Daily Papers

Direct Preference Optimization (DPO) has emerged as a prominent algorithm for the direct and robust alignment of Large Language Models (LLMs) with human preferences, offering a more straightforward alternative to the complex Reinforcement Learning from Human Feedback (RLHF). Despite its promising efficacy, DPO faces a notable drawback: "verbosity", a common over-optimization phenomenon also observed in RLHF. While previous studies mainly attributed verbosity to biased labels within the data, we propose that the issue also stems from an inherent algorithmic length reliance in DPO. Specifically, we suggest that the discrepancy between sequence-level Kullback-Leibler (KL) divergences between chosen and rejected sequences, used in DPO, results in overestimated or underestimated rewards due to varying token lengths. Empirically, we utilize datasets with different label lengths to demonstrate the presence of biased rewards. We then introduce an effective downsampling approach, named SamPO, to eliminate potential length reliance. Our experimental evaluations, conducted across three LLMs of varying scales and a diverse array of conditional and open-ended benchmarks, highlight the efficacy of SamPO in mitigating verbosity, achieving improvements of 5% to 12% over DPO through debaised rewards. Our codes can be accessed at: https://github.com/LuJunru/SamPO/.

Knowledge distillation (KD) has gained a lot of attention in the field of model compression for edge devices thanks to its effectiveness in compressing large powerful networks into smaller lower-capacity models. Online distillation, in which both the teacher and the student are learning collaboratively, has also gained much interest due to its ability to improve on the performance of the networks involved. The Kullback-Leibler (KL) divergence ensures the proper knowledge transfer between the teacher and student. However, most online KD techniques present some bottlenecks under the network capacity gap. By cooperatively and simultaneously training, the models the KL distance becomes incapable of properly minimizing the teacher's and student's distributions. Alongside accuracy, critical edge device applications are in need of well-calibrated compact networks. Confidence calibration provides a sensible way of getting trustworthy predictions. We propose BD-KD: Balancing of Divergences for online Knowledge Distillation. We show that adaptively balancing between the reverse and forward divergences shifts the focus of the training strategy to the compact student network without limiting the teacher network's learning process. We demonstrate that, by performing this balancing design at the level of the student distillation loss, we improve upon both performance accuracy and calibration of the compact student network. We conducted extensive experiments using a variety of network architectures and show improvements on multiple datasets including CIFAR-10, CIFAR-100, Tiny-ImageNet, and ImageNet. We illustrate the effectiveness of our approach through comprehensive comparisons and ablations with current state-of-the-art online and offline KD techniques.

Policy gradient algorithms have been successfully applied to enhance the reasoning capabilities of large language models (LLMs). Despite the widespread use of Kullback-Leibler (KL) regularization in policy gradient algorithms to stabilize training, the systematic exploration of how different KL divergence formulations can be estimated and integrated into surrogate loss functions for online reinforcement learning (RL) presents a nuanced and systematically explorable design space. In this paper, we propose regularized policy gradient (RPG), a systematic framework for deriving and analyzing KL-regularized policy gradient methods in the online RL setting. We derive policy gradients and corresponding surrogate loss functions for objectives regularized by both forward and reverse KL divergences, considering both normalized and unnormalized policy distributions. Furthermore, we present derivations for fully differentiable loss functions as well as REINFORCE-style gradient estimators, accommodating diverse algorithmic needs. We conduct extensive experiments on RL for LLM reasoning using these methods, showing improved or competitive results in terms of training stability and performance compared to strong baselines such as GRPO, REINFORCE++, and DAPO. The code is available at https://github.com/complex-reasoning/RPG.

Sampling from diffusion models involves a slow iterative process that hinders their practical deployment, especially for interactive applications. To accelerate generation speed, recent approaches distill a multi-step diffusion model into a single-step student generator via variational score distillation, which matches the distribution of samples generated by the student to the teacher's distribution. However, these approaches use the reverse Kullback-Leibler (KL) divergence for distribution matching which is known to be mode seeking. In this paper, we generalize the distribution matching approach using a novel f-divergence minimization framework, termed f-distill, that covers different divergences with different trade-offs in terms of mode coverage and training variance. We derive the gradient of the f-divergence between the teacher and student distributions and show that it is expressed as the product of their score differences and a weighting function determined by their density ratio. This weighting function naturally emphasizes samples with higher density in the teacher distribution, when using a less mode-seeking divergence. We observe that the popular variational score distillation approach using the reverse-KL divergence is a special case within our framework. Empirically, we demonstrate that alternative f-divergences, such as forward-KL and Jensen-Shannon divergences, outperform the current best variational score distillation methods across image generation tasks. In particular, when using Jensen-Shannon divergence, f-distill achieves current state-of-the-art one-step generation performance on ImageNet64 and zero-shot text-to-image generation on MS-COCO. Project page: https://research.nvidia.com/labs/genair/f-distill

The information bottleneck (IB) approach is popular to improve the generalization, robustness and explainability of deep neural networks. Essentially, it aims to find a minimum sufficient representation t by striking a trade-off between a compression term I(x;t) and a prediction term I(y;t), where I(cdot;cdot) refers to the mutual information (MI). MI is for the IB for the most part expressed in terms of the Kullback-Leibler (KL) divergence, which in the regression case corresponds to prediction based on mean squared error (MSE) loss with Gaussian assumption and compression approximated by variational inference. In this paper, we study the IB principle for the regression problem and develop a new way to parameterize the IB with deep neural networks by exploiting favorable properties of the Cauchy-Schwarz (CS) divergence. By doing so, we move away from MSE-based regression and ease estimation by avoiding variational approximations or distributional assumptions. We investigate the improved generalization ability of our proposed CS-IB and demonstrate strong adversarial robustness guarantees. We demonstrate its superior performance on six real-world regression tasks over other popular deep IB approaches. We additionally observe that the solutions discovered by CS-IB always achieve the best trade-off between prediction accuracy and compression ratio in the information plane. The code is available at https://github.com/SJYuCNEL/Cauchy-Schwarz-Information-Bottleneck.

Reinforcement learning (RL) is frequently employed in fine-tuning large language models (LMs), such as GPT-3, to penalize them for undesirable features of generated sequences, such as offensiveness, social bias, harmfulness or falsehood. The RL formulation involves treating the LM as a policy and updating it to maximise the expected value of a reward function which captures human preferences, such as non-offensiveness. In this paper, we analyze challenges associated with treating a language model as an RL policy and show how avoiding those challenges requires moving beyond the RL paradigm. We start by observing that the standard RL approach is flawed as an objective for fine-tuning LMs because it leads to distribution collapse: turning the LM into a degenerate distribution. Then, we analyze KL-regularised RL, a widely used recipe for fine-tuning LMs, which additionally constrains the fine-tuned LM to stay close to its original distribution in terms of Kullback-Leibler (KL) divergence. We show that KL-regularised RL is equivalent to variational inference: approximating a Bayesian posterior which specifies how to update a prior LM to conform with evidence provided by the reward function. We argue that this Bayesian inference view of KL-regularised RL is more insightful than the typically employed RL perspective. The Bayesian inference view explains how KL-regularised RL avoids the distribution collapse problem and offers a first-principles derivation for its objective. While this objective happens to be equivalent to RL (with a particular choice of parametric reward), there exist other objectives for fine-tuning LMs which are no longer equivalent to RL. That observation leads to a more general point: RL is not an adequate formal framework for problems such as fine-tuning language models. These problems are best viewed as Bayesian inference: approximating a pre-defined target distribution.

Multimodal Large Language Models (MLLMs) excel in solving text-based mathematical problems, but they struggle with mathematical diagrams since they are primarily trained on natural scene images. For humans, visual aids generally enhance problem-solving, but MLLMs perform worse as information shifts from textual to visual modality. This decline is mainly due to their shortcomings in aligning images and text. To tackle aforementioned challenges, we propose Math-PUMA, a methodology focused on Progressive Upward Multimodal Alignment. This approach is designed to improve the mathematical reasoning skills of MLLMs through a three-stage training process, with the second stage being the critical alignment stage. We first enhance the language model's mathematical reasoning capabilities with extensive set of textual mathematical problems. We then construct a multimodal dataset with varying degrees of textual and visual information, creating data pairs by presenting each problem in at least two forms. By leveraging the Kullback-Leibler (KL) divergence of next-token prediction distributions to align visual and textual modalities, consistent problem-solving abilities are ensured. Finally, we utilize multimodal instruction tuning for MLLMs with high-quality multimodal data. Experimental results on multiple mathematical reasoning benchmarks demonstrate that the MLLMs trained with Math-PUMA surpass most open-source MLLMs. Our approach effectively narrows the performance gap for problems presented in different modalities. The code and data are available at: https://github.com/wwzhuang01/Math-PUMA.

Recent advances in image-text pretraining have significantly enhanced visual understanding by aligning visual and textual representations. Contrastive Language-Image Pretraining (CLIP) has played a pivotal role in multimodal learning. However, its focus on single-label, single-granularity alignment limits its effectiveness in complex domains such as medical imaging, where images often correspond to multiple high-level labels (e.g., disease categories) across different annotation granularities (e.g., diagnostic description, clinical explanation). To address this, we propose Multi-Granular Language Learning (MGLL), a contrastive learning framework designed to improve both multi-label and cross-granularity alignment. MGLL leverages structured multi-label supervision, integrates textual descriptions across granularities, and introduces soft-label supervision with point-wise constraints to enhance alignment. MGLL employs smooth Kullback-Leibler (KL) divergence to ensure cross-granularity consistency while maintaining computational efficiency as a plug-and-play module for vision-language models. Pretrained on our constructed large-scale multi-granular datasets and evaluated across multiple datasets, MGLL outperforms other state-of-the-art methods in downstream tasks. The code is available at https://github.com/HUANGLIZI/MGLL{https://github.com/HUANGLIZI/MGLL}.

In Large Language Model (LLM) development, Reinforcement Learning from Human Feedback (RLHF) is crucial for aligning models with human values and preferences. RLHF traditionally relies on the Kullback-Leibler (KL) divergence between the current policy and a frozen initial policy as a reference, which is added as a penalty in policy optimization algorithms like Proximal Policy Optimization (PPO). While this constraint prevents models from deviating too far from the initial checkpoint, it limits exploration of the reward landscape, reducing the model's ability to discover higher-quality solutions. As a result, policy optimization is often trapped in a narrow region of the parameter space, leading to suboptimal alignment and performance. This paper presents SALSA (Soup-based Alignment Learning for Stronger Adaptation), a novel approach designed to overcome these limitations by creating a more flexible and better located reference model through weight-space averaging of two independent supervised fine-tuned (SFT) models. This model soup allows for larger deviation in KL divergence and exploring a promising region of the solution space without sacrificing stability. By leveraging this more robust reference model, SALSA fosters better exploration, achieving higher rewards and improving model robustness, out-of-distribution generalization, and performance. We validate the effectiveness of SALSA through extensive experiments on popular open models (Llama2-7B, Mistral-7B, and Gemma-2B) across various benchmarks (MT-Bench, Arena-Hard, UltraFeedback), where it consistently surpasses PPO by fostering deeper exploration and achieving superior alignment in LLMs.

