced, and a large body of empirical evidence shows that sufficiently large DNNs following standard training protocols exhibit well-behaved optimization dynamics that converge to performant solutions. This success has biased the community to use convex optimization as a mental model for learning, leading to a focus on training efficiency, either in terms of required iteration, FLOPs or wall-clock time, when improving optimizers. We argue that, while this perspective has proven extremely fruitful, another perspective specific to DNNs has received considerably less attention: the optimizer not only influences the rate of convergence, but also the qualitative properties of the learned solutions. Restated, the optimizer can and will encode inductive biases and change the effective expressivity of a given class of models. Furthermore, we believe the optimizer can be an effective way of encoding desiderata in the learning process. We contend that the community should aim at understanding the biases of already existing methods, as well as aim to build new optimizers with the explicit intent of inducing certain properties of the solution, rather than solely judging them based on their convergence rates. We hope our arguments will inspire research to improve our understanding of how the learning process can impact the type of solution we converge to, and lead to a greater recognition of optimizers design as a critical lever that complements the roles of architecture and data in shaping model outcomes.

Recent work has shown that, under certain assumptions, zero-shot reinforcement learning (RL) methods can generalise to any unseen task in an environment after reward-free pre-training. Access to Markov states is one such assumption, yet, in many real-world applications, the Markov state is only partially observable. Here, we explore how the performance of standard zero-shot RL methods degrades when subjected to partially observability, and show that, as in single-task RL, memory-based architectures are an effective remedy. We evaluate our memory-based zero-shot RL methods in domains where the states, rewards and a change in dynamics are partially observed, and show improved performance over memory-free baselines. Our code is open-sourced via: https://enjeeneer.io/projects/bfms-with-memory/.

We consider the general class of time-homogeneous stochastic dynamical systems, both discrete and continuous, and study the problem of learning a representation of the state that faithfully captures its dynamics. This is instrumental to learning the transfer operator or the generator of the system, which in turn can be used for numerous tasks, such as forecasting and interpreting the system dynamics. We show that the search for a good representation can be cast as an optimization problem over neural networks. Our approach is supported by recent results in statistical learning theory, highlighting the role of approximation error and metric distortion in the learning problem. The objective function we propose is associated with projection operators from the representation space to the data space, overcomes metric distortion, and can be empirically estimated from data. In the discrete-time setting, we further derive a relaxed objective function that is differentiable and numerically well-conditioned. We compare our method against state-of-the-art approaches on different datasets, showing better performance across the board.

AdamW has been the default optimizer for transformer pretraining. For many years, our community searches for faster and more stable optimizers with only constraint positive outcomes. In this work, we propose a single-line modification in Pytorch to any momentum-based optimizer, which we rename Cautious Optimizer, e.g. C-AdamW and C-Lion. Our theoretical result shows that this modification preserves Adam's Hamiltonian function and it does not break the convergence guarantee under the Lyapunov analysis. In addition, a whole new family of optimizers is revealed by our theoretical insight. Among them, we pick the simplest one for empirical experiments, showing speed-up on Llama and MAE pretraining up to 1.47times. Code is available at https://github.com/kyleliang919/C-Optim

Bayesian filtering approximates the true underlying behavior of a time-varying system by inverting an explicit generative model to convert noisy measurements into state estimates. This process typically requires either storage, inversion, and multiplication of large matrices or Monte Carlo estimation, neither of which are practical in high-dimensional state spaces such as the weight spaces of artificial neural networks. Here, we frame the standard Bayesian filtering problem as optimization over a time-varying objective. Instead of maintaining matrices for the filtering equations or simulating particles, we specify an optimizer that defines the Bayesian filter implicitly. In the linear-Gaussian setting, we show that every Kalman filter has an equivalent formulation using K steps of gradient descent. In the nonlinear setting, our experiments demonstrate that our framework results in filters that are effective, robust, and scalable to high-dimensional systems, comparing well against the standard toolbox of Bayesian filtering solutions. We suggest that it is easier to fine-tune an optimizer than it is to specify the correct filtering equations, making our framework an attractive option for high-dimensional filtering problems.

We present a class of novel optimisers for training neural networks that makes use of the Riemannian metric naturally induced when the loss landscape is embedded in higher-dimensional space. This is the same metric that underlies common visualisations of loss landscapes. By taking this geometric perspective literally and using the induced metric, we develop a new optimiser and compare it to existing methods, namely: SGD, Adam, AdamW, and Muon, across a range of tasks and architectures. Empirically, we conclude that this new class of optimisers is highly effective in low dimensional examples, and provides slight improvement over state-of-the-art methods for training neural networks. These new optimisers have theoretically desirable properties. In particular, the effective learning rate is automatically decreased in regions of high curvature acting as a smoothed out form of gradient clipping. Similarly, one variant of these optimisers can also be viewed as inducing an effective scheduled learning rate and decoupled weight decay is the natural choice from our geometric perspective. The basic method can be used to modify any existing preconditioning method. The new optimiser has a computational complexity comparable to that of Adam.

In recent years, various powerful policy gradient algorithms have been proposed in deep reinforcement learning. While all these algorithms build on the Policy Gradient Theorem, the specific design choices differ significantly across algorithms. We provide a holistic overview of on-policy policy gradient algorithms to facilitate the understanding of both their theoretical foundations and their practical implementations. In this overview, we include a detailed proof of the continuous version of the Policy Gradient Theorem, convergence results and a comprehensive discussion of practical algorithms. We compare the most prominent algorithms on continuous control environments and provide insights on the benefits of regularization. All code is available at https://github.com/Matt00n/PolicyGradientsJax.

In this paper, we have studied the performance and role of local optimizers in quantum variational circuits. We studied the performance of the two most popular optimizers and compared their results with some popular classical machine learning algorithms. The classical algorithms we used in our study are support vector machine (SVM), gradient boosting (GB), and random forest (RF). These were compared with a variational quantum classifier (VQC) using two sets of local optimizers viz AQGD and COBYLA. For experimenting with VQC, IBM Quantum Experience and IBM Qiskit was used while for classical machine learning models, sci-kit learn was used. The results show that machine learning on noisy immediate scale quantum machines can produce comparable results as on classical machines. For our experiments, we have used a popular restaurant sentiment analysis dataset. The extracted features from this dataset and then after applying PCA reduced the feature set into 5 features. Quantum ML models were trained using 100 epochs and 150 epochs on using EfficientSU2 variational circuit. Overall, four Quantum ML models were trained and three Classical ML models were trained. The performance of the trained models was evaluated using standard evaluation measures viz, Accuracy, Precision, Recall, F-Score. In all the cases AQGD optimizer-based model with 100 Epochs performed better than all other models. It produced an accuracy of 77% and an F-Score of 0.785 which were highest across all the trained models.

Recent Meta-learning for Black-Box Optimization (MetaBBO) methods harness neural networks to meta-learn configurations of traditional black-box optimizers. Despite their success, they are inevitably restricted by the limitations of predefined hand-crafted optimizers. In this paper, we present Symbol, a novel framework that promotes the automated discovery of black-box optimizers through symbolic equation learning. Specifically, we propose a Symbolic Equation Generator (SEG) that allows closed-form optimization rules to be dynamically generated for specific tasks and optimization steps. Within Symbol, we then develop three distinct strategies based on reinforcement learning, so as to meta-learn the SEG efficiently. Extensive experiments reveal that the optimizers generated by Symbol not only surpass the state-of-the-art BBO and MetaBBO baselines, but also exhibit exceptional zero-shot generalization abilities across entirely unseen tasks with different problem dimensions, population sizes, and optimization horizons. Furthermore, we conduct in-depth analyses of our Symbol framework and the optimization rules that it generates, underscoring its desirable flexibility and interpretability.

Recent innovations in generative large language models (LLMs) have made their applications and use-cases ubiquitous. This has led to large-scale deployments of these models, using complex, expensive, and power-hungry AI accelerators, most commonly GPUs. These developments make LLM inference efficiency an important challenge. Based on our extensive characterization, we find that there are two main phases during an LLM inference request: a compute-intensive prompt computation, and a memory-intensive token generation, each with distinct latency, throughput, memory, and power characteristics. Despite state-of-the-art batching and scheduling, the token generation phase underutilizes compute resources. Specifically, unlike compute-intensive prompt computation phases, token generation phases do not require the compute capability of the latest GPUs, and can be run with lower power and cost. With Splitwise, we propose splitting the two phases of a LLM inference request on to separate machines. This allows us to use hardware that is well-suited for each phase, and provision resources independently per phase. However, splitting an inference request across machines requires state transfer from the machine running prompt computation over to the machine generating tokens. We implement and optimize this state transfer using the fast back-plane interconnects available in today's GPU clusters. We use the Splitwise technique to design LLM inference clusters using the same or different types of machines for the prompt computation and token generation phases. Our clusters are optimized for three key objectives: throughput, cost, and power. In particular, we show that we can achieve 1.4x higher throughput at 20% lower cost than current designs. Alternatively, we can achieve 2.35x more throughput with the same cost and power budgets.

Temporal difference (TD) learning and its variants, such as multistage TD (MS-TD) learning and temporal coherence (TC) learning, have been successfully applied to 2048. These methods rely on the stochasticity of the environment of 2048 for exploration. In this paper, we propose to employ optimistic initialization (OI) to encourage exploration for 2048, and empirically show that the learning quality is significantly improved. This approach optimistically initializes the feature weights to very large values. Since weights tend to be reduced once the states are visited, agents tend to explore those states which are unvisited or visited few times. Our experiments show that both TD and TC learning with OI significantly improve the performance. As a result, the network size required to achieve the same performance is significantly reduced. With additional tunings such as expectimax search, multistage learning, and tile-downgrading technique, our design achieves the state-of-the-art performance, namely an average score of 625 377 and a rate of 72% reaching 32768 tiles. In addition, for sufficiently large tests, 65536 tiles are reached at a rate of 0.02%.

In reinforcement learning (RL), rewards of states are typically considered additive, and following the Markov assumption, they are independent of states visited previously. In many important applications, such as coverage control, experiment design and informative path planning, rewards naturally have diminishing returns, i.e., their value decreases in light of similar states visited previously. To tackle this, we propose submodular RL (SubRL), a paradigm which seeks to optimize more general, non-additive (and history-dependent) rewards modelled via submodular set functions which capture diminishing returns. Unfortunately, in general, even in tabular settings, we show that the resulting optimization problem is hard to approximate. On the other hand, motivated by the success of greedy algorithms in classical submodular optimization, we propose SubPO, a simple policy gradient-based algorithm for SubRL that handles non-additive rewards by greedily maximizing marginal gains. Indeed, under some assumptions on the underlying Markov Decision Process (MDP), SubPO recovers optimal constant factor approximations of submodular bandits. Moreover, we derive a natural policy gradient approach for locally optimizing SubRL instances even in large state- and action- spaces. We showcase the versatility of our approach by applying SubPO to several applications, such as biodiversity monitoring, Bayesian experiment design, informative path planning, and coverage maximization. Our results demonstrate sample efficiency, as well as scalability to high-dimensional state-action spaces.

We explore a data-driven approach for learning to optimize neural networks. We construct a dataset of neural network checkpoints and train a generative model on the parameters. In particular, our model is a conditional diffusion transformer that, given an initial input parameter vector and a prompted loss, error, or return, predicts the distribution over parameter updates that achieve the desired metric. At test time, it can optimize neural networks with unseen parameters for downstream tasks in just one update. We find that our approach successfully generates parameters for a wide range of loss prompts. Moreover, it can sample multimodal parameter solutions and has favorable scaling properties. We apply our method to different neural network architectures and tasks in supervised and reinforcement learning.

We introduce the framework of performative reinforcement learning where the policy chosen by the learner affects the underlying reward and transition dynamics of the environment. Following the recent literature on performative prediction~Perdomo et. al., 2020, we introduce the concept of performatively stable policy. We then consider a regularized version of the reinforcement learning problem and show that repeatedly optimizing this objective converges to a performatively stable policy under reasonable assumptions on the transition dynamics. Our proof utilizes the dual perspective of the reinforcement learning problem and may be of independent interest in analyzing the convergence of other algorithms with decision-dependent environments. We then extend our results for the setting where the learner just performs gradient ascent steps instead of fully optimizing the objective, and for the setting where the learner has access to a finite number of trajectories from the changed environment. For both settings, we leverage the dual formulation of performative reinforcement learning and establish convergence to a stable solution. Finally, through extensive experiments on a grid-world environment, we demonstrate the dependence of convergence on various parameters e.g. regularization, smoothness, and the number of samples.

In this paper, we propose feedback designs for manipulating a quantum state to a target state by performing sequential measurements. In light of Belavkin's quantum feedback control theory, for a given set of (projective or non-projective) measurements and a given time horizon, we show that finding the measurement selection policy that maximizes the probability of successful state manipulation is an optimal control problem for a controlled Markovian process. The optimal policy is Markovian and can be solved by dynamical programming. Numerical examples indicate that making use of feedback information significantly improves the success probability compared to classical scheme without taking feedback. We also consider other objective functionals including maximizing the expected fidelity to the target state as well as minimizing the expected arrival time. The connections and differences among these objectives are also discussed.

First-order optimization (FOO) algorithms are pivotal in numerous computational domains such as machine learning and signal denoising. However, their application to complex tasks like neural network training often entails significant inefficiencies due to the need for many sequential iterations for convergence. In response, we introduce first-order optimization expedited with approximately parallelized iterations (OptEx), the first framework that enhances the efficiency of FOO by leveraging parallel computing to mitigate its iterative bottleneck. OptEx employs kernelized gradient estimation to make use of gradient history for future gradient prediction, enabling parallelization of iterations -- a strategy once considered impractical because of the inherent iterative dependency in FOO. We provide theoretical guarantees for the reliability of our kernelized gradient estimation and the iteration complexity of SGD-based OptEx, confirming that estimation errors diminish to zero as historical gradients accumulate and that SGD-based OptEx enjoys an effective acceleration rate of Omega(N) over standard SGD given parallelism of N. We also use extensive empirical studies, including synthetic functions, reinforcement learning tasks, and neural network training across various datasets, to underscore the substantial efficiency improvements achieved by OptEx.

We consider the reinforcement learning (RL) problem with general utilities which consists in maximizing a function of the state-action occupancy measure. Beyond the standard cumulative reward RL setting, this problem includes as particular cases constrained RL, pure exploration and learning from demonstrations among others. For this problem, we propose a simpler single-loop parameter-free normalized policy gradient algorithm. Implementing a recursive momentum variance reduction mechanism, our algorithm achieves mathcal{O}(epsilon^{-3}) and mathcal{O}(epsilon^{-2}) sample complexities for epsilon-first-order stationarity and epsilon-global optimality respectively, under adequate assumptions. We further address the setting of large finite state action spaces via linear function approximation of the occupancy measure and show a mathcal{O}(epsilon^{-4}) sample complexity for a simple policy gradient method with a linear regression subroutine.

Policy optimization methods are popular reinforcement learning algorithms in practice. Recent works have built theoretical foundation for them by proving T regret bounds even when the losses are adversarial. Such bounds are tight in the worst case but often overly pessimistic. In this work, we show that in tabular Markov decision processes (MDPs), by properly designing the regularizer, the exploration bonus and the learning rates, one can achieve a more favorable polylog(T) regret when the losses are stochastic, without sacrificing the worst-case guarantee in the adversarial regime. To our knowledge, this is also the first time a gap-dependent polylog(T) regret bound is shown for policy optimization. Specifically, we achieve this by leveraging a Tsallis entropy or a Shannon entropy regularizer in the policy update. Then we show that under known transitions, we can further obtain a first-order regret bound in the adversarial regime by leveraging the log-barrier regularizer.

We study the autonomous exploration (AX) problem proposed by Lim & Auer (2012). In this setting, the objective is to discover a set of epsilon-optimal policies reaching a set S_L^{rightarrow} of incrementally L-controllable states. We introduce a novel layered decomposition of the set of incrementally L-controllable states that is based on the iterative application of a state-expansion operator. We leverage these results to design Layered Autonomous Exploration (LAE), a novel algorithm for AX that attains a sample complexity of mathcal{O}(LS^{rightarrow}_{L(1+epsilon)}Gamma_{L(1+epsilon)} A ln^{12}(S^{rightarrow}_{L(1+epsilon)})/epsilon^2), where S^{rightarrow}_{L(1+epsilon)} is the number of states that are incrementally L(1+epsilon)-controllable, A is the number of actions, and Gamma_{L(1+epsilon)} is the branching factor of the transitions over such states. LAE improves over the algorithm of Tarbouriech et al. (2020a) by a factor of L^2 and it is the first algorithm for AX that works in a countably-infinite state space. Moreover, we show that, under a certain identifiability assumption, LAE achieves minimax-optimal sample complexity of mathcal{O}(LS^{rightarrow}_{L}Aln^{12}(S^{rightarrow}_{L})/epsilon^2), outperforming existing algorithms and matching for the first time the lower bound proved by Cai et al. (2022) up to logarithmic factors.

We study the problem of constrained efficient global optimization, where both the objective and constraints are expensive black-box functions that can be learned with Gaussian processes. We propose CONFIG (CONstrained efFIcient Global Optimization), a simple and effective algorithm to solve it. Under certain regularity assumptions, we show that our algorithm enjoys the same cumulative regret bound as that in the unconstrained case and similar cumulative constraint violation upper bounds. For commonly used Matern and Squared Exponential kernels, our bounds are sublinear and allow us to derive a convergence rate to the optimal solution of the original constrained problem. In addition, our method naturally provides a scheme to declare infeasibility when the original black-box optimization problem is infeasible. Numerical experiments on sampled instances from the Gaussian process, artificial numerical problems, and a black-box building controller tuning problem all demonstrate the competitive performance of our algorithm. Compared to the other state-of-the-art methods, our algorithm significantly improves the theoretical guarantees, while achieving competitive empirical performance.