While large language models demonstrate remarkable capabilities, they often present challenges in terms of safety, alignment with human values, and stability during training. Here, we focus on two prevalent methods used to align these models, Supervised Fine-Tuning (SFT) and Reinforcement Learning from Human Feedback (RLHF). SFT is simple and robust, powering a host of open-source models, while RLHF is a more sophisticated method used in top-tier models like ChatGPT but also suffers from instability and susceptibility to reward hacking. We propose a novel approach, Supervised Iterative Learning from Human Feedback (SuperHF), which seeks to leverage the strengths of both methods. Our hypothesis is two-fold: that the reward model used in RLHF is critical for efficient data use and model generalization and that the use of Proximal Policy Optimization (PPO) in RLHF may not be necessary and could contribute to instability issues. SuperHF replaces PPO with a simple supervised loss and a Kullback-Leibler (KL) divergence prior. It creates its own training data by repeatedly sampling a batch of model outputs and filtering them through the reward model in an online learning regime. We then break down the reward optimization problem into three components: robustly optimizing the training rewards themselves, preventing reward hacking-exploitation of the reward model that degrades model performance-as measured by a novel METEOR similarity metric, and maintaining good performance on downstream evaluations. Our experimental results show SuperHF exceeds PPO-based RLHF on the training objective, easily and favorably trades off high reward with low reward hacking, improves downstream calibration, and performs the same on our GPT-4 based qualitative evaluation scheme all the while being significantly simpler to implement, highlighting SuperHF's potential as a competitive language model alignment technique.

Despite the widespread empirical success of ResNet, the generalization properties of deep ResNet are rarely explored beyond the lazy training regime. In this work, we investigate scaled ResNet in the limit of infinitely deep and wide neural networks, of which the gradient flow is described by a partial differential equation in the large-neural network limit, i.e., the mean-field regime. To derive the generalization bounds under this setting, our analysis necessitates a shift from the conventional time-invariant Gram matrix employed in the lazy training regime to a time-variant, distribution-dependent version. To this end, we provide a global lower bound on the minimum eigenvalue of the Gram matrix under the mean-field regime. Besides, for the traceability of the dynamic of Kullback-Leibler (KL) divergence, we establish the linear convergence of the empirical error and estimate the upper bound of the KL divergence over parameters distribution. Finally, we build the uniform convergence for generalization bound via Rademacher complexity. Our results offer new insights into the generalization ability of deep ResNet beyond the lazy training regime and contribute to advancing the understanding of the fundamental properties of deep neural networks.

The Variational Autoencoder (VAE) is a seminal approach in deep generative modeling with latent variables. Interpreting its reconstruction process as a nonlinear transformation of samples from the latent posterior distribution, we apply the Unscented Transform (UT) -- a well-known distribution approximation used in the Unscented Kalman Filter (UKF) from the field of filtering. A finite set of statistics called sigma points, sampled deterministically, provides a more informative and lower-variance posterior representation than the ubiquitous noise-scaling of the reparameterization trick, while ensuring higher-quality reconstruction. We further boost the performance by replacing the Kullback-Leibler (KL) divergence with the Wasserstein distribution metric that allows for a sharper posterior. Inspired by the two components, we derive a novel, deterministic-sampling flavor of the VAE, the Unscented Autoencoder (UAE), trained purely with regularization-like terms on the per-sample posterior. We empirically show competitive performance in Fr\'echet Inception Distance (FID) scores over closely-related models, in addition to a lower training variance than the VAE.

We introduce a new family of particle evolution samplers suitable for constrained domains and non-Euclidean geometries. Stein Variational Mirror Descent and Mirrored Stein Variational Gradient Descent minimize the Kullback-Leibler (KL) divergence to constrained target distributions by evolving particles in a dual space defined by a mirror map. Stein Variational Natural Gradient exploits non-Euclidean geometry to more efficiently minimize the KL divergence to unconstrained targets. We derive these samplers from a new class of mirrored Stein operators and adaptive kernels developed in this work. We demonstrate that these new samplers yield accurate approximations to distributions on the simplex, deliver valid confidence intervals in post-selection inference, and converge more rapidly than prior methods in large-scale unconstrained posterior inference. Finally, we establish the convergence of our new procedures under verifiable conditions on the target distribution.

Deep neural networks (DNNs) exhibit an exceptional capacity for generalization in practical applications. This work aims to capture the effect and benefits of depth for supervised learning via information-theoretic generalization bounds. We first derive two hierarchical bounds on the generalization error in terms of the Kullback-Leibler (KL) divergence or the 1-Wasserstein distance between the train and test distributions of the network internal representations. The KL divergence bound shrinks as the layer index increases, while the Wasserstein bound implies the existence of a layer that serves as a generalization funnel, which attains a minimal 1-Wasserstein distance. Analytic expressions for both bounds are derived under the setting of binary Gaussian classification with linear DNNs. To quantify the contraction of the relevant information measures when moving deeper into the network, we analyze the strong data processing inequality (SDPI) coefficient between consecutive layers of three regularized DNN models: Dropout, DropConnect, and Gaussian noise injection. This enables refining our generalization bounds to capture the contraction as a function of the network architecture parameters. Specializing our results to DNNs with a finite parameter space and the Gibbs algorithm reveals that deeper yet narrower network architectures generalize better in those examples, although how broadly this statement applies remains a question.

We introduce LLaVA-MoD, a novel framework designed to enable the efficient training of small-scale Multimodal Language Models (s-MLLM) by distilling knowledge from large-scale MLLM (l-MLLM). Our approach tackles two fundamental challenges in MLLM distillation. First, we optimize the network structure of s-MLLM by integrating a sparse Mixture of Experts (MoE) architecture into the language model, striking a balance between computational efficiency and model expressiveness. Second, we propose a progressive knowledge transfer strategy to ensure comprehensive knowledge migration. This strategy begins with mimic distillation, where we minimize the Kullback-Leibler (KL) divergence between output distributions to enable the student model to emulate the teacher network's understanding. Following this, we introduce preference distillation via Direct Preference Optimization (DPO), where the key lies in treating l-MLLM as the reference model. During this phase, the s-MLLM's ability to discriminate between superior and inferior examples is significantly enhanced beyond l-MLLM, leading to a better student that surpasses its teacher, particularly in hallucination benchmarks. Extensive experiments demonstrate that LLaVA-MoD outperforms existing models across various multimodal benchmarks while maintaining a minimal number of activated parameters and low computational costs. Remarkably, LLaVA-MoD, with only 2B activated parameters, surpasses Qwen-VL-Chat-7B by an average of 8.8% across benchmarks, using merely 0.3% of the training data and 23% trainable parameters. These results underscore LLaVA-MoD's ability to effectively distill comprehensive knowledge from its teacher model, paving the way for the development of more efficient MLLMs. The code will be available on: https://github.com/shufangxun/LLaVA-MoD.

Variational Autoencoders (VAEs) have become a cornerstone in generative modeling and representation learning within machine learning. This paper explores a nuanced aspect of VAEs, focusing on interpreting the Kullback-Leibler (KL) Divergence, a critical component within the Evidence Lower Bound (ELBO) that governs the trade-off between reconstruction accuracy and regularization. Meanwhile, the KL Divergence enforces alignment between latent variable distributions and a prior imposing a structure on the overall latent space but leaves individual variable distributions unconstrained. The proposed method redefines the ELBO with a mixture of Gaussians for the posterior probability, introduces a regularization term to prevent variance collapse, and employs a PatchGAN discriminator to enhance texture realism. Implementation details involve ResNetV2 architectures for both the Encoder and Decoder. The experiments demonstrate the ability to generate realistic faces, offering a promising solution for enhancing VAE-based generative models.

Diffusion models have emerged as a powerful paradigm for modern generative modeling, demonstrating strong potential for large language models (LLMs). Unlike conventional autoregressive (AR) models that generate tokens sequentially, diffusion models enable parallel token sampling, leading to faster generation and eliminating left-to-right generation constraints. Despite their empirical success, the theoretical understanding of diffusion model approaches remains underdeveloped. In this work, we develop convergence guarantees for diffusion language models from an information-theoretic perspective. Our analysis demonstrates that the sampling error, measured by the Kullback-Leibler (KL) divergence, decays inversely with the number of iterations T and scales linearly with the mutual information between tokens in the target text sequence. In particular, we establish matching upper and lower bounds, up to some constant factor, to demonstrate the tightness of our convergence analysis. These results offer novel theoretical insights into the practical effectiveness of diffusion language models.

Many evaluation measures are used to evaluate social biases in masked language models (MLMs). However, we find that these previously proposed evaluation measures are lacking robustness in scenarios with limited datasets. This is because these measures are obtained by comparing the pseudo-log-likelihood (PLL) scores of the stereotypical and anti-stereotypical samples using an indicator function. The disadvantage is the limited mining of the PLL score sets without capturing its distributional information. In this paper, we represent a PLL score set as a Gaussian distribution and use Kullback Leibler (KL) divergence and Jensen Shannon (JS) divergence to construct evaluation measures for the distributions of stereotypical and anti-stereotypical PLL scores. Experimental results on the publicly available datasets StereoSet (SS) and CrowS-Pairs (CP) show that our proposed measures are significantly more robust and interpretable than those proposed previously.

The prior drift is crucial in Continual Test-Time Adaptation (CTTA) methods that only use unlabeled test data, as it can cause significant error propagation. In this paper, we introduce VCoTTA, a variational Bayesian approach to measure uncertainties in CTTA. At the source stage, we transform a pre-trained deterministic model into a Bayesian Neural Network (BNN) via a variational warm-up strategy, injecting uncertainties into the model. During the testing time, we employ a mean-teacher update strategy using variational inference for the student model and exponential moving average for the teacher model. Our novel approach updates the student model by combining priors from both the source and teacher models. The evidence lower bound is formulated as the cross-entropy between the student and teacher models, along with the Kullback-Leibler (KL) divergence of the prior mixture. Experimental results on three datasets demonstrate the method's effectiveness in mitigating prior drift within the CTTA framework.

Robust Markov decision processes (MDPs) address the challenge of model uncertainty by optimizing the worst-case performance over an uncertainty set of MDPs. In this paper, we focus on the robust average-reward MDPs under the model-free setting. We first theoretically characterize the structure of solutions to the robust average-reward Bellman equation, which is essential for our later convergence analysis. We then design two model-free algorithms, robust relative value iteration (RVI) TD and robust RVI Q-learning, and theoretically prove their convergence to the optimal solution. We provide several widely used uncertainty sets as examples, including those defined by the contamination model, total variation, Chi-squared divergence, Kullback-Leibler (KL) divergence and Wasserstein distance.

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.

Variational inference (VI) seeks to approximate a target distribution pi by an element of a tractable family of distributions. Of key interest in statistics and machine learning is Gaussian VI, which approximates pi by minimizing the Kullback-Leibler (KL) divergence to pi over the space of Gaussians. In this work, we develop the (Stochastic) Forward-Backward Gaussian Variational Inference (FB-GVI) algorithm to solve Gaussian VI. Our approach exploits the composite structure of the KL divergence, which can be written as the sum of a smooth term (the potential) and a non-smooth term (the entropy) over the Bures-Wasserstein (BW) space of Gaussians endowed with the Wasserstein distance. For our proposed algorithm, we obtain state-of-the-art convergence guarantees when pi is log-smooth and log-concave, as well as the first convergence guarantees to first-order stationary solutions when pi is only log-smooth.

Variational autoencoders (VAEs) are one of the powerful unsupervised learning frameworks in NLP for latent representation learning and latent-directed generation. The classic optimization goal of VAEs is to maximize the Evidence Lower Bound (ELBo), which consists of a conditional likelihood for generation and a negative Kullback-Leibler (KL) divergence for regularization. In practice, optimizing ELBo often leads the posterior distribution of all samples converge to the same degenerated local optimum, namely posterior collapse or KL vanishing. There are effective ways proposed to prevent posterior collapse in VAEs, but we observe that they in essence make trade-offs between posterior collapse and hole problem, i.e., mismatch between the aggregated posterior distribution and the prior distribution. To this end, we introduce new training objectives to tackle both two problems through a novel regularization based on the probabilistic density gap between the aggregated posterior distribution and the prior distribution. Through experiments on language modeling, latent space visualization and interpolation, we show that our proposed method can solve both problems effectively and thus outperforms the existing methods in latent-directed generation. To the best of our knowledge, we are the first to jointly solve the hole problem and the posterior collapse.