Offline reinforcement learning (RL) can in principle synthesize more optimal behavior from a dataset consisting only of suboptimal trials. One way that this can happen is by "stitching" together the best parts of otherwise suboptimal trajectories that overlap on similar states, to create new behaviors where each individual state is in-distribution, but the overall returns are higher. However, in many interesting and complex applications, such as autonomous navigation and dialogue systems, the state is partially observed. Even worse, the state representation is unknown or not easy to define. In such cases, policies and value functions are often conditioned on observation histories instead of states. In these cases, it is not clear if the same kind of "stitching" is feasible at the level of observation histories, since two different trajectories would always have different histories, and thus "similar states" that might lead to effective stitching cannot be leveraged. Theoretically, we show that standard offline RL algorithms conditioned on observation histories suffer from poor sample complexity, in accordance with the above intuition. We then identify sufficient conditions under which offline RL can still be efficient -- intuitively, it needs to learn a compact representation of history comprising only features relevant for action selection. We introduce a bisimulation loss that captures the extent to which this happens, and propose that offline RL can explicitly optimize this loss to aid worst-case sample complexity. Empirically, we show that across a variety of tasks either our proposed loss improves performance, or the value of this loss is already minimized as a consequence of standard offline RL, indicating that it correlates well with good performance.

Behavior constrained policy optimization has been demonstrated to be a successful paradigm for tackling Offline Reinforcement Learning. By exploiting historical transitions, a policy is trained to maximize a learned value function while constrained by the behavior policy to avoid a significant distributional shift. In this paper, we propose our closed-form policy improvement operators. We make a novel observation that the behavior constraint naturally motivates the use of first-order Taylor approximation, leading to a linear approximation of the policy objective. Additionally, as practical datasets are usually collected by heterogeneous policies, we model the behavior policies as a Gaussian Mixture and overcome the induced optimization difficulties by leveraging the LogSumExp's lower bound and Jensen's Inequality, giving rise to a closed-form policy improvement operator. We instantiate offline RL algorithms with our novel policy improvement operators and empirically demonstrate their effectiveness over state-of-the-art algorithms on the standard D4RL benchmark. Our code is available at https://cfpi-icml23.github.io/.

Stateful policies play an important role in reinforcement learning, such as handling partially observable environments, enhancing robustness, or imposing an inductive bias directly into the policy structure. The conventional method for training stateful policies is Backpropagation Through Time (BPTT), which comes with significant drawbacks, such as slow training due to sequential gradient propagation and the occurrence of vanishing or exploding gradients. The gradient is often truncated to address these issues, resulting in a biased policy update. We present a novel approach for training stateful policies by decomposing the latter into a stochastic internal state kernel and a stateless policy, jointly optimized by following the stateful policy gradient. We introduce different versions of the stateful policy gradient theorem, enabling us to easily instantiate stateful variants of popular reinforcement learning and imitation learning algorithms. Furthermore, we provide a theoretical analysis of our new gradient estimator and compare it with BPTT. We evaluate our approach on complex continuous control tasks, e.g., humanoid locomotion, and demonstrate that our gradient estimator scales effectively with task complexity while offering a faster and simpler alternative to BPTT.

Optimization is a ubiquitous modeling tool and is often deployed in settings which repeatedly solve similar instances of the same problem. Amortized optimization methods use learning to predict the solutions to problems in these settings, exploiting the shared structure between similar problem instances. These methods have been crucial in variational inference and reinforcement learning and are capable of solving optimization problems many orders of magnitudes times faster than traditional optimization methods that do not use amortization. This tutorial presents an introduction to the amortized optimization foundations behind these advancements and overviews their applications in variational inference, sparse coding, gradient-based meta-learning, control, reinforcement learning, convex optimization, optimal transport, and deep equilibrium networks. The source code for this tutorial is available at https://github.com/facebookresearch/amortized-optimization-tutorial.

This paper provides a self-contained, from-scratch, exposition of key algorithms for instruction tuning of models: SFT, Rejection Sampling, REINFORCE, Trust Region Policy Optimization (TRPO), Proximal Policy Optimization (PPO), Group Relative Policy Optimization (GRPO), and Direct Preference Optimization (DPO). Explanations of these algorithms often assume prior knowledge, lack critical details, and/or are overly generalized and complex. Here, each method is discussed and developed step by step using simplified and explicit notation focused on LLMs, aiming to eliminate ambiguity and provide a clear and intuitive understanding of the concepts. By minimizing detours into the broader RL literature and connecting concepts to LLMs, we eliminate superfluous abstractions and reduce cognitive overhead. Following this exposition, we provide a literature review of new techniques and approaches beyond those detailed. Finally, new ideas for research and exploration in the form of GRAPE (Generalized Relative Advantage Policy Evolution) are presented.

Solving a linear system Ax=b is a fundamental scientific computing primitive for which numerous solvers and preconditioners have been developed. These come with parameters whose optimal values depend on the system being solved and are often impossible or too expensive to identify; thus in practice sub-optimal heuristics are used. We consider the common setting in which many related linear systems need to be solved, e.g. during a single numerical simulation. In this scenario, can we sequentially choose parameters that attain a near-optimal overall number of iterations, without extra matrix computations? We answer in the affirmative for Successive Over-Relaxation (SOR), a standard solver whose parameter omega has a strong impact on its runtime. For this method, we prove that a bandit online learning algorithm--using only the number of iterations as feedback--can select parameters for a sequence of instances such that the overall cost approaches that of the best fixed omega as the sequence length increases. Furthermore, when given additional structural information, we show that a contextual bandit method asymptotically achieves the performance of the instance-optimal policy, which selects the best omega for each instance. Our work provides the first learning-theoretic treatment of high-precision linear system solvers and the first end-to-end guarantees for data-driven scientific computing, demonstrating theoretically the potential to speed up numerical methods using well-understood learning algorithms.

Learning to Optimize (L2O), a technique that utilizes machine learning to learn an optimization algorithm automatically from data, has gained arising attention in recent years. A generic L2O approach parameterizes the iterative update rule and learns the update direction as a black-box network. While the generic approach is widely applicable, the learned model can overfit and may not generalize well to out-of-distribution test sets. In this paper, we derive the basic mathematical conditions that successful update rules commonly satisfy. Consequently, we propose a novel L2O model with a mathematics-inspired structure that is broadly applicable and generalized well to out-of-distribution problems. Numerical simulations validate our theoretical findings and demonstrate the superior empirical performance of the proposed L2O model.

Minimax problems are notoriously challenging to optimize. However, we demonstrate that the two-timescale extragradient can be a viable solution. By utilizing dynamical systems theory, we show that it converges to points that satisfy the second-order necessary condition of local minimax points, under a mild condition. This work surpasses all previous results as we eliminate a crucial assumption that the Hessian, with respect to the maximization variable, is nondegenerate.

Optimization problems are pervasive across various sectors, from manufacturing and distribution to healthcare. However, most such problems are still solved heuristically by hand rather than optimally by state-of-the-art solvers, as the expertise required to formulate and solve these problems limits the widespread adoption of optimization tools and techniques. We introduce OptiMUS, a Large Language Model (LLM)-based agent designed to formulate and solve MILP problems from their natural language descriptions. OptiMUS is capable of developing mathematical models, writing and debugging solver code, developing tests, and checking the validity of generated solutions. To benchmark our agent, we present NLP4LP, a novel dataset of linear programming (LP) and mixed integer linear programming (MILP) problems. Our experiments demonstrate that OptiMUS solves nearly twice as many problems as a basic LLM prompting strategy. OptiMUS code and NLP4LP dataset are available at https://github.com/teshnizi/OptiMUS{https://github.com/teshnizi/OptiMUS}

In coal-fired power plants, it is critical to improve the operational efficiency of boilers for sustainability. In this work, we formulate real-time boiler control as an optimization problem that looks for the best distribution of temperature in different zones and oxygen content from the flue to improve the boiler's stability and energy efficiency. We employ an efficient algorithm by integrating appropriate machine learning and optimization techniques. We obtain a large dataset collected from a real boiler for more than two months from our industry partner, and conduct extensive experiments to demonstrate the effectiveness and efficiency of the proposed algorithm.

Learned optimizers (LOs) can significantly reduce the wall-clock training time of neural networks, substantially reducing training costs. However, they often suffer from poor meta-generalization, especially when training networks larger than those seen during meta-training. To address this, we use the recently proposed Maximal Update Parametrization (muP), which allows zero-shot generalization of optimizer hyperparameters from smaller to larger models. We extend muP theory to learned optimizers, treating the meta-training problem as finding the learned optimizer under muP. Our evaluation shows that LOs meta-trained with muP substantially improve meta-generalization as compared to LOs trained under standard parametrization (SP). Notably, when applied to large-width models, our best muLO, trained for 103 GPU-hours, matches or exceeds the performance of VeLO, the largest publicly available learned optimizer, meta-trained with 4000 TPU-months of compute. Moreover, muLOs demonstrate better generalization than their SP counterparts to deeper networks and to much longer training horizons (25 times longer) than those seen during meta-training.

In dynamic programming (DP) and reinforcement learning (RL), an agent learns to act optimally in terms of expected long-term return by sequentially interacting with its environment modeled by a Markov decision process (MDP). More generally in distributional reinforcement learning (DRL), the focus is on the whole distribution of the return, not just its expectation. Although DRL-based methods produced state-of-the-art performance in RL with function approximation, they involve additional quantities (compared to the non-distributional setting) that are still not well understood. As a first contribution, we introduce a new class of distributional operators, together with a practical DP algorithm for policy evaluation, that come with a robust MDP interpretation. Indeed, our approach reformulates through an augmented state space where each state is split into a worst-case substate and a best-case substate, whose values are maximized by safe and risky policies respectively. Finally, we derive distributional operators and DP algorithms solving a new control task: How to distinguish safe from risky optimal actions in order to break ties in the space of optimal policies?

In reinforcement learning and imitation learning, an object of central importance is the state distribution induced by the policy. It plays a crucial role in the policy gradient theorem, and references to it--along with the related state-action distribution--can be found all across the literature. Despite its importance, the state distribution is mostly discussed indirectly and theoretically, rather than being modeled explicitly. The reason being an absence of appropriate density estimation tools. In this work, we investigate applications of a normalizing flow-based model for the aforementioned distributions. In particular, we use a pair of flows coupled through the optimality point of the Donsker-Varadhan representation of the Kullback-Leibler (KL) divergence, for distribution matching based imitation learning. Our algorithm, Coupled Flow Imitation Learning (CFIL), achieves state-of-the-art performance on benchmark tasks with a single expert trajectory and extends naturally to a variety of other settings, including the subsampled and state-only regimes.

We introduce Cautious Weight Decay (CWD), a one-line, optimizer-agnostic modification that applies weight decay only to parameter coordinates whose signs align with the optimizer update. Unlike standard decoupled decay, which implicitly optimizes a regularized or constrained objective, CWD preserves the original loss and admits a bilevel interpretation: it induces sliding-mode behavior upon reaching the stationary manifold, allowing it to search for locally Pareto-optimal stationary points of the unmodified objective. In practice, CWD is a drop-in change for optimizers such as AdamW, Lion, and Muon, requiring no new hyperparameters or additional tuning. For language model pre-training and ImageNet classification, CWD consistently improves final loss and accuracy at million- to billion-parameter scales.

While originally developed for continuous control problems, Proximal Policy Optimization (PPO) has emerged as the work-horse of a variety of reinforcement learning (RL) applications including the fine-tuning of generative models. Unfortunately, PPO requires multiple heuristics to enable stable convergence (e.g. value networks, clipping) and is notorious for its sensitivity to the precise implementation of these components. In response, we take a step back and ask what a minimalist RL algorithm for the era of generative models would look like. We propose REBEL, an algorithm that cleanly reduces the problem of policy optimization to regressing the relative rewards via a direct policy parameterization between two completions to a prompt, enabling strikingly lightweight implementation. In theory, we prove that fundamental RL algorithms like Natural Policy Gradient can be seen as variants of REBEL, which allows us to match the strongest known theoretical guarantees in terms of convergence and sample complexity in the RL literature. REBEL can also cleanly incorporate offline data and handle the intransitive preferences we frequently see in practice. Empirically, we find that REBEL provides a unified approach to language modeling and image generation with stronger or similar performance as PPO and DPO, all while being simpler to implement and more computationally tractable than PPO.

Actor-critic algorithms have become a cornerstone in reinforcement learning (RL), leveraging the strengths of both policy-based and value-based methods. Despite recent progress in understanding their statistical efficiency, no existing work has successfully learned an epsilon-optimal policy with a sample complexity of O(1/epsilon^2) trajectories with general function approximation when strategic exploration is necessary. We address this open problem by introducing a novel actor-critic algorithm that attains a sample-complexity of O(dH^5 log|A|/epsilon^2 + d H^4 log|F|/ epsilon^2) trajectories, and accompanying T regret when the Bellman eluder dimension d does not increase with T at more than a log T rate. Here, F is the critic function class, A is the action space, and H is the horizon in the finite horizon MDP setting. Our algorithm integrates optimism, off-policy critic estimation targeting the optimal Q-function, and rare-switching policy resets. We extend this to the setting of Hybrid RL, showing that initializing the critic with offline data yields sample efficiency gains compared to purely offline or online RL. Further, utilizing access to offline data, we provide a non-optimistic provably efficient actor-critic algorithm that only additionally requires N_{off} geq c_{off}^*dH^4/epsilon^2 in exchange for omitting optimism, where c_{off}^* is the single-policy concentrability coefficient and N_{off} is the number of offline samples. This addresses another open problem in the literature. We further provide numerical experiments to support our theoretical findings.

Past analyses of reinforcement learning from human feedback (RLHF) assume that the human fully observes the environment. What happens when human feedback is based only on partial observations? We formally define two failure cases: deception and overjustification. Modeling the human as Boltzmann-rational w.r.t. a belief over trajectories, we prove conditions under which RLHF is guaranteed to result in policies that deceptively inflate their performance, overjustify their behavior to make an impression, or both. To help address these issues, we mathematically characterize how partial observability of the environment translates into (lack of) ambiguity in the learned return function. In some cases, accounting for partial observability makes it theoretically possible to recover the return function and thus the optimal policy, while in other cases, there is irreducible ambiguity. We caution against blindly applying RLHF in partially observable settings and propose research directions to help tackle these challenges.

The maritime industry's continuous commitment to sustainability has led to a dedicated exploration of methods to reduce vessel fuel consumption. This paper undertakes this challenge through a machine learning approach, leveraging a real-world dataset spanning two years of a ferry in west coast Canada. Our focus centers on the creation of a time series forecasting model given the dynamic and static states, actions, and disturbances. This model is designed to predict dynamic states based on the actions provided, subsequently serving as an evaluative tool to assess the proficiency of the ferry's operation under the captain's guidance. Additionally, it lays the foundation for future optimization algorithms, providing valuable feedback on decision-making processes. To facilitate future studies, our code is available at https://github.com/pagand/model_optimze_vessel/tree/AAAI

This paper presents Post-Decision Proximal Policy Optimization (PDPPO), a novel variation of the leading deep reinforcement learning method, Proximal Policy Optimization (PPO). The PDPPO state transition process is divided into two steps: a deterministic step resulting in the post-decision state and a stochastic step leading to the next state. Our approach incorporates post-decision states and dual critics to reduce the problem's dimensionality and enhance the accuracy of value function estimation. Lot-sizing is a mixed integer programming problem for which we exemplify such dynamics. The objective of lot-sizing is to optimize production, delivery fulfillment, and inventory levels in uncertain demand and cost parameters. This paper evaluates the performance of PDPPO across various environments and configurations. Notably, PDPPO with a dual critic architecture achieves nearly double the maximum reward of vanilla PPO in specific scenarios, requiring fewer episode iterations and demonstrating faster and more consistent learning across different initializations. On average, PDPPO outperforms PPO in environments with a stochastic component in the state transition. These results support the benefits of using a post-decision state. Integrating this post-decision state in the value function approximation leads to more informed and efficient learning in high-dimensional and stochastic environments.

RAG systems consist of multiple modules to work together. However, these modules are usually separately trained. We argue that a system like RAG that incorporates multiple modules should be jointly optimized to achieve optimal performance. To demonstrate this, we design a specific pipeline called SmartRAG that includes a policy network and a retriever. The policy network can serve as 1) a decision maker that decides when to retrieve, 2) a query rewriter to generate a query most suited to the retriever, and 3) an answer generator that produces the final response with/without the observations. We then propose to jointly optimize the whole system using a reinforcement learning algorithm, with the reward designed to encourage the system to achieve the best performance with minimal retrieval cost. When jointly optimized, all the modules can be aware of how other modules are working and thus find the best way to work together as a complete system. Empirical results demonstrate that the jointly optimized SmartRAG can achieve better performance than separately optimized counterparts.

We consider the problem of learning the best possible policy from a fixed dataset, known as offline Reinforcement Learning (RL). A common taxonomy of existing offline RL works is policy regularization, which typically constrains the learned policy by distribution or support of the behavior policy. However, distribution and support constraints are overly conservative since they both force the policy to choose similar actions as the behavior policy when considering particular states. It will limit the learned policy's performance, especially when the behavior policy is sub-optimal. In this paper, we find that regularizing the policy towards the nearest state-action pair can be more effective and thus propose Policy Regularization with Dataset Constraint (PRDC). When updating the policy in a given state, PRDC searches the entire dataset for the nearest state-action sample and then restricts the policy with the action of this sample. Unlike previous works, PRDC can guide the policy with proper behaviors from the dataset, allowing it to choose actions that do not appear in the dataset along with the given state. It is a softer constraint but still keeps enough conservatism from out-of-distribution actions. Empirical evidence and theoretical analysis show that PRDC can alleviate offline RL's fundamentally challenging value overestimation issue with a bounded performance gap. Moreover, on a set of locomotion and navigation tasks, PRDC achieves state-of-the-art performance compared with existing methods. Code is available at https://github.com/LAMDA-RL/PRDC

We study the problem of convergence to a stationary point in zero-sum games. We propose competitive gradient optimization (CGO ), a gradient-based method that incorporates the interactions between the two players in zero-sum games for optimization updates. We provide continuous-time analysis of CGO and its convergence properties while showing that in the continuous limit, CGO predecessors degenerate to their gradient descent ascent (GDA) variants. We provide a rate of convergence to stationary points and further propose a generalized class of alpha-coherent function for which we provide convergence analysis. We show that for strictly alpha-coherent functions, our algorithm convergences to a saddle point. Moreover, we propose optimistic CGO (OCGO), an optimistic variant, for which we show convergence rate to saddle points in alpha-coherent class of functions.

Many policy optimization approaches in reinforcement learning incorporate a Kullback-Leilbler (KL) divergence to the previous policy, to prevent the policy from changing too quickly. This idea was initially proposed in a seminal paper on Conservative Policy Iteration, with approximations given by algorithms like TRPO and Munchausen Value Iteration (MVI). We continue this line of work by investigating a generalized KL divergence -- called the Tsallis KL divergence -- which use the q-logarithm in the definition. The approach is a strict generalization, as q = 1 corresponds to the standard KL divergence; q > 1 provides a range of new options. We characterize the types of policies learned under the Tsallis KL, and motivate when q >1 could be beneficial. To obtain a practical algorithm that incorporates Tsallis KL regularization, we extend MVI, which is one of the simplest approaches to incorporate KL regularization. We show that this generalized MVI(q) obtains significant improvements over the standard MVI(q = 1) across 35 Atari games.

In this paper, we present efficient quantum algorithms that are exponentially faster than classical algorithms for solving the quantum optimal control problem. This problem involves finding the control variable that maximizes a physical quantity at time T, where the system is governed by a time-dependent Schr\"odinger equation. This type of control problem also has an intricate relation with machine learning. Our algorithms are based on a time-dependent Hamiltonian simulation method and a fast gradient-estimation algorithm. We also provide a comprehensive error analysis to quantify the total error from various steps, such as the finite-dimensional representation of the control function, the discretization of the Schr\"odinger equation, the numerical quadrature, and optimization. Our quantum algorithms require fault-tolerant quantum computers.

Temporal Difference (TD) algorithms are widely used in Deep Reinforcement Learning (RL). Their performance is heavily influenced by the size of the neural network. While in supervised learning, the regime of over-parameterization and its benefits are well understood, the situation in RL is much less clear. In this paper, we present a theoretical analysis of the influence of network size and l_2-regularization on performance. We identify the ratio between the number of parameters and the number of visited states as a crucial factor and define over-parameterization as the regime when it is larger than one. Furthermore, we observe a double descent phenomenon, i.e., a sudden drop in performance around the parameter/state ratio of one. Leveraging random features and the lazy training regime, we study the regularized Least-Square Temporal Difference (LSTD) algorithm in an asymptotic regime, as both the number of parameters and states go to infinity, maintaining a constant ratio. We derive deterministic limits of both the empirical and the true Mean-Squared Bellman Error (MSBE) that feature correction terms responsible for the double descent. Correction terms vanish when the l_2-regularization is increased or the number of unvisited states goes to zero. Numerical experiments with synthetic and small real-world environments closely match the theoretical predictions.

Provably efficient Model-Based Reinforcement Learning (MBRL) based on optimism or posterior sampling (PSRL) is ensured to attain the global optimality asymptotically by introducing the complexity measure of the model. However, the complexity might grow exponentially for the simplest nonlinear models, where global convergence is impossible within finite iterations. When the model suffers a large generalization error, which is quantitatively measured by the model complexity, the uncertainty can be large. The sampled model that current policy is greedily optimized upon will thus be unsettled, resulting in aggressive policy updates and over-exploration. In this work, we propose Conservative Dual Policy Optimization (CDPO) that involves a Referential Update and a Conservative Update. The policy is first optimized under a reference model, which imitates the mechanism of PSRL while offering more stability. A conservative range of randomness is guaranteed by maximizing the expectation of model value. Without harmful sampling procedures, CDPO can still achieve the same regret as PSRL. More importantly, CDPO enjoys monotonic policy improvement and global optimality simultaneously. Empirical results also validate the exploration efficiency of CDPO.

Mirror descent value iteration (MDVI), an abstraction of Kullback-Leibler (KL) and entropy-regularized reinforcement learning (RL), has served as the basis for recent high-performing practical RL algorithms. However, despite the use of function approximation in practice, the theoretical understanding of MDVI has been limited to tabular Markov decision processes (MDPs). We study MDVI with linear function approximation through its sample complexity required to identify an varepsilon-optimal policy with probability 1-delta under the settings of an infinite-horizon linear MDP, generative model, and G-optimal design. We demonstrate that least-squares regression weighted by the variance of an estimated optimal value function of the next state is crucial to achieving minimax optimality. Based on this observation, we present Variance-Weighted Least-Squares MDVI (VWLS-MDVI), the first theoretical algorithm that achieves nearly minimax optimal sample complexity for infinite-horizon linear MDPs. Furthermore, we propose a practical VWLS algorithm for value-based deep RL, Deep Variance Weighting (DVW). Our experiments demonstrate that DVW improves the performance of popular value-based deep RL algorithms on a set of MinAtar benchmarks.

We propose a new method for optimistic planning in infinite-horizon discounted Markov decision processes based on the idea of adding regularization to the updates of an otherwise standard approximate value iteration procedure. This technique allows us to avoid contraction and monotonicity arguments typically required by existing analyses of approximate dynamic programming methods, and in particular to use approximate transition functions estimated via least-squares procedures in MDPs with linear function approximation. We use our method to recover known guarantees in tabular MDPs and to provide a computationally efficient algorithm for learning near-optimal policies in discounted linear mixture MDPs from a single stream of experience, and show it achieves near-optimal statistical guarantees.

The distribution of high-quality Greenberger-Horne-Zeilinger (GHZ) states is at the heart of many quantum communication tasks, ranging from extending the baseline of telescopes to secret sharing. They also play an important role in error-correction architectures for distributed quantum computation, where Bell pairs can be leveraged to create an entangled network of quantum computers. We investigate the creation and distillation of GHZ states out of non-perfect Bell pairs over quantum networks. In particular, we introduce a heuristic dynamic programming algorithm to optimize over a large class of protocols that create and purify GHZ states. All protocols considered use a common framework based on measurements of non-local stabilizer operators of the target state (i.e., the GHZ state), where each non-local measurement consumes another (non-perfect) entangled state as a resource. The new protocols outperform previous proposals for scenarios without decoherence and local gate noise. Furthermore, the algorithms can be applied for finding protocols for any number of parties and any number of entangled pairs involved.

Program synthesis aims to create accurate, executable code from natural language descriptions. This field has leveraged the power of reinforcement learning (RL) in conjunction with large language models (LLMs), significantly enhancing code generation capabilities. This integration focuses on directly optimizing functional correctness, transcending conventional supervised losses. While current literature predominantly favors policy-based algorithms, attributes of program synthesis suggest a natural compatibility with value-based methods. This stems from rich collection of off-policy programs developed by human programmers, and the straightforward verification of generated programs through automated unit testing (i.e. easily obtainable rewards in RL language). Diverging from the predominant use of policy-based algorithms, our work explores the applicability of value-based approaches, leading to the development of our B-Coder (pronounced Bellman coder). Yet, training value-based methods presents challenges due to the enormous search space inherent to program synthesis. To this end, we propose an initialization protocol for RL agents utilizing pre-trained LMs and a conservative Bellman operator to reduce training complexities. Moreover, we demonstrate how to leverage the learned value functions as a dual strategy to post-process generated programs. Our empirical evaluations demonstrated B-Coder's capability in achieving state-of-the-art performance compared with policy-based methods. Remarkably, this achievement is reached with minimal reward engineering effort, highlighting the effectiveness of value-based RL, independent of reward designs.

Representations are at the core of all deep reinforcement learning (RL) methods for both Markov decision processes (MDPs) and partially observable Markov decision processes (POMDPs). Many representation learning methods and theoretical frameworks have been developed to understand what constitutes an effective representation. However, the relationships between these methods and the shared properties among them remain unclear. In this paper, we show that many of these seemingly distinct methods and frameworks for state and history abstractions are, in fact, based on a common idea of self-predictive abstraction. Furthermore, we provide theoretical insights into the widely adopted objectives and optimization, such as the stop-gradient technique, in learning self-predictive representations. These findings together yield a minimalist algorithm to learn self-predictive representations for states and histories. We validate our theories by applying our algorithm to standard MDPs, MDPs with distractors, and POMDPs with sparse rewards. These findings culminate in a set of preliminary guidelines for RL practitioners.

Active learning (AL) algorithms may achieve better performance with fewer data because the model guides the data selection process. While many algorithms have been proposed, there is little study on what the optimal AL algorithm looks like, which would help researchers understand where their models fall short and iterate on the design. In this paper, we present a simulated annealing algorithm to search for this optimal oracle and analyze it for several tasks. We present qualitative and quantitative insights into the behaviors of this oracle, comparing and contrasting them with those of various heuristics. Moreover, we are able to consistently improve the heuristics using one particular insight. We hope that our findings can better inform future active learning research. The code is available at https://github.com/YilunZhou/optimal-active-learning.

Lion (Evolved Sign Momentum), a new optimizer discovered through program search, has shown promising results in training large AI models. It performs comparably or favorably to AdamW but with greater memory efficiency. As we can expect from the results of a random search program, Lion incorporates elements from several existing algorithms, including signed momentum, decoupled weight decay, Polak, and Nesterov momentum, but does not fit into any existing category of theoretically grounded optimizers. Thus, even though Lion appears to perform well as a general-purpose optimizer for a wide range of tasks, its theoretical basis remains uncertain. This lack of theoretical clarity limits opportunities to further enhance and expand Lion's efficacy. This work aims to demystify Lion. Based on both continuous-time and discrete-time analysis, we demonstrate that Lion is a theoretically novel and principled approach for minimizing a general loss function f(x) while enforcing a bound constraint |x|_infty leq 1/lambda. Lion achieves this through the incorporation of decoupled weight decay, where lambda represents the weight decay coefficient. Our analysis is made possible by the development of a new Lyapunov function for the Lion updates. It applies to a broader family of Lion-kappa algorithms, where the sign(cdot) operator in Lion is replaced by the subgradient of a convex function kappa, leading to the solution of a general composite optimization problem of min_x f(x) + kappa^*(x). Our findings provide valuable insights into the dynamics of Lion and pave the way for further improvements and extensions of Lion-related algorithms.

We present Self-Play Preference Optimization (SPO), an algorithm for reinforcement learning from human feedback. Our approach is minimalist in that it does not require training a reward model nor unstable adversarial training and is therefore rather simple to implement. Our approach is maximalist in that it provably handles non-Markovian, intransitive, and stochastic preferences while being robust to the compounding errors that plague offline approaches to sequential prediction. To achieve the preceding qualities, we build upon the concept of a Minimax Winner (MW), a notion of preference aggregation from the social choice theory literature that frames learning from preferences as a zero-sum game between two policies. By leveraging the symmetry of this game, we prove that rather than using the traditional technique of dueling two policies to compute the MW, we can simply have a single agent play against itself while maintaining strong convergence guarantees. Practically, this corresponds to sampling multiple trajectories from a policy, asking a rater or preference model to compare them, and then using the proportion of wins as the reward for a particular trajectory. We demonstrate that on a suite of continuous control tasks, we are able to learn significantly more efficiently than reward-model based approaches while maintaining robustness to the intransitive and stochastic preferences that frequently occur in practice when aggregating human judgments.

Training LLM agents in multi-turn environments with sparse rewards, where completing a single task requires 30+ turns of interaction within an episode, presents a fundamental challenge for reinforcement learning. We identify a critical failure mode unique to this setting: the exploration-exploitation cascade failure. This cascade begins with early-stage policy premature convergence, where sparse feedback causes agents to commit to flawed, low-entropy strategies. Subsequently, agents enter late-stage policy collapse, where conventional entropy regularization becomes counterproductive, promoting chaotic exploration that destabilizes training. We propose Entropy-regularized Policy Optimization (EPO), a general framework that breaks this failure cycle through three synergistic mechanisms: (1) adopting entropy regularization in multi-turn settings to enhance exploration, (2) an entropy smoothing regularizer that bounds policy entropy within historical averages to prevent abrupt fluctuations, and (3) adaptive phase-based weighting that balances exploration and exploitation across training. Our analysis justifies that EPO guarantees monotonically decreasing entropy variance while maintaining convergence. EPO achieves up to 152% performance improvement on ScienceWorld and up to 19.8% on ALFWorld. Our work demonstrates that multi-turn sparse-reward settings require fundamentally different entropy control than traditional RL, with broad implications for LLM agent training.

Modern reinforcement learning (RL) systems capture deep truths about general, human problem-solving. In domains where new data can be simulated cheaply, these systems uncover sequential decision-making policies that far exceed the ability of any human. Society faces many problems whose solutions require this skill, but they are often in domains where new data cannot be cheaply simulated. In such scenarios, we can learn simulators from existing data, but these will only ever be approximately correct, and can be pathologically incorrect when queried outside of their training distribution. As a result, a misalignment between the environments in which we train our agents and the real-world in which we wish to deploy our agents is inevitable. Dealing with this misalignment is the primary concern of zero-shot reinforcement learning, a problem setting where the agent must generalise to a new task or domain with zero practice shots. Whilst impressive progress has been made on methods that perform zero-shot RL in idealised settings, new work is needed if these results are to be replicated in real-world settings. In this thesis, we argue that doing so requires us to navigate (at least) three constraints. First, the data quality constraint: real-world datasets are small and homogeneous. Second, the observability constraint: states, dynamics and rewards in the real-world are often only partially observed. And third, the data availability constraint: a priori access to data cannot always be assumed. This work proposes a suite of methods that perform zero-shot RL subject to these constraints. In a series of empirical studies we expose the failings of existing methods, and justify our techniques for remedying them. We believe these designs take us a step closer to RL methods that can be deployed to solve real-world problems.

We present a full implementation and simulation of a novel quantum reinforcement learning method. Our work is a detailed and formal proof of concept for how quantum algorithms can be used to solve reinforcement learning problems and shows that, given access to error-free, efficient quantum realizations of the agent and environment, quantum methods can yield provable improvements over classical Monte-Carlo based methods in terms of sample complexity. Our approach shows in detail how to combine amplitude estimation and Grover search into a policy evaluation and improvement scheme. We first develop quantum policy evaluation (QPE) which is quadratically more efficient compared to an analogous classical Monte Carlo estimation and is based on a quantum mechanical realization of a finite Markov decision process (MDP). Building on QPE, we derive a quantum policy iteration that repeatedly improves an initial policy using Grover search until the optimum is reached. Finally, we present an implementation of our algorithm for a two-armed bandit MDP which we then simulate.

The Sharpness Aware Minimization (SAM) optimization algorithm has been shown to control large eigenvalues of the loss Hessian and provide generalization benefits in a variety of settings. The original motivation for SAM was a modified loss function which penalized sharp minima; subsequent analyses have also focused on the behavior near minima. However, our work reveals that SAM provides a strong regularization of the eigenvalues throughout the learning trajectory. We show that in a simplified setting, SAM dynamically induces a stabilization related to the edge of stability (EOS) phenomenon observed in large learning rate gradient descent. Our theory predicts the largest eigenvalue as a function of the learning rate and SAM radius parameters. Finally, we show that practical models can also exhibit this EOS stabilization, and that understanding SAM must account for these dynamics far away from any minima.

Autoregressively trained transformers have brought a profound revolution to the world, especially with their in-context learning (ICL) ability to address downstream tasks. Recently, several studies suggest that transformers learn a mesa-optimizer during autoregressive (AR) pretraining to implement ICL. Namely, the forward pass of the trained transformer is equivalent to optimizing an inner objective function in-context. However, whether the practical non-convex training dynamics will converge to the ideal mesa-optimizer is still unclear. Towards filling this gap, we investigate the non-convex dynamics of a one-layer linear causal self-attention model autoregressively trained by gradient flow, where the sequences are generated by an AR process x_{t+1} = W x_t. First, under a certain condition of data distribution, we prove that an autoregressively trained transformer learns W by implementing one step of gradient descent to minimize an ordinary least squares (OLS) problem in-context. It then applies the learned W for next-token prediction, thereby verifying the mesa-optimization hypothesis. Next, under the same data conditions, we explore the capability limitations of the obtained mesa-optimizer. We show that a stronger assumption related to the moments of data is the sufficient and necessary condition that the learned mesa-optimizer recovers the distribution. Besides, we conduct exploratory analyses beyond the first data condition and prove that generally, the trained transformer will not perform vanilla gradient descent for the OLS problem. Finally, our simulation results verify the theoretical results.