Domain adaptation (DA) has drawn high interest for its capacity to adapt a model trained on labeled source data to perform well on unlabeled or weakly labeled target data from a different domain. Most common DA techniques require concurrent access to the input images of both the source and target domains. However, in practice, privacy concerns often impede the availability of source images in the adaptation phase. This is a very frequent DA scenario in medical imaging, where, for instance, the source and target images could come from different clinical sites. We introduce a source-free domain adaptation for image segmentation. Our formulation is based on minimizing a label-free entropy loss defined over target-domain data, which we further guide with a domain-invariant prior on the segmentation regions. Many priors can be derived from anatomical information. Here, a class ratio prior is estimated from anatomical knowledge and integrated in the form of a Kullback Leibler (KL) divergence in our overall loss function. Furthermore, we motivate our overall loss with an interesting link to maximizing the mutual information between the target images and their label predictions. We show the effectiveness of our prior aware entropy minimization in a variety of domain-adaptation scenarios, with different modalities and applications, including spine, prostate, and cardiac segmentation. Our method yields comparable results to several state of the art adaptation techniques, despite having access to much less information, as the source images are entirely absent in our adaptation phase. Our straightforward adaptation strategy uses only one network, contrary to popular adversarial techniques, which are not applicable to a source-free DA setting. Our framework can be readily used in a breadth of segmentation problems, and our code is publicly available: https://github.com/mathilde-b/SFDA

Knowledge distillation (KD) has been widely adopted to compress large language models (LLMs). Existing KD methods investigate various divergence measures including the Kullback-Leibler (KL), reverse Kullback-Leibler (RKL), and Jensen-Shannon (JS) divergences. However, due to limitations inherent in their assumptions and definitions, these measures fail to deliver effective supervision when few distribution overlap exists between the teacher and the student. In this paper, we show that the aforementioned KL, RKL, and JS divergences respectively suffer from issues of mode-averaging, mode-collapsing, and mode-underestimation, which deteriorates logits-based KD for diverse NLP tasks. We propose the Sinkhorn Knowledge Distillation (SinKD) that exploits the Sinkhorn distance to ensure a nuanced and precise assessment of the disparity between teacher and student distributions. Besides, profit by properties of the Sinkhorn metric, we can get rid of sample-wise KD that restricts the perception of divergence in each teacher-student sample pair. Instead, we propose a batch-wise reformulation to capture geometric intricacies of distributions across samples in the high-dimensional space. Comprehensive evaluation on GLUE and SuperGLUE, in terms of comparability, validity, and generalizability, highlights our superiority over state-of-the-art methods on all kinds of LLMs with encoder-only, encoder-decoder, and decoder-only architectures.

Extracting optical flow from videos remains a core computer vision problem. Motivated by the success of large general-purpose models, we ask whether frozen self-supervised video models trained only for future frame prediction can be prompted, without fine-tuning, to output flow. Prior work reading out depth or illumination from video generators required fine-tuning, which is impractical for flow where labels are scarce and synthetic datasets suffer from a sim-to-real gap. Inspired by the Counterfactual World Model (CWM) paradigm, which can obtain point-wise correspondences by injecting a small tracer perturbation into a next-frame predictor and tracking its propagation, we extend this idea to generative video models. We explore several popular architectures and find that successful zero-shot flow extraction in this manner is aided by three model properties: (1) distributional prediction of future frames (avoiding blurry or noisy outputs); (2) factorized latents that treat each spatio-temporal patch independently; and (3) random-access decoding that can condition on any subset of future pixels. These properties are uniquely present in the recent Local Random Access Sequence (LRAS) architecture. Building on LRAS, we propose KL-tracing: a novel test-time procedure that injects a localized perturbation into the first frame, rolls out the model one step, and computes the Kullback-Leibler divergence between perturbed and unperturbed predictive distributions. Without any flow-specific fine-tuning, our method outperforms state-of-the-art models on real-world TAP-Vid DAVIS dataset (16.6% relative improvement for endpoint error) and synthetic TAP-Vid Kubric (4.7% relative improvement). Our results indicate that counterfactual prompting of controllable generative video models is a scalable and effective alternative to supervised or photometric-loss approaches for high-quality flow.

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.

For a parametric model of distributions, the closest distribution in the model to the true distribution located outside the model is considered. Measuring the closeness between two distributions with the Kullback-Leibler (K-L) divergence, the closest distribution is called the "information projection." The estimation risk of the maximum likelihood estimator (MLE) is defined as the expectation of K-L divergence between the information projection and the predictive distribution with plugged-in MLE. Here, the asymptotic expansion of the risk is derived up to n^{-2}-order, and the sufficient condition on the risk for the Bayes error rate between the true distribution and the information projection to be lower than a specified value is investigated. Combining these results, the "p-n criterion" is proposed, which determines whether the MLE is sufficiently close to the information projection for the given model and sample. In particular, the criterion for an exponential family model is relatively simple and can be used for a complex model with no explicit form of normalizing constant. This criterion can constitute a solution to the sample size or model acceptance problem. Use of the p-n criteria is demonstrated for two practical datasets. The relationship between the results and information criteria is also studied.

Kullback-Leiber divergence has been widely used in Knowledge Distillation (KD) to compress Large Language Models (LLMs). Contrary to prior assertions that reverse Kullback-Leibler (RKL) divergence is mode-seeking and thus preferable over the mean-seeking forward Kullback-Leibler (FKL) divergence, this study empirically and theoretically demonstrates that neither mode-seeking nor mean-seeking properties manifest in KD for LLMs. Instead, RKL and FKL are found to share the same optimization objective and both converge after a sufficient number of epochs. However, due to practical constraints, LLMs are seldom trained for such an extensive number of epochs. Meanwhile, we further find that RKL focuses on the tail part of the distributions, while FKL focuses on the head part at the beginning epochs. Consequently, we propose a simple yet effective Adaptive Kullback-Leiber (AKL) divergence method, which adaptively allocates weights to combine FKL and RKL. Metric-based and GPT-4-based evaluations demonstrate that the proposed AKL outperforms the baselines across various tasks and improves the diversity and quality of generated responses.

Recently, a series of papers proposed deep learning-based approaches to sample from unnormalized target densities using controlled diffusion processes. In this work, we identify these approaches as special cases of the Schr\"odinger bridge problem, seeking the most likely stochastic evolution between a given prior distribution and the specified target. We further generalize this framework by introducing a variational formulation based on divergences between path space measures of time-reversed diffusion processes. This abstract perspective leads to practical losses that can be optimized by gradient-based algorithms and includes previous objectives as special cases. At the same time, it allows us to consider divergences other than the reverse Kullback-Leibler divergence that is known to suffer from mode collapse. In particular, we propose the so-called log-variance loss, which exhibits favorable numerical properties and leads to significantly improved performance across all considered approaches.

We characterize the statistical efficiency of knowledge transfer through n samples from a teacher to a probabilistic student classifier with input space mathcal S over labels mathcal A. We show that privileged information at three progressive levels accelerates the transfer. At the first level, only samples with hard labels are known, via which the maximum likelihood estimator attains the minimax rate {|{mathcal S||{mathcal A}|}/{n}}. The second level has the teacher probabilities of sampled labels available in addition, which turns out to boost the convergence rate lower bound to {{|{mathcal S}||{mathcal A}|}/{n}}. However, under this second data acquisition protocol, minimizing a naive adaptation of the cross-entropy loss results in an asymptotically biased student. We overcome this limitation and achieve the fundamental limit by using a novel empirical variant of the squared error logit loss. The third level further equips the student with the soft labels (complete logits) on {mathcal A} given every sampled input, thereby provably enables the student to enjoy a rate {|{mathcal S}|}/{n} free of |{mathcal A}|. We find any Kullback-Leibler divergence minimizer to be optimal in the last case. Numerical simulations distinguish the four learners and corroborate our theory.

Knowledge Distillation (KD) is a promising technique for reducing the high computational demand of large language models (LLMs). However, previous KD methods are primarily applied to white-box classification models or training small models to imitate black-box model APIs like ChatGPT. How to effectively distill the knowledge from white-box generative LLMs is still under-explored, which becomes more and more important with the prosperity of LLMs. In this work, we propose MiniLLM that distills smaller language models from generative larger language models. We first replace the forward Kullback-Leibler divergence (KLD) objective in the standard KD approaches with reverse KLD, which is more suitable for KD on generative language models, to prevent the student model from overestimating the low-probability regions of the teacher distribution. Then, we derive an effective optimization approach to learn this objective. Extensive experiments in the instruction-following setting show that the MiniLLM models generate more precise responses with the higher overall quality, lower exposure bias, better calibration, and higher long-text generation performance. Our method is also scalable for different model families with 120M to 13B parameters. We will release our code and model checkpoints at https://aka.ms/MiniLLM.

This paper studies Kernel density estimation for a high-dimensional distribution rho(x). Traditional approaches have focused on the limit of large number of data points n and fixed dimension d. We analyze instead the regime where both the number n of data points y_i and their dimensionality d grow with a fixed ratio alpha=(log n)/d. Our study reveals three distinct statistical regimes for the kernel-based estimate of the density hat rho_h^{D}(x)=1{n h^d}sum_{i=1}^n Kleft(x-y_i{h}right), depending on the bandwidth h: a classical regime for large bandwidth where the Central Limit Theorem (CLT) holds, which is akin to the one found in traditional approaches. Below a certain value of the bandwidth, h_{CLT}(alpha), we find that the CLT breaks down. The statistics of hat rho_h^{D}(x) for a fixed x drawn from rho(x) is given by a heavy-tailed distribution (an alpha-stable distribution). In particular below a value h_G(alpha), we find that hat rho_h^{D}(x) is governed by extreme value statistics: only a few points in the database matter and give the dominant contribution to the density estimator. We provide a detailed analysis for high-dimensional multivariate Gaussian data. We show that the optimal bandwidth threshold based on Kullback-Leibler divergence lies in the new statistical regime identified in this paper. Our findings reveal limitations of classical approaches, show the relevance of these new statistical regimes, and offer new insights for Kernel density estimation in high-dimensional settings.

When fine-tuning pre-trained Large Language Models (LLMs) to align with human values and intentions, maximizing the estimated reward can lead to superior performance, but it also introduces potential risks due to deviations from the reference model's intended behavior. Most existing methods typically introduce KL divergence to constrain deviations between the trained model and the reference model; however, this may not be sufficient in certain applications that require tight risk control. In this paper, we introduce Risk-aware Direct Preference Optimization (Ra-DPO), a novel approach that incorporates risk-awareness by employing a class of nested risk measures. This approach formulates a constrained risk-aware advantage function maximization problem and then converts the Bradley-Terry model into a token-level representation. The objective function maximizes the likelihood of the policy while suppressing the deviation between a trained model and the reference model using a sequential risk ratio, thereby enhancing the model's risk-awareness. Experimental results across three open-source datasets: IMDb Dataset, Anthropic HH Dataset, and AlpacaEval, demonstrate the proposed method's superior performance in balancing alignment performance and model drift. Our code is opensourced at https://github.com/zlj123-max/Ra-DPO.

A simple and effective method for the alignment of generative models is the best-of-n policy, where n samples are drawn from a base policy, and ranked based on a reward function, and the highest ranking one is selected. A commonly used analytical expression in the literature claims that the KL divergence between the best-of-n policy and the base policy is equal to log (n) - (n-1)/n. We disprove the validity of this claim, and show that it is an upper bound on the actual KL divergence. We also explore the tightness of this upper bound in different regimes. Finally, we propose a new estimator for the KL divergence and empirically show that it provides a tight approximation through a few examples.