This work investigates a case study of using physical-based sonification of Quadratic Unconstrained Binary Optimization (QUBO) problems, optimized by the Variational Quantum Eigensolver (VQE) algorithm. The VQE approximates the solution of the problem by using an iterative loop between the quantum computer and a classical optimization routine. This work explores the intermediary statevectors found in each VQE iteration as the means of sonifying the optimization process itself. The implementation was realised in the form of a musical interface prototype named Variational Quantum Harmonizer (VQH), providing potential design strategies for musical applications, focusing on chords, chord progressions, and arpeggios. The VQH can be used both to enhance data visualization or to create artistic pieces. The methodology is also relevant in terms of how an artist would gain intuition towards achieving a desired musical sound by carefully designing QUBO cost functions. Flexible mapping strategies could supply a broad portfolio of sounds for QUBO and quantum-inspired musical compositions, as demonstrated in a case study composition, "Dependent Origination" by Peter Thomas and Paulo Itaborai.

Policy gradient methods hold great potential for solving complex continuous control tasks. Still, their training efficiency can be improved by exploiting structure within the optimization problem. Recent work indicates that supervised learning can be accelerated by leveraging the fact that gradients lie in a low-dimensional and slowly-changing subspace. In this paper, we conduct a thorough evaluation of this phenomenon for two popular deep policy gradient methods on various simulated benchmark tasks. Our results demonstrate the existence of such gradient subspaces despite the continuously changing data distribution inherent to reinforcement learning. These findings reveal promising directions for future work on more efficient reinforcement learning, e.g., through improving parameter-space exploration or enabling second-order optimization.

Synthesizing optimal controllers for dynamical systems often involves solving optimization problems with hard real-time constraints. These constraints determine the class of numerical methods that can be applied: computationally expensive but accurate numerical routines are replaced by fast and inaccurate methods, trading inference time for solution accuracy. This paper provides techniques to improve the quality of optimized control policies given a fixed computational budget. We achieve the above via a hypersolvers approach, which hybridizes a differential equation solver and a neural network. The performance is evaluated in direct and receding-horizon optimal control tasks in both low and high dimensions, where the proposed approach shows consistent Pareto improvements in solution accuracy and control performance.

Soft Actor-Critic is a state-of-the-art reinforcement learning algorithm for continuous action settings that is not applicable to discrete action settings. Many important settings involve discrete actions, however, and so here we derive an alternative version of the Soft Actor-Critic algorithm that is applicable to discrete action settings. We then show that, even without any hyperparameter tuning, it is competitive with the tuned model-free state-of-the-art on a selection of games from the Atari suite.

Dynamic treatment regimes (DTRs) are personalized, adaptive, multi-stage treatment plans that adapt treatment decisions both to an individual's initial features and to intermediate outcomes and features at each subsequent stage, which are affected by decisions in prior stages. Examples include personalized first- and second-line treatments of chronic conditions like diabetes, cancer, and depression, which adapt to patient response to first-line treatment, disease progression, and individual characteristics. While existing literature mostly focuses on estimating the optimal DTR from offline data such as from sequentially randomized trials, we study the problem of developing the optimal DTR in an online manner, where the interaction with each individual affect both our cumulative reward and our data collection for future learning. We term this the DTR bandit problem. We propose a novel algorithm that, by carefully balancing exploration and exploitation, is guaranteed to achieve rate-optimal regret when the transition and reward models are linear. We demonstrate our algorithm and its benefits both in synthetic experiments and in a case study of adaptive treatment of major depressive disorder using real-world data.

Offline optimization is an important task in numerous material engineering domains where online experimentation to collect data is too expensive and needs to be replaced by an in silico maximization of a surrogate of the black-box function. Although such a surrogate can be learned from offline data, its prediction might not be reliable outside the offline data regime, which happens when the surrogate has narrow prediction margin and is (therefore) sensitive to small perturbations of its parameterization. This raises the following questions: (1) how to regulate the sensitivity of a surrogate model; and (2) whether conditioning an offline optimizer with such less sensitive surrogate will lead to better optimization performance. To address these questions, we develop an optimizable sensitivity measurement for the surrogate model, which then inspires a sensitivity-informed regularizer that is applicable to a wide range of offline optimizers. This development is both orthogonal and synergistic to prior research on offline optimization, which is demonstrated in our extensive experiment benchmark.

The varying significance of distinct primitive behaviors during the policy learning process has been overlooked by prior model-free RL algorithms. Leveraging this insight, we explore the causal relationship between different action dimensions and rewards to evaluate the significance of various primitive behaviors during training. We introduce a causality-aware entropy term that effectively identifies and prioritizes actions with high potential impacts for efficient exploration. Furthermore, to prevent excessive focus on specific primitive behaviors, we analyze the gradient dormancy phenomenon and introduce a dormancy-guided reset mechanism to further enhance the efficacy of our method. Our proposed algorithm, ACE: Off-policy Actor-critic with Causality-aware Entropy regularization, demonstrates a substantial performance advantage across 29 diverse continuous control tasks spanning 7 domains compared to model-free RL baselines, which underscores the effectiveness, versatility, and efficient sample efficiency of our approach. Benchmark results and videos are available at https://ace-rl.github.io/.

In recent years, Reinforcement Learning (RL) has been applied to real-world problems with increasing success. Such applications often require to put constraints on the agent's behavior. Existing algorithms for constrained RL (CRL) rely on gradient descent-ascent, but this approach comes with a caveat. While these algorithms are guaranteed to converge on average, they do not guarantee last-iterate convergence, i.e., the current policy of the agent may never converge to the optimal solution. In practice, it is often observed that the policy alternates between satisfying the constraints and maximizing the reward, rarely accomplishing both objectives simultaneously. Here, we address this problem by introducing Reinforcement Learning with Optimistic Ascent-Descent (ReLOAD), a principled CRL method with guaranteed last-iterate convergence. We demonstrate its empirical effectiveness on a wide variety of CRL problems including discrete MDPs and continuous control. In the process we establish a benchmark of challenging CRL problems.

Reinforcement learning with human feedback~(RLHF) is critical for aligning Large Language Models (LLMs) with human preference. Compared to the widely studied offline version of RLHF, e.g. direct preference optimization (DPO), recent works have shown that the online variants achieve even better alignment. However, online alignment requires on-the-fly generation of new training data, which is costly, hard to parallelize, and suffers from varying quality and utility. In this paper, we propose a more efficient data exploration strategy for online preference tuning (OPTune), which does not rely on human-curated or pre-collected teacher responses but dynamically samples informative responses for on-policy preference alignment. During data generation, OPTune only selects prompts whose (re)generated responses can potentially provide more informative and higher-quality training signals than the existing responses. In the training objective, OPTune reweights each generated response (pair) by its utility in improving the alignment so that learning can be focused on the most helpful samples. Throughout our evaluations, OPTune'd LLMs maintain the instruction-following benefits provided by standard preference tuning whilst enjoying 1.27-1.56x faster training speed due to the efficient data exploration strategy.

For continuous action spaces, actor-critic methods are widely used in online reinforcement learning (RL). However, unlike RL algorithms for discrete actions, which generally model the optimal value function using the Bellman optimality operator, RL algorithms for continuous actions typically model Q-values for the current policy using the Bellman operator. These algorithms for continuous actions rely exclusively on policy updates for improvement, which often results in low sample efficiency. This study examines the effectiveness of incorporating the Bellman optimality operator into actor-critic frameworks. Experiments in a simple environment show that modeling optimal values accelerates learning but leads to overestimation bias. To address this, we propose an annealing approach that gradually transitions from the Bellman optimality operator to the Bellman operator, thereby accelerating learning while mitigating bias. Our method, combined with TD3 and SAC, significantly outperforms existing approaches across various locomotion and manipulation tasks, demonstrating improved performance and robustness to hyperparameters related to optimality. The code for this study is available at https://github.com/motokiomura/annealed-q-learning.

Recent advancements in meta-learning have enabled the automatic discovery of novel reinforcement learning algorithms parameterized by surrogate objective functions. To improve upon manually designed algorithms, the parameterization of this learned objective function must be expressive enough to represent novel principles of learning (instead of merely recovering already established ones) while still generalizing to a wide range of settings outside of its meta-training distribution. However, existing methods focus on discovering objective functions that, like many widely used objective functions in reinforcement learning, do not take into account the total number of steps allowed for training, or "training horizon". In contrast, humans use a plethora of different learning objectives across the course of acquiring a new ability. For instance, students may alter their studying techniques based on the proximity to exam deadlines and their self-assessed capabilities. This paper contends that ignoring the optimization time horizon significantly restricts the expressive potential of discovered learning algorithms. We propose a simple augmentation to two existing objective discovery approaches that allows the discovered algorithm to dynamically update its objective function throughout the agent's training procedure, resulting in expressive schedules and increased generalization across different training horizons. In the process, we find that commonly used meta-gradient approaches fail to discover such adaptive objective functions while evolution strategies discover highly dynamic learning rules. We demonstrate the effectiveness of our approach on a wide range of tasks and analyze the resulting learned algorithms, which we find effectively balance exploration and exploitation by modifying the structure of their learning rules throughout the agent's lifetime.

Quantum Computing has been evolving in the last years. Although nowadays quantum algorithms performance has shown superior to their classical counterparts, quantum decoherence and additional auxiliary qubits needed for error tolerance routines have been huge barriers for quantum algorithms efficient use. These restrictions lead us to search for ways to minimize algorithms costs, i.e the number of quantum logical gates and the depth of the circuit. For this, quantum circuit synthesis and quantum circuit optimization techniques are explored. We studied the viability of using Projective Simulation, a reinforcement learning technique, to tackle the problem of quantum circuit synthesis for noise quantum computers with limited number of qubits. The agent had the task of creating quantum circuits up to 5 qubits to generate GHZ states in the IBM Tenerife (IBM QX4) quantum processor. Our simulations demonstrated that the agent had a good performance but its capacity for learning new circuits decreased as the number of qubits increased.

We introduce a gradient-based approach for the problem of Bayesian optimal experimental design to learn causal models in a batch setting -- a critical component for causal discovery from finite data where interventions can be costly or risky. Existing methods rely on greedy approximations to construct a batch of experiments while using black-box methods to optimize over a single target-state pair to intervene with. In this work, we completely dispose of the black-box optimization techniques and greedy heuristics and instead propose a conceptually simple end-to-end gradient-based optimization procedure to acquire a set of optimal intervention target-state pairs. Such a procedure enables parameterization of the design space to efficiently optimize over a batch of multi-target-state interventions, a setting which has hitherto not been explored due to its complexity. We demonstrate that our proposed method outperforms baselines and existing acquisition strategies in both single-target and multi-target settings across a number of synthetic datasets.

We investigate the fidelity of Grover's search algorithm by implementing it on an open quantum system. In particular, we study with what accuracy one can estimate that the algorithm would deliver the searched state. In reality, every system has some influence of its environment. We include the environmental effects on the system dynamics by using a recently reported fluctuation-regulated quantum master equation (FRQME). The FRQME indicates that in addition to the regular relaxation due to system-environment coupling, the applied drive also causes dissipation in the system dynamics. As a result, the fidelity is found to depend on both the drive-induced dissipative terms and the relaxation terms and we find that there exists a competition between them, leading to an optimum value of the drive amplitude for which the fidelity becomes maximum. For efficient implementation of the search algorithm, precise knowledge of this optimum drive amplitude is essential.

In many environments only a tiny subset of all states yield high reward. In these cases, few of the interactions with the environment provide a relevant learning signal. Hence, we may want to preferentially train on those high-reward states and the probable trajectories leading to them. To this end, we advocate for the use of a backtracking model that predicts the preceding states that terminate at a given high-reward state. We can train a model which, starting from a high value state (or one that is estimated to have high value), predicts and sample for which the (state, action)-tuples may have led to that high value state. These traces of (state, action) pairs, which we refer to as Recall Traces, sampled from this backtracking model starting from a high value state, are informative as they terminate in good states, and hence we can use these traces to improve a policy. We provide a variational interpretation for this idea and a practical algorithm in which the backtracking model samples from an approximate posterior distribution over trajectories which lead to large rewards. Our method improves the sample efficiency of both on- and off-policy RL algorithms across several environments and tasks.

This paper aims to overcome a major obstacle in scaling RL for reasoning with LLMs, namely the collapse of policy entropy. Such phenomenon is consistently observed across vast RL runs without entropy intervention, where the policy entropy dropped sharply at the early training stage, this diminished exploratory ability is always accompanied with the saturation of policy performance. In practice, we establish a transformation equation R=-a*e^H+b between entropy H and downstream performance R. This empirical law strongly indicates that, the policy performance is traded from policy entropy, thus bottlenecked by its exhaustion, and the ceiling is fully predictable H=0, R=-a+b. Our finding necessitates entropy management for continuous exploration toward scaling compute for RL. To this end, we investigate entropy dynamics both theoretically and empirically. Our derivation highlights that, the change in policy entropy is driven by the covariance between action probability and the change in logits, which is proportional to its advantage when using Policy Gradient-like algorithms. Empirical study shows that, the values of covariance term and entropy differences matched exactly, supporting the theoretical conclusion. Moreover, the covariance term stays mostly positive throughout training, further explaining why policy entropy would decrease monotonically. Through understanding the mechanism behind entropy dynamics, we motivate to control entropy by restricting the update of high-covariance tokens. Specifically, we propose two simple yet effective techniques, namely Clip-Cov and KL-Cov, which clip and apply KL penalty to tokens with high covariances respectively. Experiments show that these methods encourage exploration, thus helping policy escape entropy collapse and achieve better downstream performance.

We introduce Feasible Learning (FL), a sample-centric learning paradigm where models are trained by solving a feasibility problem that bounds the loss for each training sample. In contrast to the ubiquitous Empirical Risk Minimization (ERM) framework, which optimizes for average performance, FL demands satisfactory performance on every individual data point. Since any model that meets the prescribed performance threshold is a valid FL solution, the choice of optimization algorithm and its dynamics play a crucial role in shaping the properties of the resulting solutions. In particular, we study a primal-dual approach which dynamically re-weights the importance of each sample during training. To address the challenge of setting a meaningful threshold in practice, we introduce a relaxation of FL that incorporates slack variables of minimal norm. Our empirical analysis, spanning image classification, age regression, and preference optimization in large language models, demonstrates that models trained via FL can learn from data while displaying improved tail behavior compared to ERM, with only a marginal impact on average performance.

Preference optimization methods typically begin training with a well-trained SFT model as a reference model. In RLHF and DPO, a regularization term is used during the preference optimization process to prevent the policy model from deviating too far from the reference model's distribution, thereby avoiding the generation of anomalous responses. When the reference model is already well-aligned with the given data or only requires slight adjustments, this approach can produce a well-aligned model. However, if the reference model is not aligned with the given data and requires significant deviation from its current state, a regularization term may actually hinder the model alignment. In this study, we propose Modulated Intervention Preference Optimization (MIPO) to address this issue. MIPO modulates the degree of intervention from the reference model based on how well the given data is aligned with it. If the data is well-aligned, the intervention is increased to prevent the policy model from diverging significantly from reference model. Conversely, if the alignment is poor, the interference is reduced to facilitate more extensive training. We compare the performance of MIPO and DPO using Mistral-7B and Llama3-8B in Alpaca Eval 2.0 and MT-Bench. The experimental results demonstrate that MIPO consistently outperforms DPO across various evaluation scenarios.

On-policy reinforcement learning (RL) has become a popular framework for solving sequential decision problems due to its computational efficiency and theoretical simplicity. Some on-policy methods guarantee every policy update is constrained to a trust region relative to the prior policy to ensure training stability. These methods often require computationally intensive non-linear optimization or require a particular form of action distribution. In this work, we show that applying KL penalization alone is nearly sufficient to enforce such trust regions. Then, we show that introducing a "fixup" phase is sufficient to guarantee a trust region is enforced on every policy update while adding fewer than 5% additional gradient steps in practice. The resulting algorithm, which we call FixPO, is able to train a variety of policy architectures and action spaces, is easy to implement, and produces results competitive with other trust region methods.

Symbolic world modeling requires inferring and representing an environment's transitional dynamics as an executable program. Prior work has focused on largely deterministic environments with abundant interaction data, simple mechanics, and human guidance. We address a more realistic and challenging setting, learning in a complex, stochastic environment where the agent has only "one life" to explore a hostile environment without human guidance. We introduce OneLife, a framework that models world dynamics through conditionally-activated programmatic laws within a probabilistic programming framework. Each law operates through a precondition-effect structure, activating in relevant world states. This creates a dynamic computation graph that routes inference and optimization only through relevant laws, avoiding scaling challenges when all laws contribute to predictions about a complex, hierarchical state, and enabling the learning of stochastic dynamics even with sparse rule activation. To evaluate our approach under these demanding constraints, we introduce a new evaluation protocol that measures (a) state ranking, the ability to distinguish plausible future states from implausible ones, and (b) state fidelity, the ability to generate future states that closely resemble reality. We develop and evaluate our framework on Crafter-OO, our reimplementation of the Crafter environment that exposes a structured, object-oriented symbolic state and a pure transition function that operates on that state alone. OneLife can successfully learn key environment dynamics from minimal, unguided interaction, outperforming a strong baseline on 16 out of 23 scenarios tested. We also test OneLife's planning ability, with simulated rollouts successfully identifying superior strategies. Our work establishes a foundation for autonomously constructing programmatic world models of unknown, complex environments.