We give an improved theoretical analysis of score-based generative modeling. Under a score estimate with small L^2 error (averaged across timesteps), we provide efficient convergence guarantees for any data distribution with second-order moment, by either employing early stopping or assuming smoothness condition on the score function of the data distribution. Our result does not rely on any log-concavity or functional inequality assumption and has a logarithmic dependence on the smoothness. In particular, we show that under only a finite second moment condition, approximating the following in reverse KL divergence in epsilon-accuracy can be done in tilde Oleft(d log (1/delta){epsilon}right) steps: 1) the variance-delta Gaussian perturbation of any data distribution; 2) data distributions with 1/delta-smooth score functions. Our analysis also provides a quantitative comparison between different discrete approximations and may guide the choice of discretization points in practice.

Recent advances in large language model (LLM) post-training have leveraged two distinct paradigms to enhance reasoning capabilities: reinforcement learning (RL) and knowledge distillation (KD). While RL enables the emergence of complex reasoning behaviors, it often suffers from low sample efficiency when the initial policy struggles to explore high-reward trajectories. Conversely, KD improves learning efficiency via mimicking the teacher model but tends to generalize poorly to out-of-domain scenarios. In this work, we present KDRL, a unified post-training framework that jointly optimizes a reasoning model through teacher supervision (KD) and self-exploration (RL). Specifically, KDRL leverages policy gradient optimization to simultaneously minimize the reverse Kullback-Leibler divergence (RKL) between the student and teacher distributions while maximizing the expected rule-based rewards. We first formulate a unified objective that integrates GRPO and KD, and systematically explore how different KL approximations, KL coefficients, and reward-guided KD strategies affect the overall post-training dynamics and performance. Empirical results on multiple reasoning benchmarks demonstrate that KDRL outperforms GRPO and various KD baselines while achieving a favorable balance between performance and reasoning token efficiency. These findings indicate that integrating KD and RL serves as an effective and efficient strategy to train reasoning LLMs.

Autoregressive (AR) models remain the standard for natural language generation but still suffer from high latency due to strictly sequential decoding. Recent diffusion-inspired approaches, such as LlaDA and Dream, mitigate this by generating in parallel, yet they suffer from two core limitations: information loss, as predictive distributions for non-finalized tokens are discarded at each step, and premature commitment, where local decisions are made without sufficient global coordination. We introduce Latent Refinement Decoding (LRD), a two-stage framework with Latent Refinement and a Predictive Feedback Loop. The first stage maintains masked positions as distributional mixtures of predicted tokens and the mask embedding, allowing the model to establish more globally consistent beliefs. The second stage progressively finalizes confident tokens while retaining uncertain ones for iterative feedback. KL-divergence dynamics provide a principled and reliable criterion for convergence and early stopping. Experiments across coding (HumanEval +6.3, MBPP +2.6) and reasoning (GSM8K +2.9, MATH500 +3.8) show that LRD improves accuracy while delivering speedups of up to 10.6x, making it a strong and versatile alternative for parallel sequence generation.

Knowledge distillation is a model compression technique in which a compact "student" network is trained to replicate the predictive behavior of a larger "teacher" network. In logit-based knowledge distillation, it has become the de facto approach to augment cross-entropy with a distillation term. Typically, this term is either a KL divergence that matches marginal probabilities or a correlation-based loss that captures intra- and inter-class relationships. In every case, it acts as an additional term to cross-entropy. This term has its own weight, which must be carefully tuned. In this paper, we adopt a choice-theoretic perspective and recast knowledge distillation under the Plackett-Luce model by interpreting teacher logits as "worth" scores. We introduce "Plackett-Luce Distillation (PLD)", a weighted list-wise ranking loss. In PLD, the teacher model transfers knowledge of its full ranking of classes, weighting each ranked choice by its own confidence. PLD directly optimizes a single "teacher-optimal" ranking. The true label is placed first, followed by the remaining classes in descending teacher confidence. This process yields a convex and translation-invariant surrogate that subsumes weighted cross-entropy. Empirically, across CIFAR-100, ImageNet-1K, and MS-COCO, PLD achieves consistent gains across diverse architectures and distillation objectives, including divergence-based, correlation-based, and feature-based methods, in both homogeneous and heterogeneous teacher-student pairs.

In this work we present a new method for the estimation of Mutual Information (MI) between random variables. Our approach is based on an original interpretation of the Girsanov theorem, which allows us to use score-based diffusion models to estimate the Kullback Leibler divergence between two densities as a difference between their score functions. As a by-product, our method also enables the estimation of the entropy of random variables. Armed with such building blocks, we present a general recipe to measure MI, which unfolds in two directions: one uses conditional diffusion process, whereas the other uses joint diffusion processes that allow simultaneous modelling of two random variables. Our results, which derive from a thorough experimental protocol over all the variants of our approach, indicate that our method is more accurate than the main alternatives from the literature, especially for challenging distributions. Furthermore, our methods pass MI self-consistency tests, including data processing and additivity under independence, which instead are a pain-point of existing methods.

The problem of estimating an unknown discrete distribution from its samples is a fundamental tenet of statistical learning. Over the past decade, it attracted significant research effort and has been solved for a variety of divergence measures. Surprisingly, an equally important problem, estimating an unknown Markov chain from its samples, is still far from understood. We consider two problems related to the min-max risk (expected loss) of estimating an unknown k-state Markov chain from its n sequential samples: predicting the conditional distribution of the next sample with respect to the KL-divergence, and estimating the transition matrix with respect to a natural loss induced by KL or a more general f-divergence measure. For the first measure, we determine the min-max prediction risk to within a linear factor in the alphabet size, showing it is Omega(kloglog n / n) and O(k^2loglog n / n). For the second, if the transition probabilities can be arbitrarily small, then only trivial uniform risk upper bounds can be derived. We therefore consider transition probabilities that are bounded away from zero, and resolve the problem for essentially all sufficiently smooth f-divergences, including KL-, L_2-, Chi-squared, Hellinger, and Alpha-divergences.

In mixtures-of-experts (ME) model, where a number of submodels (experts) are combined, there have been two longstanding problems: (i) how many experts should be chosen, given the size of the training data? (ii) given the total number of parameters, is it better to use a few very complex experts, or is it better to combine many simple experts? In this paper, we try to provide some insights to these problems through a theoretic study on a ME structure where m experts are mixed, with each expert being related to a polynomial regression model of order k. We study the convergence rate of the maximum likelihood estimator (MLE), in terms of how fast the Kullback-Leibler divergence of the estimated density converges to the true density, when the sample size n increases. The convergence rate is found to be dependent on both m and k, and certain choices of m and k are found to produce optimal convergence rates. Therefore, these results shed light on the two aforementioned important problems: on how to choose m, and on how m and k should be compromised, for achieving good convergence rates.

Recent Reinforcement Learning (RL) algorithms making use of Kullback-Leibler (KL) regularization as a core component have shown outstanding performance. Yet, only little is understood theoretically about why KL regularization helps, so far. We study KL regularization within an approximate value iteration scheme and show that it implicitly averages q-values. Leveraging this insight, we provide a very strong performance bound, the very first to combine two desirable aspects: a linear dependency to the horizon (instead of quadratic) and an error propagation term involving an averaging effect of the estimation errors (instead of an accumulation effect). We also study the more general case of an additional entropy regularizer. The resulting abstract scheme encompasses many existing RL algorithms. Some of our assumptions do not hold with neural networks, so we complement this theoretical analysis with an extensive empirical study.

In the field of large language models (LLMs), Knowledge Distillation (KD) is a critical technique for transferring capabilities from teacher models to student models. However, existing KD methods face limitations and challenges in distillation of LLMs, including efficiency and insufficient measurement capabilities of traditional KL divergence. It is shown that LLMs can serve as an implicit reward function, which we define as a supplement to KL divergence. In this work, we propose Direct Preference Knowledge Distillation (DPKD) for LLMs. DPKD utilizes distribution divergence to represent the preference loss and implicit reward function. We re-formulate KD of LLMs into two stages: first optimizing and objective consisting of implicit reward and reverse KL divergence and then improving the preference probability of teacher outputs over student outputs. We conducted experiments and analysis on various datasets with LLM parameters ranging from 120M to 13B and demonstrate the broad applicability and effectiveness of our DPKD approach. Meanwhile, we prove the value and effectiveness of the introduced implicit reward and output preference in KD through experiments and theoretical analysis. The DPKD method outperforms the baseline method in both output response precision and exact match percentage. Code and data are available at https://aka.ms/dpkd.

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.

To compare autoregressive language models at scale, we propose using log-likelihood vectors computed on a predefined text set as model features. This approach has a solid theoretical basis: when treated as model coordinates, their squared Euclidean distance approximates the Kullback-Leibler divergence of text-generation probabilities. Our method is highly scalable, with computational cost growing linearly in both the number of models and text samples, and is easy to implement as the required features are derived from cross-entropy loss. Applying this method to over 1,000 language models, we constructed a "model map," providing a new perspective on large-scale model analysis.

Knowledge distillation (KD) is widely used for compressing a teacher model to a smaller student model, reducing its inference cost and memory footprint while preserving model capabilities. However, current KD methods for auto-regressive sequence models (e.g., large language models) suffer from missing a standardized objective function. Moreover, the recent use of student-generated outputs to address training-inference mismatches has significantly escalated computational costs. To tackle these issues, we introduce DistiLLM, a more effective and efficient KD framework for auto-regressive language models. DistiLLM comprises two components: (1) a novel skew Kullback-Leibler divergence loss, where we unveil and leverage its theoretical properties, and (2) an adaptive off-policy approach designed to enhance the efficiency in utilizing student-generated outputs. Extensive experiments, including instruction-following tasks, demonstrate the effectiveness of DistiLLM in building high-performing student models while achieving up to 4.3times speedup compared to recent KD methods.

A central paradox in fine-tuning Large Language Models (LLMs) with Reinforcement Learning with Verifiable Reward (RLVR) is the frequent degradation of multi-attempt performance (Pass@k) despite improvements in single-attempt accuracy (Pass@1). This is often accompanied by catastrophic forgetting, where models lose previously acquired skills. While various methods have been proposed, the choice and function of the divergence term have been surprisingly unexamined as a proactive solution. We argue that standard RLVR objectives -- both those using the mode-seeking reverse KL-divergence and those forgoing a divergence term entirely -- lack a crucial mechanism for knowledge retention. The reverse-KL actively accelerates this decay by narrowing the policy, while its absence provides no safeguard against the model drifting from its diverse knowledge base. We propose a fundamental shift in perspective: using the divergence term itself as the solution. Our framework, Diversity-Preserving Hybrid RL (DPH-RL), leverages mass-covering f-divergences (like forward-KL and JS-divergence) to function as a rehearsal mechanism. By continuously referencing the initial policy, this approach forces the model to maintain broad solution coverage. Extensive experiments on math and SQL generation demonstrate that DPH-RL not only resolves the Pass@k degradation but improves both Pass@1 and Pass@k in- and out-of-domain. Additionally, DPH-RL is more training-efficient because it computes f-divergence using generator functions, requiring only sampling from the initial policy and no online reference model. Our work highlights a crucial, overlooked axis for improving RLVR, demonstrating that the proper selection of a divergence measure is a powerful tool for building more general and diverse reasoning models.

The increasing capabilities of large language models (LLMs) raise opportunities for artificial general intelligence but concurrently amplify safety concerns, such as potential misuse of AI systems, necessitating effective AI alignment. Reinforcement Learning from Human Feedback (RLHF) has emerged as a promising pathway towards AI alignment but brings forth challenges due to its complexity and dependence on a separate reward model. Direct Preference Optimization (DPO) has been proposed as an alternative, and it remains equivalent to RLHF under the reverse KL regularization constraint. This paper presents f-DPO, a generalized approach to DPO by incorporating diverse divergence constraints. We show that under certain f-divergences, including Jensen-Shannon divergence, forward KL divergences and alpha-divergences, the complex relationship between the reward and optimal policy can also be simplified by addressing the Karush-Kuhn-Tucker conditions. This eliminates the need for estimating the normalizing constant in the Bradley-Terry model and enables a tractable mapping between the reward function and the optimal policy. Our approach optimizes LLMs to align with human preferences in a more efficient and supervised manner under a broad set of divergence constraints. Empirically, adopting these divergences ensures a balance between alignment performance and generation diversity. Importantly, f-DPO outperforms PPO-based methods in divergence efficiency, and divergence constraints directly influence expected calibration error (ECE).

A fundamental question in modern machine learning is why large, over-parameterized models, such as deep neural networks and transformers, tend to generalize well, even when their number of parameters far exceeds the number of training samples. We investigate this phenomenon through the lens of information theory, grounded in universal learning theory. Specifically, we study a Bayesian mixture learner with log-loss and (almost) uniform prior over an expansive hypothesis class. Our key result shows that the learner's regret is not determined by the overall size of the hypothesis class, but rather by the cumulative probability of all models that are close, in Kullback-Leibler divergence distance, to the true data-generating process. We refer to this cumulative probability as the weight of the hypothesis. This leads to a natural notion of model simplicity: simple models are those with large weight and thus require fewer samples to generalize, while complex models have small weight and need more data. This perspective provides a rigorous and intuitive explanation for why over-parameterized models often avoid overfitting: the presence of simple hypotheses allows the posterior to concentrate on them when supported by the data. We further bridge theory and practice by recalling that stochastic gradient descent with Langevin dynamics samples from the correct posterior distribution, enabling our theoretical learner to be approximated using standard machine learning methods combined with ensemble learning. Our analysis yields non-uniform regret bounds and aligns with key practical concepts such as flat minima and model distillation. The results apply broadly across online, batch, and supervised learning settings, offering a unified and principled understanding of the generalization behavior of modern AI systems.

In the loss function of Variational Autoencoders there is a well known tension between two components: the reconstruction loss, improving the quality of the resulting images, and the Kullback-Leibler divergence, acting as a regularizer of the latent space. Correctly balancing these two components is a delicate issue, easily resulting in poor generative behaviours. In a recent work, Dai and Wipf obtained a sensible improvement by allowing the network to learn the balancing factor during training, according to a suitable loss function. In this article, we show that learning can be replaced by a simple deterministic computation, helping to understand the underlying mechanism, and resulting in a faster and more accurate behaviour. On typical datasets such as Cifar and Celeba, our technique sensibly outperforms all previous VAE architectures.

The controllable generation of diffusion models aims to steer the model to generate samples that optimize some given objective functions. It is desirable for a variety of applications including image generation, molecule generation, and DNA/sequence generation. Reinforcement Learning (RL) based fine-tuning of the base model is a popular approach but it can overfit the reward function while requiring significant resources. We frame controllable generation as a problem of finding a distribution that optimizes a KL-regularized objective function. We present SLCD -- Supervised Learning based Controllable Diffusion, which iteratively generates online data and trains a small classifier to guide the generation of the diffusion model. Similar to the standard classifier-guided diffusion, SLCD's key computation primitive is classification and does not involve any complex concepts from RL or control. Via a reduction to no-regret online learning analysis, we show that under KL divergence, the output from SLCD provably converges to the optimal solution of the KL-regularized objective. Further, we empirically demonstrate that SLCD can generate high quality samples with nearly the same inference time as the base model in both image generation with continuous diffusion and biological sequence generation with discrete diffusion. Our code is available at https://github.com/Owen-Oertell/slcd

Aligning language models with preferences can be posed as approximating a target distribution representing some desired behavior. Existing approaches differ both in the functional form of the target distribution and the algorithm used to approximate it. For instance, Reinforcement Learning from Human Feedback (RLHF) corresponds to minimizing a reverse KL from an implicit target distribution arising from a KL penalty in the objective. On the other hand, Generative Distributional Control (GDC) has an explicit target distribution and minimizes a forward KL from it using the Distributional Policy Gradient (DPG) algorithm. In this paper, we propose a new approach, f-DPG, which allows the use of any f-divergence to approximate any target distribution that can be evaluated. f-DPG unifies both frameworks (RLHF, GDC) and the approximation methods (DPG, RL with KL penalties). We show the practical benefits of various choices of divergence objectives and demonstrate that there is no universally optimal objective but that different divergences present different alignment and diversity trade-offs. We show that Jensen-Shannon divergence strikes a good balance between these objectives, and frequently outperforms forward KL divergence by a wide margin, leading to significant improvements over prior work. These distinguishing characteristics between divergences persist as the model size increases, highlighting the importance of selecting appropriate divergence objectives.

We propose a goodness-of-fit measure for probability densities modeling observations with varying dimensionality, such as text documents of differing lengths or variable-length sequences. The proposed measure is an instance of the kernel Stein discrepancy (KSD), which has been used to construct goodness-of-fit tests for unnormalized densities. The KSD is defined by its Stein operator: current operators used in testing apply to fixed-dimensional spaces. As our main contribution, we extend the KSD to the variable-dimension setting by identifying appropriate Stein operators, and propose a novel KSD goodness-of-fit test. As with the previous variants, the proposed KSD does not require the density to be normalized, allowing the evaluation of a large class of models. Our test is shown to perform well in practice on discrete sequential data benchmarks.

Moment restrictions and their conditional counterparts emerge in many areas of machine learning and statistics ranging from causal inference to reinforcement learning. Estimators for these tasks, generally called methods of moments, include the prominent generalized method of moments (GMM) which has recently gained attention in causal inference. GMM is a special case of the broader family of empirical likelihood estimators which are based on approximating a population distribution by means of minimizing a varphi-divergence to an empirical distribution. However, the use of varphi-divergences effectively limits the candidate distributions to reweightings of the data samples. We lift this long-standing limitation and provide a method of moments that goes beyond data reweighting. This is achieved by defining an empirical likelihood estimator based on maximum mean discrepancy which we term the kernel method of moments (KMM). We provide a variant of our estimator for conditional moment restrictions and show that it is asymptotically first-order optimal for such problems. Finally, we show that our method achieves competitive performance on several conditional moment restriction tasks.

The primary goal of knowledge distillation (KD) is to encapsulate the information of a model learned from a teacher network into a student network, with the latter being more compact than the former. Existing work, e.g., using Kullback-Leibler divergence for distillation, may fail to capture important structural knowledge in the teacher network and often lacks the ability for feature generalization, particularly in situations when teacher and student are built to address different classification tasks. We propose Wasserstein Contrastive Representation Distillation (WCoRD), which leverages both primal and dual forms of Wasserstein distance for KD. The dual form is used for global knowledge transfer, yielding a contrastive learning objective that maximizes the lower bound of mutual information between the teacher and the student networks. The primal form is used for local contrastive knowledge transfer within a mini-batch, effectively matching the distributions of features between the teacher and the student networks. Experiments demonstrate that the proposed WCoRD method outperforms state-of-the-art approaches on privileged information distillation, model compression and cross-modal transfer.

We show that deep belief networks with binary hidden units can approximate any multivariate probability density under very mild integrability requirements on the parental density of the visible nodes. The approximation is measured in the L^q-norm for qin[1,infty] (q=infty corresponding to the supremum norm) and in Kullback-Leibler divergence. Furthermore, we establish sharp quantitative bounds on the approximation error in terms of the number of hidden units.

The variational autoencoder (VAE) framework is a popular option for training unsupervised generative models, featuring ease of training and latent representation of data. The objective function of VAE does not guarantee to achieve the latter, however, and failure to do so leads to a frequent failure mode called posterior collapse. Even in successful cases, VAEs often result in low-precision reconstructions and generated samples. The introduction of the KL-divergence weight beta can help steer the model clear of posterior collapse, but its tuning is often a trial-and-error process with no guiding metrics. Here we test the idea of using the total VAE loss of generated samples (generated loss) as the proxy metric for generation quality, the related hypothesis that VAE reconstruction from the mean latent vector tends to be a more typical example of its class than the original, and the idea of exploiting this property by augmenting training data with generated variants (augmented training). The results are mixed, but repeated encoding and decoding indeed result in qualitatively and quantitatively more typical examples from both convolutional and fully-connected MNIST VAEs, suggesting that it may be an inherent property of the VAE framework.

Knowledge distillation (KD) is essential for training non-autoregressive translation (NAT) models by reducing the complexity of the raw data with an autoregressive teacher model. In this study, we empirically show that as a side effect of this training, the lexical choice errors on low-frequency words are propagated to the NAT model from the teacher model. To alleviate this problem, we propose to expose the raw data to NAT models to restore the useful information of low-frequency words, which are missed in the distilled data. To this end, we introduce an extra Kullback-Leibler divergence term derived by comparing the lexical choice of NAT model and that embedded in the raw data. Experimental results across language pairs and model architectures demonstrate the effectiveness and universality of the proposed approach. Extensive analyses confirm our claim that our approach improves performance by reducing the lexical choice errors on low-frequency words. Encouragingly, our approach pushes the SOTA NAT performance on the WMT14 English-German and WMT16 Romanian-English datasets up to 27.8 and 33.8 BLEU points, respectively. The source code will be released.

Happiness computing based on large-scale online web data and machine learning methods is an emerging research topic that underpins a range of issues, from personal growth to social stability. Many advanced Machine Learning (ML) models with explanations are used to compute the happiness online assessment while maintaining high accuracy of results. However, domain knowledge constraints, such as the primary and secondary relations of happiness factors, are absent from these models, which limits the association between computing results and the right reasons for why they occurred. This article attempts to provide new insights into the explanation consistency from an empirical study perspective. Then we study how to represent and introduce domain knowledge constraints to make ML models more trustworthy. We achieve this through: (1) proving that multiple prediction models with additive factor attributions will have the desirable property of primary and secondary relations consistency, and (2) showing that factor relations with quantity can be represented as an importance distribution for encoding domain knowledge. Factor explanation difference is penalized by the Kullback-Leibler divergence-based loss among computing models. Experimental results using two online web datasets show that domain knowledge of stable factor relations exists. Using this knowledge not only improves happiness computing accuracy but also reveals more significative happiness factors for assisting decisions well.

Kernelized Stein discrepancy (KSD) is a score-based discrepancy widely used in goodness-of-fit tests. It can be applied even when the target distribution has an unknown normalising factor, such as in Bayesian analysis. We show theoretically and empirically that the KSD test can suffer from low power when the target and the alternative distributions have the same well-separated modes but differ in mixing proportions. We propose to perturb the observed sample via Markov transition kernels, with respect to which the target distribution is invariant. This allows us to then employ the KSD test on the perturbed sample. We provide numerical evidence that with suitably chosen transition kernels the proposed approach can lead to substantially higher power than the KSD test.

Reinforcement learning with verifiable rewards (RLVR) has delivered impressive gains in mathematical and multimodal reasoning and has become a standard post-training paradigm for contemporary language and vision-language models. However, the RLVR recipe introduces a significant risk of capability regression, where models forget foundational skills after prolonged training without employing regularization strategies. We empirically confirm this concern, observing that open-source reasoning models suffer performance degradation on core capabilities such as perception and faithfulness. While imposing regularization terms like KL divergence can help prevent deviation from the base model, these terms are calculated on the current task, thus they do not guarantee broader knowledge. Meanwhile, commonly used experience replay across heterogeneous domains makes it nontrivial to decide how much training focus each objective should receive. To address this, we propose RECAP-a replay strategy with dynamic objective reweighting for general knowledge preservation. Our reweighting mechanism adapts in an online manner using short-horizon signals of convergence and instability, shifting the post-training focus away from saturated objectives and toward underperforming or volatile ones. Our method is end-to-end and readily applicable to existing RLVR pipelines without training additional models or heavy tuning. Extensive experiments on benchmarks based on Qwen2.5-VL-3B and Qwen2.5-VL-7B demonstrate the effectiveness of our method, which not only preserves general capabilities but also improves reasoning by enabling more flexible trade-offs among in-task rewards.