We consider online learning in multi-player smooth monotone games. Existing algorithms have limitations such as (1) being only applicable to strongly monotone games; (2) lacking the no-regret guarantee; (3) having only asymptotic or slow O(1{T}) last-iterate convergence rate to a Nash equilibrium. While the O(1{T}) rate is tight for a large class of algorithms including the well-studied extragradient algorithm and optimistic gradient algorithm, it is not optimal for all gradient-based algorithms. We propose the accelerated optimistic gradient (AOG) algorithm, the first doubly optimal no-regret learning algorithm for smooth monotone games. Namely, our algorithm achieves both (i) the optimal O(T) regret in the adversarial setting under smooth and convex loss functions and (ii) the optimal O(1{T}) last-iterate convergence rate to a Nash equilibrium in multi-player smooth monotone games. As a byproduct of the accelerated last-iterate convergence rate, we further show that each player suffers only an O(log T) individual worst-case dynamic regret, providing an exponential improvement over the previous state-of-the-art O(T) bound.

In this paper, we study the predict-then-optimize problem where the output of a machine learning prediction task is used as the input of some downstream optimization problem, say, the objective coefficient vector of a linear program. The problem is also known as predictive analytics or contextual linear programming. The existing approaches largely suffer from either (i) optimization intractability (a non-convex objective function)/statistical inefficiency (a suboptimal generalization bound) or (ii) requiring strong condition(s) such as no constraint or loss calibration. We develop a new approach to the problem called maximum optimality margin which designs the machine learning loss function by the optimality condition of the downstream optimization. The max-margin formulation enjoys both computational efficiency and good theoretical properties for the learning procedure. More importantly, our new approach only needs the observations of the optimal solution in the training data rather than the objective function, which makes it a new and natural approach to the inverse linear programming problem under both contextual and context-free settings; we also analyze the proposed method under both offline and online settings, and demonstrate its performance using numerical experiments.

Quantum machine learning implemented by variational quantum circuits (VQCs) is considered a promising concept for the noisy intermediate-scale quantum computing era. Focusing on applications in quantum reinforcement learning, we propose a specific action decoding procedure for a quantum policy gradient approach. We introduce a novel quality measure that enables us to optimize the classical post-processing required for action selection, inspired by local and global quantum measurements. The resulting algorithm demonstrates a significant performance improvement in several benchmark environments. With this technique, we successfully execute a full training routine on a 5-qubit hardware device. Our method introduces only negligible classical overhead and has the potential to improve VQC-based algorithms beyond the field of quantum reinforcement learning.

Multiobjective optimization plays an increasingly important role in modern applications, where several criteria are often of equal importance. The task in multiobjective optimization and multiobjective optimal control is therefore to compute the set of optimal compromises (the Pareto set) between the conflicting objectives. The advances in algorithms and the increasing interest in Pareto-optimal solutions have led to a wide range of new applications related to optimal and feedback control - potentially with non-smoothness both on the level of the objectives or in the system dynamics. This results in new challenges such as dealing with expensive models (e.g., governed by partial differential equations (PDEs)) and developing dedicated algorithms handling the non-smoothness. Since in contrast to single-objective optimization, the Pareto set generally consists of an infinite number of solutions, the computational effort can quickly become challenging, which is particularly problematic when the objectives are costly to evaluate or when a solution has to be presented very quickly. This article gives an overview of recent developments in the field of multiobjective optimization of non-smooth PDE-constrained problems. In particular we report on the advances achieved within Project 2 "Multiobjective Optimization of Non-Smooth PDE-Constrained Problems - Switches, State Constraints and Model Order Reduction" of the DFG Priority Programm 1962 "Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization".

Test-time scaling has emerged as a prominent research direction in machine learning, enabling models to enhance their expressive capabilities during inference.Transformers, renowned for striking a delicate balance between efficiency and expressiveness, have benefited from test-time scaling techniques that leverage an expanding key-value (KV) cache to significantly improve performance.In this paper, we introduce a novel state-based approach to test-time scaling, which we term state tuning, tailored to the RNN-based RWKV-7 model.By exploiting the unique strengths of RWKV-7, our method achieves state-of-the-art performance on the target task without altering the model's pre-trained weights. Our approach centers on three key innovations. First, we develop an observer framework that allows a smaller model to replicate and learn the state dynamics of the RWKV-7 model. Second, we employ a kernel method to dynamically upscale the state size, enhancing the model's capacity to capture intricate patterns. Third, we integrate Decorrelated Backpropagation (DBP) to optimize the upscaled state matrix, thereby improving convergence and expressivity. By tuning only the state matrix, we demonstrate that a smaller model can outperform larger models on the given task. This method preserves the efficiency of the original RWKV-7 architecture while harnessing the power of test-time scaling to deliver superior results. Our findings underscore the potential of state tuning as an effective strategy for advancing model performance in resource-constrained settings. Our code is https://github.com/TorchRWKV/flash-linear-attention.

Adversarial imitation learning (AIL) has stood out as a dominant framework across various imitation learning (IL) applications, with Discriminator Actor Critic (DAC) (Kostrikov et al.,, 2019) demonstrating the effectiveness of off-policy learning algorithms in improving sample efficiency and scalability to higher-dimensional observations. Despite DAC's empirical success, the original AIL objective is on-policy and DAC's ad-hoc application of off-policy training does not guarantee successful imitation (Kostrikov et al., 2019; 2020). Follow-up work such as ValueDICE (Kostrikov et al., 2020) tackles this issue by deriving a fully off-policy AIL objective. Instead in this work, we develop a novel and principled AIL algorithm via the framework of boosting. Like boosting, our new algorithm, AILBoost, maintains an ensemble of properly weighted weak learners (i.e., policies) and trains a discriminator that witnesses the maximum discrepancy between the distributions of the ensemble and the expert policy. We maintain a weighted replay buffer to represent the state-action distribution induced by the ensemble, allowing us to train discriminators using the entire data collected so far. In the weighted replay buffer, the contribution of the data from older policies are properly discounted with the weight computed based on the boosting framework. Empirically, we evaluate our algorithm on both controller state-based and pixel-based environments from the DeepMind Control Suite. AILBoost outperforms DAC on both types of environments, demonstrating the benefit of properly weighting replay buffer data for off-policy training. On state-based environments, DAC outperforms ValueDICE and IQ-Learn (Gary et al., 2021), achieving competitive performance with as little as one expert trajectory.

We introduce a modern Hopfield network with continuous states and a corresponding update rule. The new Hopfield network can store exponentially (with the dimension of the associative space) many patterns, retrieves the pattern with one update, and has exponentially small retrieval errors. It has three types of energy minima (fixed points of the update): (1) global fixed point averaging over all patterns, (2) metastable states averaging over a subset of patterns, and (3) fixed points which store a single pattern. The new update rule is equivalent to the attention mechanism used in transformers. This equivalence enables a characterization of the heads of transformer models. These heads perform in the first layers preferably global averaging and in higher layers partial averaging via metastable states. The new modern Hopfield network can be integrated into deep learning architectures as layers to allow the storage of and access to raw input data, intermediate results, or learned prototypes. These Hopfield layers enable new ways of deep learning, beyond fully-connected, convolutional, or recurrent networks, and provide pooling, memory, association, and attention mechanisms. We demonstrate the broad applicability of the Hopfield layers across various domains. Hopfield layers improved state-of-the-art on three out of four considered multiple instance learning problems as well as on immune repertoire classification with several hundreds of thousands of instances. On the UCI benchmark collections of small classification tasks, where deep learning methods typically struggle, Hopfield layers yielded a new state-of-the-art when compared to different machine learning methods. Finally, Hopfield layers achieved state-of-the-art on two drug design datasets. The implementation is available at: https://github.com/ml-jku/hopfield-layers

The past decade has seen vast progress in deep reinforcement learning (RL) on the back of algorithms manually designed by human researchers. Recently, it has been shown that it is possible to meta-learn update rules, with the hope of discovering algorithms that can perform well on a wide range of RL tasks. Despite impressive initial results from algorithms such as Learned Policy Gradient (LPG), there remains a generalization gap when these algorithms are applied to unseen environments. In this work, we examine how characteristics of the meta-training distribution impact the generalization performance of these algorithms. Motivated by this analysis and building on ideas from Unsupervised Environment Design (UED), we propose a novel approach for automatically generating curricula to maximize the regret of a meta-learned optimizer, in addition to a novel approximation of regret, which we name algorithmic regret (AR). The result is our method, General RL Optimizers Obtained Via Environment Design (GROOVE). In a series of experiments, we show that GROOVE achieves superior generalization to LPG, and evaluate AR against baseline metrics from UED, identifying it as a critical component of environment design in this setting. We believe this approach is a step towards the discovery of truly general RL algorithms, capable of solving a wide range of real-world environments.

To minimize the average of a set of log-convex functions, the stochastic Newton method iteratively updates its estimate using subsampled versions of the full objective's gradient and Hessian. We contextualize this optimization problem as sequential Bayesian inference on a latent state-space model with a discriminatively-specified observation process. Applying Bayesian filtering then yields a novel optimization algorithm that considers the entire history of gradients and Hessians when forming an update. We establish matrix-based conditions under which the effect of older observations diminishes over time, in a manner analogous to Polyak's heavy ball momentum. We illustrate various aspects of our approach with an example and review other relevant innovations for the stochastic Newton method.

Reinforcement learning (RL) has recently become the core paradigm for aligning and strengthening large language models (LLMs). Yet, applying RL in off-policy settings--where stale data from past policies are used for training--improves sample efficiency, but remains challenging: policy entropy declines sharply, optimization often becomes unstable and may even collapse. Through theoretical and empirical analysis, we identify two key insights: (i) an imbalance in optimization, where negative-advantage samples dominate the policy gradient, suppressing useful behaviors and risking gradient explosions; and (ii) the derived Entropy-Clip Rule, which reveals that the fixed clipping mechanism in PPO-like objectives systematically blocks entropy-increasing updates, thereby driving the policy toward over-exploitation at the expense of exploration. Building on these insights, we propose BAlanced Policy Optimization with Adaptive Clipping (BAPO), a simple yet effective method that dynamically adjusts clipping bounds to adaptively re-balance positive and negative contributions, preserve entropy, and stabilize RL optimization. Across diverse off-policy scenarios--including sample replay and partial rollout--BAPO achieves fast, stable, and data-efficient training. On AIME 2024 and AIME 2025 benchmarks, our 7B BAPO model surpasses open-source counterparts such as SkyWork-OR1-7B, while our 32B BAPO model not only achieves state-of-the-art results among models of the same scale but also outperforms leading proprietary systems like o3-mini and Gemini-2.5-Flash-Thinking.

Reinforcement Learning has emerged as the fundamental technique for enhancing reasoning in LLMs. However, existing algorithms apply uniform optimization to all tokens, ignoring their different roles in reasoning process. To address this limitation, we introduce Heterogeneous Adaptive Policy Optimization (HAPO), a comprehensive token-aware algorithm that dynamically adapts optimization based on token entropy. For rollout sampling, we propose Adaptive Temperature Sampling, which adjusts sampling temperature in real time, promoting exploration at high-entropy tokens while preserving coherence at low-entropy ones. For advantage calculation, we introduce Token Level Group Average that normalizes advantages at token level, jointly accounting for sequence-length as in token-mean loss while preserving non-biased treatment. We then develop Differential Advantage Redistribution that leverages entropy and importance ratios to modulate rewards-adjusting updates for tokens with clear signals. For clipping loss, we design Asymmetric Adaptive Clipping, allowing aggressive probability reduction for noisy low-entropy tokens while enabling exploration for high-entropy tokens. Through systematic investigation between entropy and training dynamics, we embedded token-level treatment into every stages to achieve fine-grained control. Extensive experiments demonstrate that HAPO consistently outperforms DAPO across multiple model scales. Our code can be found in https://github.com/starriver030515/HAPO.

Flow-based generative models, including diffusion models, excel at modeling continuous distributions in high-dimensional spaces. In this work, we introduce Flow Policy Optimization (FPO), a simple on-policy reinforcement learning algorithm that brings flow matching into the policy gradient framework. FPO casts policy optimization as maximizing an advantage-weighted ratio computed from the conditional flow matching loss, in a manner compatible with the popular PPO-clip framework. It sidesteps the need for exact likelihood computation while preserving the generative capabilities of flow-based models. Unlike prior approaches for diffusion-based reinforcement learning that bind training to a specific sampling method, FPO is agnostic to the choice of diffusion or flow integration at both training and inference time. We show that FPO can train diffusion-style policies from scratch in a variety of continuous control tasks. We find that flow-based models can capture multimodal action distributions and achieve higher performance than Gaussian policies, particularly in under-conditioned settings.

We propose a quantum version of a generative diffusion model. In this algorithm, artificial neural networks are replaced with parameterized quantum circuits, in order to directly generate quantum states. We present both a full quantum and a latent quantum version of the algorithm; we also present a conditioned version of these models. The models' performances have been evaluated using quantitative metrics complemented by qualitative assessments. An implementation of a simplified version of the algorithm has been executed on real NISQ quantum hardware.

Model-free reinforcement learning algorithms have seen remarkable progress, but key challenges remain. Trust Region Policy Optimization (TRPO) is known for ensuring monotonic policy improvement through conservative updates within a trust region, backed by strong theoretical guarantees. However, its reliance on complex second-order optimization limits its practical efficiency. Proximal Policy Optimization (PPO) addresses this by simplifying TRPO's approach using ratio clipping, improving efficiency but sacrificing some theoretical robustness. This raises a natural question: Can we combine the strengths of both methods? In this paper, we introduce Simple Policy Optimization (SPO), a novel unconstrained first-order algorithm. By slightly modifying the policy loss used in PPO, SPO can achieve the best of both worlds. Our new objective improves upon ratio clipping, offering stronger theoretical properties and better constraining the probability ratio within the trust region. Empirical results demonstrate that SPO outperforms PPO with a simple implementation, particularly for training large, complex network architectures end-to-end.

We propose a new method for count-based exploration in high-dimensional state spaces. Unlike previous work which relies on density models, we show that counts can be derived by averaging samples from the Rademacher distribution (or coin flips). This insight is used to set up a simple supervised learning objective which, when optimized, yields a state's visitation count. We show that our method is significantly more effective at deducing ground-truth visitation counts than previous work; when used as an exploration bonus for a model-free reinforcement learning algorithm, it outperforms existing approaches on most of 9 challenging exploration tasks, including the Atari game Montezuma's Revenge.

Policy optimization methods are powerful algorithms in Reinforcement Learning (RL) for their flexibility to deal with policy parameterization and ability to handle model misspecification. However, these methods usually suffer from slow convergence rates and poor sample complexity. Hence it is important to design provably sample efficient algorithms for policy optimization. Yet, recent advances for this problems have only been successful in tabular and linear setting, whose benign structures cannot be generalized to non-linearly parameterized policies. In this paper, we address this problem by leveraging recent advances in value-based algorithms, including bounded eluder-dimension and online sensitivity sampling, to design a low-switching sample-efficient policy optimization algorithm, LPO, with general non-linear function approximation. We show that, our algorithm obtains an varepsilon-optimal policy with only O(text{poly(d)}{varepsilon^3}) samples, where varepsilon is the suboptimality gap and d is a complexity measure of the function class approximating the policy. This drastically improves previously best-known sample bound for policy optimization algorithms, O(text{poly(d)}{varepsilon^8}). Moreover, we empirically test our theory with deep neural nets to show the benefits of the theoretical inspiration.

Recently, there has been significant interest in replacing the reward model in Reinforcement Learning with Human Feedback (RLHF) methods for Large Language Models (LLMs), such as Direct Preference Optimization (DPO) and its variants. These approaches commonly use a binary cross-entropy mechanism on pairwise samples, i.e., minimizing and maximizing the loss based on preferred or dis-preferred responses, respectively. However, while this training strategy omits the reward model, it also overlooks the varying preference degrees within different responses. We hypothesize that this is a key factor hindering LLMs from sufficiently understanding human preferences. To address this problem, we propose a novel Self-supervised Preference Optimization (SPO) framework, which constructs a self-supervised preference degree loss combined with the alignment loss, thereby helping LLMs improve their ability to understand the degree of preference. Extensive experiments are conducted on two widely used datasets of different tasks. The results demonstrate that SPO can be seamlessly integrated with existing preference optimization methods and significantly boost their performance to achieve state-of-the-art performance. We also conduct detailed analyses to offer comprehensive insights into SPO, which verifies its effectiveness. The code is available at https://github.com/lijian16/SPO.

Mirror descent (MD), a well-known first-order method in constrained convex optimization, has recently been shown as an important tool to analyze trust-region algorithms in reinforcement learning (RL). However, there remains a considerable gap between such theoretically analyzed algorithms and the ones used in practice. Inspired by this, we propose an efficient RL algorithm, called {\em mirror descent policy optimization} (MDPO). MDPO iteratively updates the policy by {\em approximately} solving a trust-region problem, whose objective function consists of two terms: a linearization of the standard RL objective and a proximity term that restricts two consecutive policies to be close to each other. Each update performs this approximation by taking multiple gradient steps on this objective function. We derive {\em on-policy} and {\em off-policy} variants of MDPO, while emphasizing important design choices motivated by the existing theory of MD in RL. We highlight the connections between on-policy MDPO and two popular trust-region RL algorithms: TRPO and PPO, and show that explicitly enforcing the trust-region constraint is in fact {\em not} a necessity for high performance gains in TRPO. We then show how the popular soft actor-critic (SAC) algorithm can be derived by slight modifications of off-policy MDPO. Overall, MDPO is derived from the MD principles, offers a unified approach to viewing a number of popular RL algorithms, and performs better than or on-par with TRPO, PPO, and SAC in a number of continuous control tasks. Code is available at https://github.com/manantomar/Mirror-Descent-Policy-Optimization.