Efficient fine-tuning of large language models for task-specific applications is imperative, yet the vast number of parameters in these models makes their training increasingly challenging. Despite numerous proposals for effective methods, a substantial memory overhead remains for gradient computations during updates. Can we fine-tune a series of task-specific small models and transfer their knowledge directly to a much larger model without additional training? In this paper, we explore weak-to-strong specialization using logit arithmetic, facilitating a direct answer to this question. Existing weak-to-strong methods often employ a static knowledge transfer ratio and a single small model for transferring complex knowledge, which leads to suboptimal performance. % To address this, To surmount these limitations, we propose a dynamic logit fusion approach that works with a series of task-specific small models, each specialized in a different task. This method adaptively allocates weights among these models at each decoding step, learning the weights through Kullback-Leibler divergence constrained optimization problems. We conduct extensive experiments across various benchmarks in both single-task and multi-task settings, achieving leading results. By transferring expertise from the 7B model to the 13B model, our method closes the performance gap by 96.4\% in single-task scenarios and by 86.3\% in multi-task scenarios compared to full fine-tuning of the 13B model. Notably, we achieve surpassing performance on unseen tasks. Moreover, we further demonstrate that our method can effortlessly integrate in-context learning for single tasks and task arithmetic for multi-task scenarios. (Our implementation is available in https://github.com/Facico/Dynamic-Logit-Fusion.)

As the field of representation learning grows, there has been a proliferation of different loss functions to solve different classes of problems. We introduce a single information-theoretic equation that generalizes a large collection of modern loss functions in machine learning. In particular, we introduce a framework that shows that several broad classes of machine learning methods are precisely minimizing an integrated KL divergence between two conditional distributions: the supervisory and learned representations. This viewpoint exposes a hidden information geometry underlying clustering, spectral methods, dimensionality reduction, contrastive learning, and supervised learning. This framework enables the development of new loss functions by combining successful techniques from across the literature. We not only present a wide array of proofs, connecting over 23 different approaches, but we also leverage these theoretical results to create state-of-the-art unsupervised image classifiers that achieve a +8% improvement over the prior state-of-the-art on unsupervised classification on ImageNet-1K. We also demonstrate that I-Con can be used to derive principled debiasing methods which improve contrastive representation learners.

Traditional Reinforcement Learning from Human Feedback (RLHF) often relies on reward models, frequently assuming preference structures like the Bradley-Terry model, which may not accurately capture the complexities of real human preferences (e.g., intransitivity). Nash Learning from Human Feedback (NLHF) offers a more direct alternative by framing the problem as finding a Nash equilibrium of a game defined by these preferences. In this work, we introduce Nash Mirror Prox (Nash-MP), an online NLHF algorithm that leverages the Mirror Prox optimization scheme to achieve fast and stable convergence to the Nash equilibrium. Our theoretical analysis establishes that Nash-MP exhibits last-iterate linear convergence towards the beta-regularized Nash equilibrium. Specifically, we prove that the KL-divergence to the optimal policy decreases at a rate of order (1+2beta)^{-N/2}, where N is a number of preference queries. We further demonstrate last-iterate linear convergence for the exploitability gap and uniformly for the span semi-norm of log-probabilities, with all these rates being independent of the size of the action space. Furthermore, we propose and analyze an approximate version of Nash-MP where proximal steps are estimated using stochastic policy gradients, making the algorithm closer to applications. Finally, we detail a practical implementation strategy for fine-tuning large language models and present experiments that demonstrate its competitive performance and compatibility with existing methods.

We introduce a novel method for solving density-based topology optimization problems: Sigmoidal Mirror descent with a Projected Latent variable (SiMPL). The SiMPL method (pronounced as ``the simple method'') optimizes a design using only first-order derivative information of the objective function. The bound constraints on the density field are enforced with the help of the (negative) Fermi--Dirac entropy, which is also used to define a non-symmetric distance function called a Bregman divergence on the set of admissible designs. This Bregman divergence leads to a simple update rule that is further simplified with the help of a so-called latent variable. Because the SiMPL method involves discretizing the latent variable, it produces a sequence of pointwise-feasible iterates, even when high-order finite elements are used in the discretization. Numerical experiments demonstrate that the method outperforms other popular first-order optimization algorithms. To outline the general applicability of the technique, we include examples with (self-load) compliance minimization and compliant mechanism optimization problems.

Numerous capability and safety techniques of Large Language Models (LLMs), including RLHF, automated red-teaming, prompt engineering, and infilling, can be cast as sampling from an unnormalized target distribution defined by a given reward or potential function over the full sequence. In this work, we leverage the rich toolkit of Sequential Monte Carlo (SMC) for these probabilistic inference problems. In particular, we use learned twist functions to estimate the expected future value of the potential at each timestep, which enables us to focus inference-time computation on promising partial sequences. We propose a novel contrastive method for learning the twist functions, and establish connections with the rich literature of soft reinforcement learning. As a complementary application of our twisted SMC framework, we present methods for evaluating the accuracy of language model inference techniques using novel bidirectional SMC bounds on the log partition function. These bounds can be used to estimate the KL divergence between the inference and target distributions in both directions. We apply our inference evaluation techniques to show that twisted SMC is effective for sampling undesirable outputs from a pretrained model (a useful component of harmlessness training and automated red-teaming), generating reviews with varied sentiment, and performing infilling tasks.

Distributed representations of words encode lexical semantic information, but what type of information is encoded and how? Focusing on the skip-gram with negative-sampling method, we found that the squared norm of static word embedding encodes the information gain conveyed by the word; the information gain is defined by the Kullback-Leibler divergence of the co-occurrence distribution of the word to the unigram distribution. Our findings are explained by the theoretical framework of the exponential family of probability distributions and confirmed through precise experiments that remove spurious correlations arising from word frequency. This theory also extends to contextualized word embeddings in language models or any neural networks with the softmax output layer. We also demonstrate that both the KL divergence and the squared norm of embedding provide a useful metric of the informativeness of a word in tasks such as keyword extraction, proper-noun discrimination, and hypernym discrimination.

To achieve scalable and accurate inference for latent Gaussian processes, we propose a variational approximation based on a family of Gaussian distributions whose covariance matrices have sparse inverse Cholesky (SIC) factors. We combine this variational approximation of the posterior with a similar and efficient SIC-restricted Kullback-Leibler-optimal approximation of the prior. We then focus on a particular SIC ordering and nearest-neighbor-based sparsity pattern resulting in highly accurate prior and posterior approximations. For this setting, our variational approximation can be computed via stochastic gradient descent in polylogarithmic time per iteration. We provide numerical comparisons showing that the proposed double-Kullback-Leibler-optimal Gaussian-process approximation (DKLGP) can sometimes be vastly more accurate for stationary kernels than alternative approaches such as inducing-point and mean-field approximations at similar computational complexity.

We identify empirical scaling laws for the cross-entropy loss in four domains: generative image modeling, video modeling, multimodal imageleftrightarrowtext models, and mathematical problem solving. In all cases autoregressive Transformers smoothly improve in performance as model size and compute budgets increase, following a power-law plus constant scaling law. The optimal model size also depends on the compute budget through a power-law, with exponents that are nearly universal across all data domains. The cross-entropy loss has an information theoretic interpretation as S(True) + D_{KL}(True||Model), and the empirical scaling laws suggest a prediction for both the true data distribution's entropy and the KL divergence between the true and model distributions. With this interpretation, billion-parameter Transformers are nearly perfect models of the YFCC100M image distribution downsampled to an 8times 8 resolution, and we can forecast the model size needed to achieve any given reducible loss (ie D_{KL}) in nats/image for other resolutions. We find a number of additional scaling laws in specific domains: (a) we identify a scaling relation for the mutual information between captions and images in multimodal models, and show how to answer the question "Is a picture worth a thousand words?"; (b) in the case of mathematical problem solving, we identify scaling laws for model performance when extrapolating beyond the training distribution; (c) we finetune generative image models for ImageNet classification and find smooth scaling of the classification loss and error rate, even as the generative loss levels off. Taken together, these results strengthen the case that scaling laws have important implications for neural network performance, including on downstream tasks.

Top-k decoding is a widely used method for sampling from LLMs: at each token, only the largest k next-token-probabilities are kept, and the next token is sampled after re-normalizing them to sum to unity. Top-k and other sampling methods are motivated by the intuition that true next-token distributions are sparse, and the noisy LLM probabilities need to be truncated. However, to our knowledge, a precise theoretical motivation for the use of top-k decoding is missing. In this work, we develop a theoretical framework that both explains and generalizes top-k decoding. We view decoding at a fixed token as the recovery of a sparse probability distribution. We consider Bregman decoders obtained by minimizing a separable Bregman divergence (for both the primal and dual cases) with a sparsity-inducing ell_0 regularization. Despite the combinatorial nature of the objective, we show how to optimize it efficiently for a large class of divergences. We show that the optimal decoding strategies are greedy, and further that the loss function is discretely convex in k, so that binary search provably and efficiently finds the optimal k. We show that top-k decoding arises as a special case for the KL divergence, and identify new decoding strategies that have distinct behaviors (e.g., non-linearly up-weighting larger probabilities after re-normalization).

The kernel thinning (KT) algorithm of Dwivedi and Mackey (2021) compresses a probability distribution more effectively than independent sampling by targeting a reproducing kernel Hilbert space (RKHS) and leveraging a less smooth square-root kernel. Here we provide four improvements. First, we show that KT applied directly to the target RKHS yields tighter, dimension-free guarantees for any kernel, any distribution, and any fixed function in the RKHS. Second, we show that, for analytic kernels like Gaussian, inverse multiquadric, and sinc, target KT admits maximum mean discrepancy (MMD) guarantees comparable to or better than those of square-root KT without making explicit use of a square-root kernel. Third, we prove that KT with a fractional power kernel yields better-than-Monte-Carlo MMD guarantees for non-smooth kernels, like Laplace and Mat\'ern, that do not have square-roots. Fourth, we establish that KT applied to a sum of the target and power kernels (a procedure we call KT+) simultaneously inherits the improved MMD guarantees of power KT and the tighter individual function guarantees of target KT. In our experiments with target KT and KT+, we witness significant improvements in integration error even in 100 dimensions and when compressing challenging differential equation posteriors.

Ensembles of independently trained neural networks are a state-of-the-art approach to estimate predictive uncertainty in Deep Learning, and can be interpreted as an approximation of the posterior distribution via a mixture of delta functions. The training of ensembles relies on non-convexity of the loss landscape and random initialization of their individual members, making the resulting posterior approximation uncontrolled. This paper proposes a novel and principled method to tackle this limitation, minimizing an f-divergence between the true posterior and a kernel density estimator (KDE) in a function space. We analyze this objective from a combinatorial point of view, and show that it is submodular with respect to mixture components for any f. Subsequently, we consider the problem of greedy ensemble construction. From the marginal gain on the negative f-divergence, which quantifies an improvement in posterior approximation yielded by adding a new component into the KDE, we derive a novel diversity term for ensemble methods. The performance of our approach is demonstrated on computer vision out-of-distribution detection benchmarks in a range of architectures trained on multiple datasets. The source code of our method is made publicly available at https://github.com/Oulu-IMEDS/greedy_ensembles_training.