This manuscript portrays optimization as a process. In many practical applications the environment is so complex that it is infeasible to lay out a comprehensive theoretical model and use classical algorithmic theory and mathematical optimization. It is necessary as well as beneficial to take a robust approach, by applying an optimization method that learns as one goes along, learning from experience as more aspects of the problem are observed. This view of optimization as a process has become prominent in varied fields and has led to some spectacular success in modeling and systems that are now part of our daily lives.

Much has been said about observability in system theory and control; however, it has been recently that observability in complex networks has seriously attracted the attention of researchers. This paper examines the state-of-the-art and discusses some issues raised due to "complexity" and "stochasticity". These unresolved issues call for a new practical methodology. For stochastic systems, a degree of observability may be defined and the observability problem is not a binary (i.e., yes-no) question anymore. Here, we propose to employ a goal-seeking system to play a supervisory role in the network. Hence, improving the degree of observability would be a valid objective for the supervisory system. Towards this goal, the supervisor dynamically optimizes the observation process by reconfiguring the sensory parts in the network. A cognitive dynamic system is suggested as a proper choice for the supervisory system. In this framework, the network itself is viewed as the environment with which the cognitive dynamic system interacts. Computer experiments confirm the potential of the proposed approach for addressing some of the issues raised in networks due to complexity and stochasticity.

In a compound AI system, components such as an LLM call, a retriever, a code interpreter, or tools are interconnected. The system's behavior is primarily driven by parameters such as instructions or tool definitions. Recent advancements enable end-to-end optimization of these parameters using an LLM. Notably, leveraging an LLM as an optimizer is particularly efficient because it avoids gradient computation and can generate complex code and instructions. This paper presents a survey of the principles and emerging trends in LLM-based optimization of compound AI systems. It covers archetypes of compound AI systems, approaches to LLM-based end-to-end optimization, and insights into future directions and broader impacts. Importantly, this survey uses concepts from program analysis to provide a unified view of how an LLM optimizer is prompted to optimize a compound AI system. The exhaustive list of paper is provided at https://github.com/linyuhongg/LLM-based-Optimization-of-Compound-AI-Systems.

Offline optimization has recently emerged as an increasingly popular approach to mitigate the prohibitively expensive cost of online experimentation. The key idea is to learn a surrogate of the black-box function that underlines the target experiment using a static (offline) dataset of its previous input-output queries. Such an approach is, however, fraught with an out-of-distribution issue where the learned surrogate becomes inaccurate outside the offline data regimes. To mitigate this, existing offline optimizers have proposed numerous conditioning techniques to prevent the learned surrogate from being too erratic. Nonetheless, such conditioning strategies are often specific to particular surrogate or search models, which might not generalize to a different model choice. This motivates us to develop a model-agnostic approach instead, which incorporates a notion of model sharpness into the training loss of the surrogate as a regularizer. Our approach is supported by a new theoretical analysis demonstrating that reducing surrogate sharpness on the offline dataset provably reduces its generalized sharpness on unseen data. Our analysis extends existing theories from bounding generalized prediction loss (on unseen data) with loss sharpness to bounding the worst-case generalized surrogate sharpness with its empirical estimate on training data, providing a new perspective on sharpness regularization. Our extensive experimentation on a diverse range of optimization tasks also shows that reducing surrogate sharpness often leads to significant improvement, marking (up to) a noticeable 9.6% performance boost. Our code is publicly available at https://github.com/cuong-dm/IGNITE

Representation learning and exploration are among the key challenges for any deep reinforcement learning agent. In this work, we provide a singular value decomposition based method that can be used to obtain representations that preserve the underlying transition structure in the domain. Perhaps interestingly, we show that these representations also capture the relative frequency of state visitations, thereby providing an estimate for pseudo-counts for free. To scale this decomposition method to large-scale domains, we provide an algorithm that never requires building the transition matrix, can make use of deep networks, and also permits mini-batch training. Further, we draw inspiration from predictive state representations and extend our decomposition method to partially observable environments. With experiments on multi-task settings with partially observable domains, we show that the proposed method can not only learn useful representation on DM-Lab-30 environments (that have inputs involving language instructions, pixel images, and rewards, among others) but it can also be effective at hard exploration tasks in DM-Hard-8 environments.

Auditing large language models for unexpected behaviors is critical to preempt catastrophic deployments, yet remains challenging. In this work, we cast auditing as an optimization problem, where we automatically search for input-output pairs that match a desired target behavior. For example, we might aim to find a non-toxic input that starts with "Barack Obama" that a model maps to a toxic output. This optimization problem is difficult to solve as the set of feasible points is sparse, the space is discrete, and the language models we audit are non-linear and high-dimensional. To combat these challenges, we introduce a discrete optimization algorithm, ARCA, that jointly and efficiently optimizes over inputs and outputs. Our approach automatically uncovers derogatory completions about celebrities (e.g. "Barack Obama is a legalized unborn" -> "child murderer"), produces French inputs that complete to English outputs, and finds inputs that generate a specific name. Our work offers a promising new tool to uncover models' failure-modes before deployment.

Selective state space models (SSM), such as Mamba, have gained prominence for their effectiveness in modeling sequential data. Despite their outstanding empirical performance, a comprehensive theoretical understanding of deep selective SSM remains elusive, hindering their further development and adoption for applications that need high fidelity. In this paper, we investigate the dynamical properties of tokens in a pre-trained Mamba model. In particular, we derive the dynamical system governing the continuous-time limit of the Mamba model and characterize the asymptotic behavior of its solutions. In the one-dimensional case, we prove that only one of the following two scenarios happens: either all tokens converge to zero, or all tokens diverge to infinity. We provide criteria based on model parameters to determine when each scenario occurs. For the convergent scenario, we empirically verify that this scenario negatively impacts the model's performance. For the divergent scenario, we prove that different tokens will diverge to infinity at different rates, thereby contributing unequally to the updates during model training. Based on these investigations, we propose two refinements for the model: excluding the convergent scenario and reordering tokens based on their importance scores, both aimed at improving practical performance. Our experimental results validate these refinements, offering insights into enhancing Mamba's effectiveness in real-world applications.

Reinforcement Learning from Human Feedback (RLHF) has emerged as a pivotal tool for aligning large language models (LLMs) with human preferences. Direct Preference Optimization (DPO), one of the most popular approaches, formulates RLHF as a policy optimization problem without explicitly estimating the reward function. It overcomes the stability and efficiency issues of two-step approaches, which typically involve first estimating the reward function and then optimizing the policy via proximal policy optimization (PPO). Since RLHF is essentially an optimization problem, and it is well-known that momentum techniques can accelerate optimization both theoretically and empirically, a natural question arises: Can RLHF be accelerated by momentum? This paper answers this question in the affirmative. In detail, we first show that the iterative preference optimization method can be viewed as a proximal point method. Based on this observation, we propose a general Accelerated Preference Optimization (APO) framework, which unifies many existing preference optimization algorithms and employs Nesterov's momentum technique to speed up the alignment of LLMs. Theoretically, we demonstrate that APO can achieve a faster convergence rate than the standard iterative preference optimization methods, including DPO and Self-Play Preference Optimization (SPPO). Empirically, we show the superiority of APO over DPO, iterative DPO, and other strong baselines for RLHF on the AlpacaEval 2.0 benchmark.

The alignment of language models with human preferences is vital for their application in real-world tasks. The problem is formulated as optimizing the model's policy to maximize the expected reward that reflects human preferences with minimal deviation from the initial policy. While considered as a straightforward solution, reinforcement learning (RL) suffers from high variance in policy updates, which impedes efficient policy improvement. Recently, direct preference optimization (DPO) was proposed to directly optimize the policy from preference data. Though simple to implement, DPO is derived based on the optimal policy that is not assured to be achieved in practice, which undermines its convergence to the intended solution. In this paper, we propose efficient exact optimization (EXO) of the alignment objective. We prove that EXO is guaranteed to optimize in the same direction as the RL algorithms asymptotically for arbitary parametrization of the policy, while enables efficient optimization by circumventing the complexities associated with RL algorithms. We compare our method to DPO with both theoretical and empirical analyses, and further demonstrate the advantages of our method over existing approaches on realistic human preference data.

Machine learning has successfully framed many sequential decision making problems as either supervised prediction, or optimal decision-making policy identification via reinforcement learning. In data-constrained offline settings, both approaches may fail as they assume fully optimal behavior or rely on exploring alternatives that may not exist. We introduce an inherently different approach that identifies possible "dead-ends" of a state space. We focus on the condition of patients in the intensive care unit, where a "medical dead-end" indicates that a patient will expire, regardless of all potential future treatment sequences. We postulate "treatment security" as avoiding treatments with probability proportional to their chance of leading to dead-ends, present a formal proof, and frame discovery as an RL problem. We then train three independent deep neural models for automated state construction, dead-end discovery and confirmation. Our empirical results discover that dead-ends exist in real clinical data among septic patients, and further reveal gaps between secure treatments and those that were administered.

When applying offline reinforcement learning (RL) in healthcare scenarios, the out-of-distribution (OOD) issues pose significant risks, as inappropriate generalization beyond clinical expertise can result in potentially harmful recommendations. While existing methods like conservative Q-learning (CQL) attempt to address the OOD issue, their effectiveness is limited by only constraining action selection by suppressing uncertain actions. This action-only regularization imitates clinician actions that prioritize short-term rewards, but it fails to regulate downstream state trajectories, thereby limiting the discovery of improved long-term treatment strategies. To safely improve policy beyond clinician recommendations while ensuring that state-action trajectories remain in-distribution, we propose Offline Guarded Safe Reinforcement Learning (OGSRL), a theoretically grounded model-based offline RL framework. OGSRL introduces a novel dual constraint mechanism for improving policy with reliability and safety. First, the OOD guardian is established to specify clinically validated regions for safe policy exploration. By constraining optimization within these regions, it enables the reliable exploration of treatment strategies that outperform clinician behavior by leveraging the full patient state history, without drifting into unsupported state-action trajectories. Second, we introduce a safety cost constraint that encodes medical knowledge about physiological safety boundaries, providing domain-specific safeguards even in areas where training data might contain potentially unsafe interventions. Notably, we provide theoretical guarantees on safety and near-optimality: policies that satisfy these constraints remain in safe and reliable regions and achieve performance close to the best possible policy supported by the data.

The higher-order correlation clustering problem is an expressive model, and recently, local search heuristics have been proposed for several applications. Certifying optimality, however, is NP-hard and practically hampered already by the complexity of the problem statement. Here, we focus on establishing partial optimality conditions for the special case of complete graphs and cubic objective functions. In addition, we define and implement algorithms for testing these conditions and examine their effect numerically, on two datasets.

We study regret minimization for reinforcement learning (RL) in Latent Markov Decision Processes (LMDPs) with context in hindsight. We design a novel model-based algorithmic framework which can be instantiated with both a model-optimistic and a value-optimistic solver. We prove an O(mathsf{Var^star M Gamma S A K}) regret bound where O hides logarithm factors, M is the number of contexts, S is the number of states, A is the number of actions, K is the number of episodes, Gamma le S is the maximum transition degree of any state-action pair, and Var^star is a variance quantity describing the determinism of the LMDP. The regret bound only scales logarithmically with the planning horizon, thus yielding the first (nearly) horizon-free regret bound for LMDP. This is also the first problem-dependent regret bound for LMDP. Key in our proof is an analysis of the total variance of alpha vectors (a generalization of value functions), which is handled with a truncation method. We complement our positive result with a novel Omega(mathsf{Var^star M S A K}) regret lower bound with Gamma = 2, which shows our upper bound minimax optimal when Gamma is a constant for the class of variance-bounded LMDPs. Our lower bound relies on new constructions of hard instances and an argument inspired by the symmetrization technique from theoretical computer science, both of which are technically different from existing lower bound proof for MDPs, and thus can be of independent interest.

Much of the excitement in modern AI is driven by the observation that scaling up existing systems leads to better performance. But does better performance necessarily imply better internal representations? While the representational optimist assumes it must, this position paper challenges that view. We compare neural networks evolved through an open-ended search process to networks trained via conventional stochastic gradient descent (SGD) on the simple task of generating a single image. This minimal setup offers a unique advantage: each hidden neuron's full functional behavior can be easily visualized as an image, thus revealing how the network's output behavior is internally constructed neuron by neuron. The result is striking: while both networks produce the same output behavior, their internal representations differ dramatically. The SGD-trained networks exhibit a form of disorganization that we term fractured entangled representation (FER). Interestingly, the evolved networks largely lack FER, even approaching a unified factored representation (UFR). In large models, FER may be degrading core model capacities like generalization, creativity, and (continual) learning. Therefore, understanding and mitigating FER could be critical to the future of representation learning.

We report an optical method of generating arbitrary polarization states by manipulating the thicknesses of a pair of uniaxial birefringent plates, the optical axes of which are set at a crossing angle of {\pi}/4. The method has the remarkable feature of being able to generate a distribution of arbitrary polarization states in a group of highly discrete spectra without spatially separating the individual spectral components. The target polarization-state distribution is obtained as an optimal solution through an exploration. Within a realistic exploration range, a sufficient number of near-optimal solutions are found. This property is also reproduced well by a concise model based on a distribution of exploration points on a Poincar\'e sphere, showing that the number of near-optimal solutions behaves according to a power law with respect to the number of spectral components of concern. As a typical example of an application, by applying this method to a set of phase-locked highly discrete spectra, we numerically demonstrate the continuous generation of a vector-like optical electric field waveform, the helicity of which is alternated within a single optical cycle in the time domain.

Reinforcement learning (RL) has emerged as a powerful approach for tackling complex problems. The recent introduction of multi-objective reinforcement learning (MORL) has further expanded the scope of RL by enabling agents to make trade-offs among multiple objectives. This advancement not only has broadened the range of problems that can be tackled but also created numerous opportunities for exploration and advancement. Yet, the effectiveness of RL agents heavily relies on appropriately setting their hyperparameters. In practice, this task often proves to be challenging, leading to unsuccessful deployments of these techniques in various instances. Hence, prior research has explored hyperparameter optimization in RL to address this concern. This paper presents an initial investigation into the challenge of hyperparameter optimization specifically for MORL. We formalize the problem, highlight its distinctive challenges, and propose a systematic methodology to address it. The proposed methodology is applied to a well-known environment using a state-of-the-art MORL algorithm, and preliminary results are reported. Our findings indicate that the proposed methodology can effectively provide hyperparameter configurations that significantly enhance the performance of MORL agents. Furthermore, this study identifies various future research opportunities to further advance the field of hyperparameter optimization for MORL.

Code optimization is a daunting task that requires a significant level of expertise from experienced programmers. This level of expertise is not sufficient when compared to the rapid development of new hardware architectures. Towards advancing the whole code optimization process, recent approaches rely on machine learning and artificial intelligence techniques. This paper introduces a new framework to decrease the complexity of code optimization. The proposed framework builds on large language models (LLMs) and reinforcement learning (RL) and enables LLMs to receive feedback from their environment (i.e., unit tests) during the fine-tuning process. We compare our framework with existing state-of-the-art models and show that it is more efficient with respect to speed and computational usage, as a result of the decrement in training steps and its applicability to models with fewer parameters. Additionally, our framework reduces the possibility of logical and syntactical errors. Toward evaluating our approach, we run several experiments on the PIE dataset using a CodeT5 language model and RRHF, a new reinforcement learning algorithm. We adopt a variety of evaluation metrics with regards to optimization quality, and speedup. The evaluation results demonstrate that the proposed framework has similar results in comparison with existing models using shorter training times and smaller pre-trained models. In particular, we accomplish an increase of 5.6% and 2.2 over the baseline models concerning the %OP T and SP metrics.

Reinforcement learning (RL) has emerged as a powerful tool for fine-tuning large language models (LLMs) to improve complex reasoning abilities. However, state-of-the-art policy optimization methods often suffer from high computational overhead and memory consumption, primarily due to the need for multiple generations per prompt and the reliance on critic networks or advantage estimates of the current policy. In this paper, we propose A*-PO, a novel two-stage policy optimization framework that directly approximates the optimal advantage function and enables efficient training of LLMs for reasoning tasks. In the first stage, we leverage offline sampling from a reference policy to estimate the optimal value function V*, eliminating the need for costly online value estimation. In the second stage, we perform on-policy updates using a simple least-squares regression loss with only a single generation per prompt. Theoretically, we establish performance guarantees and prove that the KL-regularized RL objective can be optimized without requiring complex exploration strategies. Empirically, A*-PO achieves competitive performance across a wide range of mathematical reasoning benchmarks, while reducing training time by up to 2times and peak memory usage by over 30% compared to PPO, GRPO, and REBEL. Implementation of A*-PO can be found at https://github.com/ZhaolinGao/A-PO.

Exploration bonuses in reinforcement learning guide long-horizon exploration by defining custom intrinsic objectives. Several exploration objectives like count-based bonuses, pseudo-counts, and state-entropy maximization are non-stationary and hence are difficult to optimize for the agent. While this issue is generally known, it is usually omitted and solutions remain under-explored. The key contribution of our work lies in transforming the original non-stationary rewards into stationary rewards through an augmented state representation. For this purpose, we introduce the Stationary Objectives For Exploration (SOFE) framework. SOFE requires identifying sufficient statistics for different exploration bonuses and finding an efficient encoding of these statistics to use as input to a deep network. SOFE is based on proposing state augmentations that expand the state space but hold the promise of simplifying the optimization of the agent's objective. We show that SOFE improves the performance of several exploration objectives, including count-based bonuses, pseudo-counts, and state-entropy maximization. Moreover, SOFE outperforms prior methods that attempt to stabilize the optimization of intrinsic objectives. We demonstrate the efficacy of SOFE in hard-exploration problems, including sparse-reward tasks, pixel-based observations, 3D navigation, and procedurally generated environments.