Sparse Autoencoders (SAEs) have emerged as a promising approach for interpreting neural network representations by learning sparse, human-interpretable features from dense activations. We investigate whether incorporating variational methods into SAE architectures can improve feature organization and interpretability. We introduce the Variational Sparse Autoencoder (vSAE), which replaces deterministic ReLU gating with stochastic sampling from learned Gaussian posteriors and incorporates KL divergence regularization toward a standard normal prior. Our hypothesis is that this probabilistic sampling creates dispersive pressure, causing features to organize more coherently in the latent space while avoiding overlap. We evaluate a TopK vSAE against a standard TopK SAE on Pythia-70M transformer residual stream activations using comprehensive benchmarks including SAE Bench, individual feature interpretability analysis, and global latent space visualization through t-SNE. The vSAE underperforms standard SAE across core evaluation metrics, though excels at feature independence and ablation metrics. The KL divergence term creates excessive regularization pressure that substantially reduces the fraction of living features, leading to observed performance degradation. While vSAE features demonstrate improved robustness, they exhibit many more dead features than baseline. Our findings suggest that naive application of variational methods to SAEs does not improve feature organization or interpretability.

Multimodal Large Language Models (MLLMs) are powerful at integrating diverse data, but they often struggle with complex reasoning. While Reinforcement learning (RL) can boost reasoning in LLMs, applying it to MLLMs is tricky. Common issues include a drop in performance on general tasks and the generation of overly detailed or "overthinking" reasoning. Our work investigates how the KL penalty and overthinking affect RL training in MLLMs. We propose Asymmetric Policy Optimization (APO) to address these issues, which divides the sampled responses into positive and negative groups. For positive samples, Difficulty-Adaptive Divergence Shaping (DADS) is introduced to dynamically adjust the KL divergence weight based on their difficulty. This method prevents policy entropy from dropping sharply, improves training stability, utilizes samples better, and preserves the model's existing knowledge. For negative samples, Suboptimal Trajectory Complexity Regularization (STCR) is proposed to penalize overly long responses. This helps mitigate overthinking and encourages more concise reasoning while preserving the model's explorative capacity. We apply our method to Qwen2.5-VL-3B, creating View-R1-3B. View-R1-3B significantly enhances reasoning capabilities, showing an average 7\% gain over the base model and outperforming larger MLLMs (7-11B) on various reasoning benchmarks. Importantly, unlike other reasoning-tuned MLLMs that often degrade on general tasks, View-R1-3B maintains consistent improvement, demonstrating superior generalization. These results highlight the effectiveness and broad applicability of our DADS and STCR techniques for advancing complex multimodal reasoning in MLLMs. The code will be made available at https://github.com/Indolent-Kawhi/View-R1.

Large pre-trained language models often struggle to incorporate new domain-specific terminology when fine-tuned on small, specialized corpora. In this work, we address the challenge of vocabulary expansion in frozen LLMs by introducing a mathematically grounded method for knowledge distillation via KL divergence, even when the original and extended models use different tokenizations. This allows the student model to inherit distributional knowledge from the teacher despite differing vocabularies. We compare our KL-based distillation approach to conventional cross-entropy training, evaluating both methods across multiple strategies for initializing new token embeddings. After embedding initialization, models are further fine-tuned to integrate the new vocabulary. Each trained model is benchmarked on approximately 2000 code-generation tasks, where our approach achieves the best performance across the board. Finally, through mechanistic interpretability, we analyze how models learn representations for the new tokens, providing an explanation for the observed gains and offering insight into the structure of embedding space during vocabulary expansion.

We propose a direct estimation method for R\'{e}nyi and f-divergence measures based on a new graph theoretical interpretation. Suppose that we are given two sample sets X and Y, respectively with N and M samples, where eta:=M/N is a constant value. Considering the k-nearest neighbor (k-NN) graph of Y in the joint data set (X,Y), we show that the average powered ratio of the number of X points to the number of Y points among all k-NN points is proportional to R\'{e}nyi divergence of X and Y densities. A similar method can also be used to estimate f-divergence measures. We derive bias and variance rates, and show that for the class of gamma-H\"{o}lder smooth functions, the estimator achieves the MSE rate of O(N^{-2gamma/(gamma+d)}). Furthermore, by using a weighted ensemble estimation technique, for density functions with continuous and bounded derivatives of up to the order d, and some extra conditions at the support set boundary, we derive an ensemble estimator that achieves the parametric MSE rate of O(1/N). Our estimators are more computationally tractable than other competing estimators, which makes them appealing in many practical applications.

Discrete diffusion models have emerged as a powerful generative modeling framework for discrete data with successful applications spanning from text generation to image synthesis. However, their deployment faces challenges due to the high dimensionality of the state space, necessitating the development of efficient inference algorithms. Current inference approaches mainly fall into two categories: exact simulation and approximate methods such as tau-leaping. While exact methods suffer from unpredictable inference time and redundant function evaluations, tau-leaping is limited by its first-order accuracy. In this work, we advance the latter category by tailoring the first extension of high-order numerical inference schemes to discrete diffusion models, enabling larger step sizes while reducing error. We rigorously analyze the proposed schemes and establish the second-order accuracy of the theta-trapezoidal method in KL divergence. Empirical evaluations on GPT-2 level text and ImageNet-level image generation tasks demonstrate that our method achieves superior sample quality compared to existing approaches under equivalent computational constraints.

As Large Language Models (LLMs) become increasingly computationally complex, developing efficient deployment strategies, such as quantization, becomes crucial. State-of-the-art Post-training Quantization (PTQ) techniques often rely on calibration processes to maintain the accuracy of these models. However, while these calibration techniques can enhance performance in certain domains, they may not be as effective in others. This paper aims to draw attention to robust statistical approaches that can mitigate such issues. We propose a weight-adaptive PTQ method that can be considered a precursor to calibration-based PTQ methods, guiding the quantization process to preserve the distribution of weights by minimizing the Kullback-Leibler divergence between the quantized weights and the originally trained weights. This minimization ensures that the quantized model retains the Shannon information content of the original model to a great extent, guaranteeing robust and efficient deployment across many tasks. As such, our proposed approach can perform on par with most common calibration-based PTQ methods, establishing a new pre-calibration step for further adjusting the quantized weights with calibration. We show that our pre-calibration results achieve the same accuracy as some existing calibration-based PTQ methods on various LLMs.

Dirichlet Process mixture models (DPMM) in combination with Gaussian kernels have been an important modeling tool for numerous data domains arising from biological, physical, and social sciences. However, this versatility in applications does not extend to strong theoretical guarantees for the underlying parameter estimates, for which only a logarithmic rate is achieved. In this work, we (re)introduce and investigate a metric, named Orlicz-Wasserstein distance, in the study of the Bayesian contraction behavior for the parameters. We show that despite the overall slow convergence guarantees for all the parameters, posterior contraction for parameters happens at almost polynomial rates in outlier regions of the parameter space. Our theoretical results provide new insight in understanding the convergence behavior of parameters arising from various settings of hierarchical Bayesian nonparametric models. In addition, we provide an algorithm to compute the metric by leveraging Sinkhorn divergences and validate our findings through a simulation study.

Given alphain(0,1] and pin[1,+infty], we define the space DM^{alpha,p}(mathbb R^n) of L^p vector fields whose alpha-divergence is a finite Radon measure, extending the theory of divergence-measure vector fields to the distributional fractional setting. Our main results concern the absolute continuity properties of the alpha-divergence-measure with respect to the Hausdorff measure and fractional analogues of the Leibniz rule and the Gauss-Green formula. The sharpness of our results is discussed via some explicit examples.

We propose a gradient flow procedure for generative modeling by transporting particles from an initial source distribution to a target distribution, where the gradient field on the particles is given by a noise-adaptive Wasserstein Gradient of the Maximum Mean Discrepancy (MMD). The noise-adaptive MMD is trained on data distributions corrupted by increasing levels of noise, obtained via a forward diffusion process, as commonly used in denoising diffusion probabilistic models. The result is a generalization of MMD Gradient Flow, which we call Diffusion-MMD-Gradient Flow or DMMD. The divergence training procedure is related to discriminator training in Generative Adversarial Networks (GAN), but does not require adversarial training. We obtain competitive empirical performance in unconditional image generation on CIFAR10, MNIST, CELEB-A (64 x64) and LSUN Church (64 x 64). Furthermore, we demonstrate the validity of the approach when MMD is replaced by a lower bound on the KL divergence.

Estimating truncated density models is difficult, as these models have intractable normalising constants and hard to satisfy boundary conditions. Score matching can be adapted to solve the truncated density estimation problem, but requires a continuous weighting function which takes zero at the boundary and is positive elsewhere. Evaluation of such a weighting function (and its gradient) often requires a closed-form expression of the truncation boundary and finding a solution to a complicated optimisation problem. In this paper, we propose approximate Stein classes, which in turn leads to a relaxed Stein identity for truncated density estimation. We develop a novel discrepancy measure, truncated kernelised Stein discrepancy (TKSD), which does not require fixing a weighting function in advance, and can be evaluated using only samples on the boundary. We estimate a truncated density model by minimising the Lagrangian dual of TKSD. Finally, experiments show the accuracy of our method to be an improvement over previous works even without the explicit functional form of the boundary.

The trade-off between robustness and accuracy has been widely studied in the adversarial literature. Although still controversial, the prevailing view is that this trade-off is inherent, either empirically or theoretically. Thus, we dig for the origin of this trade-off in adversarial training and find that it may stem from the improperly defined robust error, which imposes an inductive bias of local invariance -- an overcorrection towards smoothness. Given this, we advocate employing local equivariance to describe the ideal behavior of a robust model, leading to a self-consistent robust error named SCORE. By definition, SCORE facilitates the reconciliation between robustness and accuracy, while still handling the worst-case uncertainty via robust optimization. By simply substituting KL divergence with variants of distance metrics, SCORE can be efficiently minimized. Empirically, our models achieve top-rank performance on RobustBench under AutoAttack. Besides, SCORE provides instructive insights for explaining the overfitting phenomenon and semantic input gradients observed on robust models. Code is available at https://github.com/P2333/SCORE.

We rigorously quantify the improvement in the sample complexity of variational divergence estimations for group-invariant distributions. In the cases of the Wasserstein-1 metric and the Lipschitz-regularized alpha-divergences, the reduction of sample complexity is proportional to an ambient-dimension-dependent power of the group size. For the maximum mean discrepancy (MMD), the improvement of sample complexity is more nuanced, as it depends on not only the group size but also the choice of kernel. Numerical simulations verify our theories.

Weight-sharing neural architecture search (NAS) is an effective technique for automating efficient neural architecture design. Weight-sharing NAS builds a supernet that assembles all the architectures as its sub-networks and jointly trains the supernet with the sub-networks. The success of weight-sharing NAS heavily relies on distilling the knowledge of the supernet to the sub-networks. However, we find that the widely used distillation divergence, i.e., KL divergence, may lead to student sub-networks that over-estimate or under-estimate the uncertainty of the teacher supernet, leading to inferior performance of the sub-networks. In this work, we propose to improve the supernet training with a more generalized alpha-divergence. By adaptively selecting the alpha-divergence, we simultaneously prevent the over-estimation or under-estimation of the uncertainty of the teacher model. We apply the proposed alpha-divergence based supernets training to both slimmable neural networks and weight-sharing NAS, and demonstrate significant improvements. Specifically, our discovered model family, AlphaNet, outperforms prior-art models on a wide range of FLOPs regimes, including BigNAS, Once-for-All networks, and AttentiveNAS. We achieve ImageNet top-1 accuracy of 80.0% with only 444M FLOPs. Our code and pretrained models are available at https://github.com/facebookresearch/AlphaNet.