We study a K-armed bandit with delayed feedback and intermediate observations. We consider a model where intermediate observations have a form of a finite state, which is observed immediately after taking an action, whereas the loss is observed after an adversarially chosen delay. We show that the regime of the mapping of states to losses determines the complexity of the problem, irrespective of whether the mapping of actions to states is stochastic or adversarial. If the mapping of states to losses is adversarial, then the regret rate is of order (K+d)T (within log factors), where T is the time horizon and d is a fixed delay. This matches the regret rate of a K-armed bandit with delayed feedback and without intermediate observations, implying that intermediate observations are not helpful. However, if the mapping of states to losses is stochastic, we show that the regret grows at a rate of big(K+min{|mathcal{S|,d}big)T} (within log factors), implying that if the number |S| of states is smaller than the delay, then intermediate observations help. We also provide refined high-probability regret upper bounds for non-uniform delays, together with experimental validation of our algorithms.

In Reinforcement Learning (RL), regularization has emerged as a popular tool both in theory and practice, typically based either on an entropy bonus or a Kullback-Leibler divergence that constrains successive policies. In practice, these approaches have been shown to improve exploration, robustness and stability, giving rise to popular Deep RL algorithms such as SAC and TRPO. Policy Mirror Descent (PMD) is a theoretical framework that solves this general regularized policy optimization problem, however the closed-form solution involves the sum of all past Q-functions, which is intractable in practice. We propose and analyze PMD-like algorithms that only keep the last M Q-functions in memory, and show that for finite and large enough M, a convergent algorithm can be derived, introducing no error in the policy update, unlike prior deep RL PMD implementations. StaQ, the resulting algorithm, enjoys strong theoretical guarantees and is competitive with deep RL baselines, while exhibiting less performance oscillation, paving the way for fully stable deep RL algorithms and providing a testbed for experimentation with Policy Mirror Descent.

Self-play alignment algorithms have been developed as effective methods for fine-tuning large language models (LLMs), formulating preference optimization as a two-player game. However, the regularization with respect to the reference policy, which is crucial for mitigating over-optimization, has been insufficiently investigated in self-play alignment. In this paper, we show that our regularization method can improve the unregularized self-play significantly. To study the impact of different regularizations in self-play alignment, we propose Regularized Self-Play Policy Optimization (RSPO). This generalized framework regularizes the self-play by simply adding a chosen regularization term into the loss while maintaining provable last-iterate convergence to the Nash Equilibrium of the corresponding regularized game. Surprisingly, empirical evaluations using the Mistral-7B-Instruct base model reveal that forward KL divergence regularization reduces response length in RSPO, whereas reverse KL divergence markedly improves raw win rates. RSPO with a linear combination of forward and reverse KL divergence regularization substantially increases the length-controlled win rate in AlpacaEval-2, elevating the unregularized self-play alignment method (SPPO) from 28.53% to 35.44%. Finally, we show that RSPO also improves the response diversity.

Offline preference optimization is a key method for enhancing and controlling the quality of Large Language Model (LLM) outputs. Typically, preference optimization is approached as an offline supervised learning task using manually-crafted convex loss functions. While these methods are based on theoretical insights, they are inherently constrained by human creativity, so the large search space of possible loss functions remains under explored. We address this by performing LLM-driven objective discovery to automatically discover new state-of-the-art preference optimization algorithms without (expert) human intervention. Specifically, we iteratively prompt an LLM to propose and implement new preference optimization loss functions based on previously-evaluated performance metrics. This process leads to the discovery of previously-unknown and performant preference optimization algorithms. The best performing of these we call Discovered Preference Optimization (DiscoPOP), a novel algorithm that adaptively blends logistic and exponential losses. Experiments demonstrate the state-of-the-art performance of DiscoPOP and its successful transfer to held-out tasks.

Amid mounting concern about the reliability and credibility of machine learning research, we present a principled framework for making robust and generalizable claims: the multiverse analysis. Our framework builds upon the multiverse analysis (Steegen et al., 2016) introduced in response to psychology's own reproducibility crisis. To efficiently explore high-dimensional and often continuous ML search spaces, we model the multiverse with a Gaussian Process surrogate and apply Bayesian experimental design. Our framework is designed to facilitate drawing robust scientific conclusions about model performance, and thus our approach focuses on exploration rather than conventional optimization. In the first of two case studies, we investigate disputed claims about the relative merit of adaptive optimizers. Second, we synthesize conflicting research on the effect of learning rate on the large batch training generalization gap. For the machine learning community, the multiverse analysis is a simple and effective technique for identifying robust claims, for increasing transparency, and a step toward improved reproducibility.

We study offline multi-agent reinforcement learning (RL) in Markov games, where the goal is to learn an approximate equilibrium -- such as Nash equilibrium and (Coarse) Correlated Equilibrium -- from an offline dataset pre-collected from the game. Existing works consider relatively restricted tabular or linear models and handle each equilibria separately. In this work, we provide the first framework for sample-efficient offline learning in Markov games under general function approximation, handling all 3 equilibria in a unified manner. By using Bellman-consistent pessimism, we obtain interval estimation for policies' returns, and use both the upper and the lower bounds to obtain a relaxation on the gap of a candidate policy, which becomes our optimization objective. Our results generalize prior works and provide several additional insights. Importantly, we require a data coverage condition that improves over the recently proposed "unilateral concentrability". Our condition allows selective coverage of deviation policies that optimally trade-off between their greediness (as approximate best responses) and coverage, and we show scenarios where this leads to significantly better guarantees. As a new connection, we also show how our algorithmic framework can subsume seemingly different solution concepts designed for the special case of two-player zero-sum games.

Simulated annealing (SA) is a stochastic global optimisation technique applicable to a wide range of discrete and continuous variable problems. Despite its simplicity, the development of an effective SA optimiser for a given problem hinges on a handful of carefully handpicked components; namely, neighbour proposal distribution and temperature annealing schedule. In this work, we view SA from a reinforcement learning perspective and frame the proposal distribution as a policy, which can be optimised for higher solution quality given a fixed computational budget. We demonstrate that this Neural SA with such a learnt proposal distribution, parametrised by small equivariant neural networks, outperforms SA baselines on a number of problems: Rosenbrock's function, the Knapsack problem, the Bin Packing problem, and the Travelling Salesperson problem. We also show that Neural SA scales well to large problems - generalising to significantly larger problems than the ones seen during training - while achieving comparable performance to popular off-the-shelf solvers and other machine learning methods in terms of solution quality and wall-clock time.

Offline-to-online reinforcement learning (RL) is a training paradigm that combines pre-training on a pre-collected dataset with fine-tuning in an online environment. However, the incorporation of online fine-tuning can intensify the well-known distributional shift problem. Existing solutions tackle this problem by imposing a policy constraint on the policy improvement objective in both offline and online learning. They typically advocate a single balance between policy improvement and constraints across diverse data collections. This one-size-fits-all manner may not optimally leverage each collected sample due to the significant variation in data quality across different states. To this end, we introduce Family Offline-to-Online RL (FamO2O), a simple yet effective framework that empowers existing algorithms to determine state-adaptive improvement-constraint balances. FamO2O utilizes a universal model to train a family of policies with different improvement/constraint intensities, and a balance model to select a suitable policy for each state. Theoretically, we prove that state-adaptive balances are necessary for achieving a higher policy performance upper bound. Empirically, extensive experiments show that FamO2O offers a statistically significant improvement over various existing methods, achieving state-of-the-art performance on the D4RL benchmark. Codes are available at https://github.com/LeapLabTHU/FamO2O.

Having explored an environment, intelligent agents should be able to transfer their knowledge to most downstream tasks within that environment. Referred to as "zero-shot learning," this ability remains elusive for general-purpose reinforcement learning algorithms. While recent works have attempted to produce zero-shot RL agents, they make assumptions about the nature of the tasks or the structure of the MDP. We present Proto Successor Measure: the basis set for all possible solutions of Reinforcement Learning in a dynamical system. We provably show that any possible policy can be represented using an affine combination of these policy independent basis functions. Given a reward function at test time, we simply need to find the right set of linear weights to combine these basis corresponding to the optimal policy. We derive a practical algorithm to learn these basis functions using only interaction data from the environment and show that our approach can produce the optimal policy at test time for any given reward function without additional environmental interactions. Project page: https://agarwalsiddhant10.github.io/projects/psm.html.

In the post-training of large language models (LLMs), Reinforcement Learning from Human Feedback (RLHF) is an effective approach to achieve generation aligned with human preferences. Direct Preference Optimization (DPO) allows for policy training with a simple binary cross-entropy loss without a reward model. The objective of DPO is regularized by reverse KL divergence that encourages mode-seeking fitting to the reference policy. Nonetheless, we indicate that minimizing reverse KL divergence could fail to capture a mode of the reference distribution, which may hurt the policy's performance. Based on this observation, we propose a simple modification to DPO, H-DPO, which allows for control over the entropy of the resulting policy, enhancing the distribution's sharpness and thereby enabling mode-seeking fitting more effectively. In our experiments, we show that H-DPO outperformed DPO across various tasks, demonstrating superior results in pass@k evaluations for mathematical tasks. Moreover, H-DPO is simple to implement, requiring only minor modifications to the loss calculation of DPO, which makes it highly practical and promising for wide-ranging applications in the training of LLMs.

As first-order optimization methods become the method of choice for solving large-scale optimization problems, optimization solvers based on first-order algorithms are being built. Such general-purpose solvers must robustly detect infeasible or misspecified problem instances, but the computational complexity of first-order methods for doing so has yet to be formally studied. In this work, we characterize the optimal accelerated rate of infeasibility detection. We show that the standard fixed-point iteration achieves a O(1/k^2) and O(1/k) rates, respectively, on the normalized iterates and the fixed-point residual converging to the infimal displacement vector, while the accelerated fixed-point iteration achieves O(1/k^2) and mathcal{O}(1/k^2) rates. We then provide a matching complexity lower bound to establish that Theta(1/k^2) is indeed the optimal accelerated rate.

Multi-objective reinforcement learning (MORL) excels at handling rapidly changing preferences in tasks that involve multiple criteria, even for unseen preferences. However, previous dominating MORL methods typically generate a fixed policy set or preference-conditioned policy through multiple training iterations exclusively for sampled preference vectors, and cannot ensure the efficient discovery of the Pareto front. Furthermore, integrating preferences into the input of policy or value functions presents scalability challenges, in particular as the dimension of the state and preference space grow, which can complicate the learning process and hinder the algorithm's performance on more complex tasks. To address these issues, we propose a two-stage Pareto front discovery algorithm called Constrained MORL (C-MORL), which serves as a seamless bridge between constrained policy optimization and MORL. Concretely, a set of policies is trained in parallel in the initialization stage, with each optimized towards its individual preference over the multiple objectives. Then, to fill the remaining vacancies in the Pareto front, the constrained optimization steps are employed to maximize one objective while constraining the other objectives to exceed a predefined threshold. Empirically, compared to recent advancements in MORL methods, our algorithm achieves more consistent and superior performances in terms of hypervolume, expected utility, and sparsity on both discrete and continuous control tasks, especially with numerous objectives (up to nine objectives in our experiments).

In a recent work, Laforgue et al. introduce the model of last switch dependent (LSD) bandits, in an attempt to capture nonstationary phenomena induced by the interaction between the player and the environment. Examples include satiation, where consecutive plays of the same action lead to decreased performance, or deprivation, where the payoff of an action increases after an interval of inactivity. In this work, we take a step towards understanding the approximability of planning LSD bandits, namely, the (NP-hard) problem of computing an optimal arm-pulling strategy under complete knowledge of the model. In particular, we design the first efficient constant approximation algorithm for the problem and show that, under a natural monotonicity assumption on the payoffs, its approximation guarantee (almost) matches the state-of-the-art for the special and well-studied class of recharging bandits (also known as delay-dependent). In this attempt, we develop new tools and insights for this class of problems, including a novel higher-dimensional relaxation and the technique of mirroring the evolution of virtual states. We believe that these novel elements could potentially be used for approaching richer classes of action-induced nonstationary bandits (e.g., special instances of restless bandits). In the case where the model parameters are initially unknown, we develop an online learning adaptation of our algorithm for which we provide sublinear regret guarantees against its full-information counterpart.

We consider minimizing functions for which it is expensive to compute the (possibly stochastic) gradient. Such functions are prevalent in reinforcement learning, imitation learning and adversarial training. Our target optimization framework uses the (expensive) gradient computation to construct surrogate functions in a target space (e.g. the logits output by a linear model for classification) that can be minimized efficiently. This allows for multiple parameter updates to the model, amortizing the cost of gradient computation. In the full-batch setting, we prove that our surrogate is a global upper-bound on the loss, and can be (locally) minimized using a black-box optimization algorithm. We prove that the resulting majorization-minimization algorithm ensures convergence to a stationary point of the loss. Next, we instantiate our framework in the stochastic setting and propose the SSO algorithm, which can be viewed as projected stochastic gradient descent in the target space. This connection enables us to prove theoretical guarantees for SSO when minimizing convex functions. Our framework allows the use of standard stochastic optimization algorithms to construct surrogates which can be minimized by any deterministic optimization method. To evaluate our framework, we consider a suite of supervised learning and imitation learning problems. Our experiments indicate the benefits of target optimization and the effectiveness of SSO.

Optimization in machine learning, both theoretical and applied, is presently dominated by first-order gradient methods such as stochastic gradient descent. Second-order optimization methods, that involve second derivatives and/or second order statistics of the data, are far less prevalent despite strong theoretical properties, due to their prohibitive computation, memory and communication costs. In an attempt to bridge this gap between theoretical and practical optimization, we present a scalable implementation of a second-order preconditioned method (concretely, a variant of full-matrix Adagrad), that along with several critical algorithmic and numerical improvements, provides significant convergence and wall-clock time improvements compared to conventional first-order methods on state-of-the-art deep models. Our novel design effectively utilizes the prevalent heterogeneous hardware architecture for training deep models, consisting of a multicore CPU coupled with multiple accelerator units. We demonstrate superior performance compared to state-of-the-art on very large learning tasks such as machine translation with Transformers, language modeling with BERT, click-through rate prediction on Criteo, and image classification on ImageNet with ResNet-50.

In this paper, we investigate the long-term memory learning capabilities of state-space models (SSMs) from the perspective of parameterization. We prove that state-space models without any reparameterization exhibit a memory limitation similar to that of traditional RNNs: the target relationships that can be stably approximated by state-space models must have an exponential decaying memory. Our analysis identifies this "curse of memory" as a result of the recurrent weights converging to a stability boundary, suggesting that a reparameterization technique can be effective. To this end, we introduce a class of reparameterization techniques for SSMs that effectively lift its memory limitations. Besides improving approximation capabilities, we further illustrate that a principled choice of reparameterization scheme can also enhance optimization stability. We validate our findings using synthetic datasets and language models.

Given a stationary state-space model that relates a sequence of hidden states and corresponding measurements or observations, Bayesian filtering provides a principled statistical framework for inferring the posterior distribution of the current state given all measurements up to the present time. For example, the Apollo lunar module implemented a Kalman filter to infer its location from a sequence of earth-based radar measurements and land safely on the moon. To perform Bayesian filtering, we require a measurement model that describes the conditional distribution of each observation given state. The Kalman filter takes this measurement model to be linear, Gaussian. Here we show how a nonlinear, Gaussian approximation to the distribution of state given observation can be used in conjunction with Bayes' rule to build a nonlinear, non-Gaussian measurement model. The resulting approach, called the Discriminative Kalman Filter (DKF), retains fast closed-form updates for the posterior. We argue there are many cases where the distribution of state given measurement is better-approximated as Gaussian, especially when the dimensionality of measurements far exceeds that of states and the Bernstein-von Mises theorem applies. Online neural decoding for brain-computer interfaces provides a motivating example, where filtering incorporates increasingly detailed measurements of neural activity to provide users control over external devices. Within the BrainGate2 clinical trial, the DKF successfully enabled three volunteers with quadriplegia to control an on-screen cursor in real-time using mental imagery alone. Participant "T9" used the DKF to type out messages on a tablet PC.

Before the deep learning revolution, many perception algorithms were based on runtime optimization in conjunction with a strong prior/regularization penalty. A prime example of this in computer vision is optical and scene flow. Supervised learning has largely displaced the need for explicit regularization. Instead, they rely on large amounts of labeled data to capture prior statistics, which are not always readily available for many problems. Although optimization is employed to learn the neural network, the weights of this network are frozen at runtime. As a result, these learning solutions are domain-specific and do not generalize well to other statistically different scenarios. This paper revisits the scene flow problem that relies predominantly on runtime optimization and strong regularization. A central innovation here is the inclusion of a neural scene flow prior, which uses the architecture of neural networks as a new type of implicit regularizer. Unlike learning-based scene flow methods, optimization occurs at runtime, and our approach needs no offline datasets -- making it ideal for deployment in new environments such as autonomous driving. We show that an architecture based exclusively on multilayer perceptrons (MLPs) can be used as a scene flow prior. Our method attains competitive -- if not better -- results on scene flow benchmarks. Also, our neural prior's implicit and continuous scene flow representation allows us to estimate dense long-term correspondences across a sequence of point clouds. The dense motion information is represented by scene flow fields where points can be propagated through time by integrating motion vectors. We demonstrate such a capability by accumulating a sequence of lidar point clouds.

In this study, we investigate the DIstribution Correction Estimation (DICE) methods, an important line of work in offline reinforcement learning (RL) and imitation learning (IL). DICE-based methods impose state-action-level behavior constraint, which is an ideal choice for offline learning. However, they typically perform much worse than current state-of-the-art (SOTA) methods that solely use action-level behavior constraint. After revisiting DICE-based methods, we find there exist two gradient terms when learning the value function using true-gradient update: forward gradient (taken on the current state) and backward gradient (taken on the next state). Using forward gradient bears a large similarity to many offline RL methods, and thus can be regarded as applying action-level constraint. However, directly adding the backward gradient may degenerate or cancel out its effect if these two gradients have conflicting directions. To resolve this issue, we propose a simple yet effective modification that projects the backward gradient onto the normal plane of the forward gradient, resulting in an orthogonal-gradient update, a new learning rule for DICE-based methods. We conduct thorough theoretical analyses and find that the projected backward gradient brings state-level behavior regularization, which reveals the mystery of DICE-based methods: the value learning objective does try to impose state-action-level constraint, but needs to be used in a corrected way. Through toy examples and extensive experiments on complex offline RL and IL tasks, we demonstrate that DICE-based methods using orthogonal-gradient updates (O-DICE) achieve SOTA performance and great robustness.

Quantum algorithms for probing ground-state properties of quantum systems require good initial states. Projection-based methods such as eigenvalue filtering rely on inputs that have a significant overlap with the low-energy subspace, which can be challenging for large, strongly-correlated systems. This issue has motivated the study of physically-inspired dynamical approaches such as thermodynamic cooling. In this work, we introduce a ground-state preparation algorithm based on the simulation of quantum dynamics. Our main insight is to transform the Hamiltonian by a shifted sign function via quantum signal processing, effectively mapping eigenvalues into positive and negative subspaces separated by a large gap. This automatically ensures that all states within each subspace conserve energy with respect to the transformed Hamiltonian. Subsequent time-evolution with a perturbed Hamiltonian induces transitions to lower-energy states while preventing unwanted jumps to higher energy states. The approach does not rely on a priori knowledge of energy gaps and requires no additional qubits to model a bath. Furthermore, it makes mathcal{O}(d^{,3/2}/epsilon) queries to the time-evolution operator of the system and mathcal{O}(d^{,3/2}) queries to a block-encoding of the perturbation, for d cooling steps and an epsilon-accurate energy resolution. Our results provide a framework for combining quantum signal processing and Hamiltonian simulation to design heuristic quantum algorithms for ground-state preparation.

Offline reinforcement learning enables agents to leverage large pre-collected datasets of environment transitions to learn control policies, circumventing the need for potentially expensive or unsafe online data collection. Significant progress has been made recently in offline model-based reinforcement learning, approaches which leverage a learned dynamics model. This typically involves constructing a probabilistic model, and using the model uncertainty to penalize rewards where there is insufficient data, solving for a pessimistic MDP that lower bounds the true MDP. Existing methods, however, exhibit a breakdown between theory and practice, whereby pessimistic return ought to be bounded by the total variation distance of the model from the true dynamics, but is instead implemented through a penalty based on estimated model uncertainty. This has spawned a variety of uncertainty heuristics, with little to no comparison between differing approaches. In this paper, we compare these heuristics, and design novel protocols to investigate their interaction with other hyperparameters, such as the number of models, or imaginary rollout horizon. Using these insights, we show that selecting these key hyperparameters using Bayesian Optimization produces superior configurations that are vastly different to those currently used in existing hand-tuned state-of-the-art methods, and result in drastically stronger performance.

A common pipeline in learning-based control is to iteratively estimate a model of system dynamics, and apply a trajectory optimization algorithm - e.g.~iLQR - on the learned model to minimize a target cost. This paper conducts a rigorous analysis of a simplified variant of this strategy for general nonlinear systems. We analyze an algorithm which iterates between estimating local linear models of nonlinear system dynamics and performing iLQR-like policy updates. We demonstrate that this algorithm attains sample complexity polynomial in relevant problem parameters, and, by synthesizing locally stabilizing gains, overcomes exponential dependence in problem horizon. Experimental results validate the performance of our algorithm, and compare to natural deep-learning baselines.

Policy optimization methods with function approximation are widely used in multi-agent reinforcement learning. However, it remains elusive how to design such algorithms with statistical guarantees. Leveraging a multi-agent performance difference lemma that characterizes the landscape of multi-agent policy optimization, we find that the localized action value function serves as an ideal descent direction for each local policy. Motivated by the observation, we present a multi-agent PPO algorithm in which the local policy of each agent is updated similarly to vanilla PPO. We prove that with standard regularity conditions on the Markov game and problem-dependent quantities, our algorithm converges to the globally optimal policy at a sublinear rate. We extend our algorithm to the off-policy setting and introduce pessimism to policy evaluation, which aligns with experiments. To our knowledge, this is the first provably convergent multi-agent PPO algorithm in cooperative Markov games.

State Space Models (SSMs) have emerged as efficient alternatives to Transformers, mitigating their quadratic computational cost. However, the application of Parameter-Efficient Fine-Tuning (PEFT) methods to SSMs remains largely unexplored. In particular, prompt-based methods like Prompt Tuning and Prefix-Tuning, which are widely used in Transformers, do not perform well on SSMs. To address this, we propose state-based methods as a superior alternative to prompt-based methods. This new family of methods naturally stems from the architectural characteristics of SSMs. State-based methods adjust state-related features directly instead of depending on external prompts. Furthermore, we introduce a novel state-based PEFT method: State-offset Tuning. At every timestep, our method directly affects the state at the current step, leading to more effective adaptation. Through extensive experiments across diverse datasets, we demonstrate the effectiveness of our method. Code is available at https://github.com/furiosa-ai/ssm-state-tuning.

POMDPs capture a broad class of decision making problems, but hardness results suggest that learning is intractable even in simple settings due to the inherent partial observability. However, in many realistic problems, more information is either revealed or can be computed during some point of the learning process. Motivated by diverse applications ranging from robotics to data center scheduling, we formulate a Hindsight Observable Markov Decision Process (HOMDP) as a POMDP where the latent states are revealed to the learner in hindsight and only during training. We introduce new algorithms for the tabular and function approximation settings that are provably sample-efficient with hindsight observability, even in POMDPs that would otherwise be statistically intractable. We give a lower bound showing that the tabular algorithm is optimal in its dependence on latent state and observation cardinalities.

We propose a reinforcement learning (RL) framework under a broad class of risk objectives, characterized by convex scoring functions. This class covers many common risk measures, such as variance, Expected Shortfall, entropic Value-at-Risk, and mean-risk utility. To resolve the time-inconsistency issue, we consider an augmented state space and an auxiliary variable and recast the problem as a two-state optimization problem. We propose a customized Actor-Critic algorithm and establish some theoretical approximation guarantees. A key theoretical contribution is that our results do not require the Markov decision process to be continuous. Additionally, we propose an auxiliary variable sampling method inspired by the alternating minimization algorithm, which is convergent under certain conditions. We validate our approach in simulation experiments with a financial application in statistical arbitrage trading, demonstrating the effectiveness of the algorithm.

We study reinforcement learning with function approximation for large-scale Partially Observable Markov Decision Processes (POMDPs) where the state space and observation space are large or even continuous. Particularly, we consider Hilbert space embeddings of POMDP where the feature of latent states and the feature of observations admit a conditional Hilbert space embedding of the observation emission process, and the latent state transition is deterministic. Under the function approximation setup where the optimal latent state-action Q-function is linear in the state feature, and the optimal Q-function has a gap in actions, we provide a computationally and statistically efficient algorithm for finding the exact optimal policy. We show our algorithm's computational and statistical complexities scale polynomially with respect to the horizon and the intrinsic dimension of the feature on the observation space. Furthermore, we show both the deterministic latent transitions and gap assumptions are necessary to avoid statistical complexity exponential in horizon or dimension. Since our guarantee does not have an explicit dependence on the size of the state and observation spaces, our algorithm provably scales to large-scale POMDPs.

Preconditioned gradient methods are among the most general and powerful tools in optimization. However, preconditioning requires storing and manipulating prohibitively large matrices. We describe and analyze a new structure-aware preconditioning algorithm, called Shampoo, for stochastic optimization over tensor spaces. Shampoo maintains a set of preconditioning matrices, each of which operates on a single dimension, contracting over the remaining dimensions. We establish convergence guarantees in the stochastic convex setting, the proof of which builds upon matrix trace inequalities. Our experiments with state-of-the-art deep learning models show that Shampoo is capable of converging considerably faster than commonly used optimizers. Although it involves a more complex update rule, Shampoo's runtime per step is comparable to that of simple gradient methods such as SGD, AdaGrad, and Adam.

Automatic program repair seeks to generate correct code from buggy programs, with most approaches searching the correct program in a discrete, symbolic space of source code tokens. This symbolic search is fundamentally limited by its inability to directly reason about program behavior. We introduce Gradient-Based Program Repair (GBPR), a new paradigm that reframes program repair as continuous optimization in a differentiable numerical program space. Our core insight is to compile symbolic programs into differentiable numerical representations, enabling search in the numerical program space directly guided by program behavior. To evaluate GBPR, we present RaspBugs, a new benchmark of 1,466 buggy symbolic RASP programs and their respective numerical representations. Our experiments demonstrate that GBPR can effectively repair buggy symbolic programs by gradient-based optimization in the numerical program space, with convincing repair trajectories. To our knowledge, we are the first to state program repair as continuous optimization in a numerical program space. Our work establishes a new direction for program repair research, bridging two rich worlds: continuous optimization and program behavior.

The Koopman operator provides a powerful framework for modeling dynamical systems and has attracted growing interest from the machine learning community. However, its infinite-dimensional nature makes identifying suitable finite-dimensional subspaces challenging, especially for deep architectures. We argue that these difficulties come from suboptimal representation learning, where latent variables fail to balance expressivity and simplicity. This tension is closely related to the information bottleneck (IB) dilemma: constructing compressed representations that are both compact and predictive. Rethinking Koopman learning through this lens, we demonstrate that latent mutual information promotes simplicity, yet an overemphasis on simplicity may cause latent space to collapse onto a few dominant modes. In contrast, expressiveness is sustained by the von Neumann entropy, which prevents such collapse and encourages mode diversity. This insight leads us to propose an information-theoretic Lagrangian formulation that explicitly balances this tradeoff. Furthermore, we propose a new algorithm based on the Lagrangian formulation that encourages both simplicity and expressiveness, leading to a stable and interpretable Koopman representation. Beyond quantitative evaluations, we further visualize the learned manifolds under our representations, observing empirical results consistent with our theoretical predictions. Finally, we validate our approach across a diverse range of dynamical systems, demonstrating improved performance over existing Koopman learning methods. The implementation is publicly available at https://github.com/Wenxuan52/InformationKoopman.

Direct Preference Optimization (DPO) simplifies reinforcement learning from human feedback (RLHF) for large language models (LLMs) by directly optimizing human preferences without an explicit reward model. We find that during DPO training, the reference model plays the role of a data weight adjuster. However, the common practice of initializing the policy and reference models identically in DPO can lead to inefficient data utilization and impose a performance ceiling. Meanwhile, the lack of a reference model in Simple Preference Optimization (SimPO) reduces training robustness and necessitates stricter conditions to prevent catastrophic forgetting. In this work, we propose Pre-DPO, a simple yet effective DPO-based training paradigm that enhances preference optimization performance by leveraging a guiding reference model. This reference model provides foresight into the optimal policy state achievable through the training preference data, serving as a guiding mechanism that adaptively assigns higher weights to samples more suitable for the model and lower weights to those less suitable. Extensive experiments on AlpacaEval 2.0 and Arena-Hard v0.1 benchmarks demonstrate that Pre-DPO consistently improves the performance of both DPO and SimPO, without relying on external models or additional data.

With the widespread adoption of machine learning systems, the need to curtail their behavior has become increasingly apparent. This is evidenced by recent advancements towards developing models that satisfy robustness, safety, and fairness requirements. These requirements can be imposed (with generalization guarantees) by formulating constrained learning problems that can then be tackled by dual ascent algorithms. Yet, though these algorithms converge in objective value, even in non-convex settings, they cannot guarantee that their outcome is feasible. Doing so requires randomizing over all iterates, which is impractical in virtually any modern applications. Still, final iterates have been observed to perform well in practice. In this work, we address this gap between theory and practice by characterizing the constraint violation of Lagrangian minimizers associated with optimal dual variables, despite lack of convexity. To do this, we leverage the fact that non-convex, finite-dimensional constrained learning problems can be seen as parametrizations of convex, functional problems. Our results show that rich parametrizations effectively mitigate the issue of feasibility in dual methods, shedding light on prior empirical successes of dual learning. We illustrate our findings in fair learning tasks.

Reinforcement learning from human feedback (RLHF) is a standard approach for fine-tuning large language models to follow instructions. As part of this process, learned reward models are used to approximately model human preferences. However, as imperfect representations of the "true" reward, these learned reward models are susceptible to overoptimization. Gao et al. (2023) studied this phenomenon in a synthetic human feedback setup with a significantly larger "gold" reward model acting as the true reward (instead of humans) and showed that overoptimization remains a persistent problem regardless of the size of the proxy reward model and training data used. Using a similar setup, we conduct a systematic study to evaluate the efficacy of using ensemble-based conservative optimization objectives, specifically worst-case optimization (WCO) and uncertainty-weighted optimization (UWO), for mitigating reward model overoptimization when using two optimization methods: (a) best-of-n sampling (BoN) (b) proximal policy optimization (PPO). We additionally extend the setup of Gao et al. (2023) to include 25% label noise to better mirror real-world conditions. Both with and without label noise, we find that conservative optimization practically eliminates overoptimization and improves performance by up to 70% for BoN sampling. For PPO, ensemble-based conservative optimization always reduces overoptimization and outperforms single reward model optimization. Moreover, combining it with a small KL penalty successfully prevents overoptimization at no performance cost. Overall, our results demonstrate that ensemble-based conservative optimization can effectively counter overoptimization.

Generalization in Reinforcement Learning (RL) aims to learn an agent during training that generalizes to the target environment. This paper studies RL generalization from a theoretical aspect: how much can we expect pre-training over training environments to be helpful? When the interaction with the target environment is not allowed, we certify that the best we can obtain is a near-optimal policy in an average sense, and we design an algorithm that achieves this goal. Furthermore, when the agent is allowed to interact with the target environment, we give a surprising result showing that asymptotically, the improvement from pre-training is at most a constant factor. On the other hand, in the non-asymptotic regime, we design an efficient algorithm and prove a distribution-based regret bound in the target environment that is independent of the state-action space.

The Quantum Approximate Optimization Algorithm (QAOA) is a prominent variational algorithm used for solving combinatorial optimization problems such as the Max-Cut problem. A key challenge in QAOA lies in efficiently identifying suitable parameters (gamma, beta) that lead to high-quality solutions. In this paper, we propose a framework that combines Fully Informed Particle Swarm Optimization (FIPSO) with adaptive gradient correction using the Adam Optimizer to navigate the QAOA parameter space. This approach aims to avoid issues such as barren plateaus and convergence to local minima. The proposed algorithm is evaluated against two classes of graph instances, Erdos Renyi and Watts-Strogatz. Experimental results across multiple QAOA depths consistently demonstrate superior performance compared to random initialization, underscoring the effectiveness and robustness of the proposed optimization framework.

In nonstationary bandit learning problems, the decision-maker must continually gather information and adapt their action selection as the latent state of the environment evolves. In each time period, some latent optimal action maximizes expected reward under the environment state. We view the optimal action sequence as a stochastic process, and take an information-theoretic approach to analyze attainable performance. We bound limiting per-period regret in terms of the entropy rate of the optimal action process. The bound applies to a wide array of problems studied in the literature and reflects the problem's information structure through its information-ratio.

Bayesian optimization is an approach to optimizing objective functions that take a long time (minutes or hours) to evaluate. It is best-suited for optimization over continuous domains of less than 20 dimensions, and tolerates stochastic noise in function evaluations. It builds a surrogate for the objective and quantifies the uncertainty in that surrogate using a Bayesian machine learning technique, Gaussian process regression, and then uses an acquisition function defined from this surrogate to decide where to sample. In this tutorial, we describe how Bayesian optimization works, including Gaussian process regression and three common acquisition functions: expected improvement, entropy search, and knowledge gradient. We then discuss more advanced techniques, including running multiple function evaluations in parallel, multi-fidelity and multi-information source optimization, expensive-to-evaluate constraints, random environmental conditions, multi-task Bayesian optimization, and the inclusion of derivative information. We conclude with a discussion of Bayesian optimization software and future research directions in the field. Within our tutorial material we provide a generalization of expected improvement to noisy evaluations, beyond the noise-free setting where it is more commonly applied. This generalization is justified by a formal decision-theoretic argument, standing in contrast to previous ad hoc modifications.

In meta reinforcement learning (meta RL), an agent seeks a Bayes-optimal policy -- the optimal policy when facing an unknown task that is sampled from some known task distribution. Previous approaches tackled this problem by inferring a belief over task parameters, using variational inference methods. Motivated by recent successes of contrastive learning approaches in RL, such as contrastive predictive coding (CPC), we investigate whether contrastive methods can be used for learning Bayes-optimal behavior. We begin by proving that representations learned by CPC are indeed sufficient for Bayes optimality. Based on this observation, we propose a simple meta RL algorithm that uses CPC in lieu of variational belief inference. Our method, ContraBAR, achieves comparable performance to state-of-the-art in domains with state-based observation and circumvents the computational toll of future observation reconstruction, enabling learning in domains with image-based observations. It can also be combined with image augmentations for domain randomization and used seamlessly in both online and offline meta RL settings.