The main goal of post-training quantization (PTQ) is to produced a compressed model whose output distribution is as close to the original model's as possible. To do this tractably, almost all LLM PTQ algorithms quantize linear layers by independently minimizing the immediate activation error. However, this localized objective ignores the effect of subsequent layers, so reducing it does not necessarily give a closer model. In this work, we introduce Yet Another Quantization Algorithm (YAQA), an adaptive rounding algorithm that uses Kronecker-factored approximations of each linear layer's Hessian with respect to the full model KL divergence. YAQA consists of two components: Kronecker-factored sketches of the full layerwise Hessian that can be tractably computed for hundred-billion parameter LLMs, and a quantizer-independent rounding algorithm that uses these sketches and comes with theoretical guarantees. Across a wide range of models and quantizers, YAQA empirically reduces the KL divergence to the original model by approx 30% while achieving state of the art performance on downstream tasks.

Large language models (LLMs) have shown remarkable capabilities in code generation. However, the effects of hallucinations (e.g., output noise) make it particularly challenging for LLMs to generate high-quality code in one pass. In this work, we propose a simple and effective uncertainty-aware selective contrastive decoding (USCD) mechanism to improve the quality of one-pass code generation in LLMs and reduce the impact of output noise. To be specific, we first elaborately designed a negative prompt (namely lame prompt) to output noise by removing input-output examples from the standard few-shot prompt. Our preliminary study shows that the Jensen-Shannon divergence (JS divergence) between token distribution uncertainty and the output noise is relatively low (approximately 0.25), indicating their high relevance. Then, we selectively eliminate output noise induced by lame prompts based on the uncertainty of the prediction distribution from the standard prompt. Notably, our proposed plug-and-play mechanism is an inference-only method, enjoying appealing flexibility. Extensive experiments on widely used benchmarks, e.g., HumanEval, MBPP, and MultiPL-E, upon several LLMs (i.e., Inocder-6b, CodeLlama-7b, WizardCoder-15b, StarCoder, and Llama2-7b), demonstrate that our proposed USCD significantly improves one-pass code generation, with an average pass@1 scores increase of 16.59\%. We will release code and data on GitHub.

We provide a theoretical framework for Reinforcement Learning with Human Feedback (RLHF). Our analysis shows that when the true reward function is linear, the widely used maximum likelihood estimator (MLE) converges under both the Bradley-Terry-Luce (BTL) model and the Plackett-Luce (PL) model. However, we show that when training a policy based on the learned reward model, MLE fails while a pessimistic MLE provides policies with improved performance under certain coverage assumptions. Additionally, we demonstrate that under the PL model, the true MLE and an alternative MLE that splits the K-wise comparison into pairwise comparisons both converge. Moreover, the true MLE is asymptotically more efficient. Our results validate the empirical success of existing RLHF algorithms in InstructGPT and provide new insights for algorithm design. Furthermore, our results unify the problem of RLHF and max-entropy Inverse Reinforcement Learning (IRL), and provide the first sample complexity bound for max-entropy IRL.

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.

Let p denote a generative language model. Let r denote a reward model that returns a scalar that captures the degree at which a draw from p is preferred. The goal of language model alignment is to alter p to a new distribution phi that results in a higher expected reward while keeping phi close to p. A popular alignment method is the KL-constrained reinforcement learning (RL), which chooses a distribution phi_Delta that maximizes E_{phi_{Delta}} r(y) subject to a relative entropy constraint KL(phi_Delta || p) leq Delta. Another simple alignment method is best-of-N, where N samples are drawn from p and one with highest reward is selected. In this paper, we offer a closed-form characterization of the optimal KL-constrained RL solution. We demonstrate that any alignment method that achieves a comparable trade-off between KL divergence and reward must approximate the optimal KL-constrained RL solution in terms of relative entropy. To further analyze the properties of alignment methods, we introduce two simplifying assumptions: we let the language model be memoryless, and the reward model be linear. Although these assumptions may not reflect complex real-world scenarios, they enable a precise characterization of the asymptotic behavior of both the best-of-N alignment, and the KL-constrained RL method, in terms of information-theoretic quantities. We prove that the reward of the optimal KL-constrained RL solution satisfies a large deviation principle, and we fully characterize its rate function. We also show that the rate of growth of the scaled cumulants of the reward is characterized by a proper Renyi cross entropy. Finally, we show that best-of-N is asymptotically equivalent to KL-constrained RL solution by proving that their expected rewards are asymptotically equal, and concluding that the two distributions must be close in KL divergence.

Knowledge distillation~(KD) has proven to be a highly effective approach for enhancing model performance through a teacher-student training scheme. However, most existing distillation methods are designed under the assumption that the teacher and student models belong to the same model family, particularly the hint-based approaches. By using centered kernel alignment (CKA) to compare the learned features between heterogeneous teacher and student models, we observe significant feature divergence. This divergence illustrates the ineffectiveness of previous hint-based methods in cross-architecture distillation. To tackle the challenge in distilling heterogeneous models, we propose a simple yet effective one-for-all KD framework called OFA-KD, which significantly improves the distillation performance between heterogeneous architectures. Specifically, we project intermediate features into an aligned latent space such as the logits space, where architecture-specific information is discarded. Additionally, we introduce an adaptive target enhancement scheme to prevent the student from being disturbed by irrelevant information. Extensive experiments with various architectures, including CNN, Transformer, and MLP, demonstrate the superiority of our OFA-KD framework in enabling distillation between heterogeneous architectures. Specifically, when equipped with our OFA-KD, the student models achieve notable performance improvements, with a maximum gain of 8.0% on the CIFAR-100 dataset and 0.7% on the ImageNet-1K dataset. PyTorch code and checkpoints can be found at https://github.com/Hao840/OFAKD.

Knowledge distillation (KD) is a widely adopted approach for compressing large neural networks by transferring knowledge from a large teacher model to a smaller student model. In the context of large language models, token level KD, typically minimizing the KL divergence between student output distribution and teacher output distribution, has shown strong empirical performance. However, prior work assumes student output distribution and teacher output distribution share the same optimal representation space, a premise that may not hold in many cases. To solve this problem, we propose Delta Knowledge Distillation (Delta-KD), a novel extension of token level KD that encourages the student to approximate an optimal representation space by explicitly preserving the distributional shift Delta introduced during the teacher's supervised finetuning (SFT). Empirical results on ROUGE metrics demonstrate that Delta KD substantially improves student performance while preserving more of the teacher's knowledge.

The ability to achieve long-term goals is a key challenge in the current development of large language models (LLMs). To address this, pre-trained LLMs can be fine-tuned with reinforcement learning (RL) to explore solutions that optimize a given goal. However, exploration with LLMs is difficult, as a balance has to be struck between discovering new solutions and staying close enough to the pre-trained model, so as not to degrade basic capabilities. This is typically controlled with a Kullback-Leibler (KL) penalty. In this paper, we investigate the exploration dynamics of a small language model on a simple arithmetic task. We show how varying degrees of pre-training influence exploration and demonstrate the importance of "critical tokens" which have a dramatic impact on the final outcome. Consequently, we introduce a simple modification to the KL penalty that favors exploration on critical tokens, increasing the efficiency of the RL fine-tuning stage.

Assigning importance weights to adversarial data has achieved great success in training adversarially robust networks under limited model capacity. However, existing instance-reweighted adversarial training (AT) methods heavily depend on heuristics and/or geometric interpretations to determine those importance weights, making these algorithms lack rigorous theoretical justification/guarantee. Moreover, recent research has shown that adversarial training suffers from a severe non-uniform robust performance across the training distribution, e.g., data points belonging to some classes can be much more vulnerable to adversarial attacks than others. To address both issues, in this paper, we propose a novel doubly-robust instance reweighted AT framework, which allows to obtain the importance weights via exploring distributionally robust optimization (DRO) techniques, and at the same time boosts the robustness on the most vulnerable examples. In particular, our importance weights are obtained by optimizing the KL-divergence regularized loss function, which allows us to devise new algorithms with a theoretical convergence guarantee. Experiments on standard classification datasets demonstrate that our proposed approach outperforms related state-of-the-art baseline methods in terms of average robust performance, and at the same time improves the robustness against attacks on the weakest data points. Codes will be available soon.

Machine unlearning -- efficiently removing the effect of a small "forget set" of training data on a pre-trained machine learning model -- has recently attracted significant research interest. Despite this interest, however, recent work shows that existing machine unlearning techniques do not hold up to thorough evaluation in non-convex settings. In this work, we introduce a new machine unlearning technique that exhibits strong empirical performance even in such challenging settings. Our starting point is the perspective that the goal of unlearning is to produce a model whose outputs are statistically indistinguishable from those of a model re-trained on all but the forget set. This perspective naturally suggests a reduction from the unlearning problem to that of data attribution, where the goal is to predict the effect of changing the training set on a model's outputs. Thus motivated, we propose the following meta-algorithm, which we call Datamodel Matching (DMM): given a trained model, we (a) use data attribution to predict the output of the model if it were re-trained on all but the forget set points; then (b) fine-tune the pre-trained model to match these predicted outputs. In a simple convex setting, we show how this approach provably outperforms a variety of iterative unlearning algorithms. Empirically, we use a combination of existing evaluations and a new metric based on the KL-divergence to show that even in non-convex settings, DMM achieves strong unlearning performance relative to existing algorithms. An added benefit of DMM is that it is a meta-algorithm, in the sense that future advances in data attribution translate directly into better unlearning algorithms, pointing to a clear direction for future progress in unlearning.

The present work explores the theoretical limits of Machine Learning (ML) within the framework of Kolmogorov's theory of Algorithmic Probability, which clarifies the notion of entropy as Expected Kolmogorov Complexity and formalizes other fundamental concepts such as Occam's razor via Levin's Universal Distribution. As a fundamental application, we develop Maximum Entropy methods that allow us to derive the Erdos--Kac Law in Probabilistic Number Theory, and establish the impossibility of discovering a formula for primes using Machine Learning via the Prime Coding Theorem.

Knowledge Distillation (KD) has emerged as a promising approach for transferring knowledge from a larger, more complex teacher model to a smaller student model. Traditionally, KD involves training the student to mimic the teacher's output probabilities, while more advanced techniques have explored guiding the student to adopt the teacher's internal representations. Despite its widespread success, the performance of KD in binary classification and few-class problems has been less satisfactory. This is because the information about the teacher model's generalization patterns scales directly with the number of classes. Moreover, several sophisticated distillation methods may not be universally applicable or effective for data types beyond Computer Vision. Consequently, effective distillation techniques remain elusive for a range of key real-world applications, such as sentiment analysis, search query understanding, and advertisement-query relevance assessment. Taking these observations into account, we introduce a novel method for distilling knowledge from the teacher's model representations, which we term Learning Embedding Linear Projections (LELP). Inspired by recent findings about the structure of final-layer representations, LELP works by identifying informative linear subspaces in the teacher's embedding space, and splitting them into pseudo-subclasses. The student model is then trained to replicate these pseudo-classes. Our experimental evaluation on large-scale NLP benchmarks like Amazon Reviews and Sentiment140 demonstrate the LELP is consistently competitive with, and typically superior to, existing state-of-the-art distillation algorithms for binary and few-class problems, where most KD methods suffer.

Audio language models have recently emerged as a promising approach for various audio generation tasks, relying on audio tokenizers to encode waveforms into sequences of discrete symbols. Audio tokenization often poses a necessary compromise between code bitrate and reconstruction accuracy. When dealing with low-bitrate audio codes, language models are constrained to process only a subset of the information embedded in the audio, which in turn restricts their generative capabilities. To circumvent these issues, we propose encoding audio as vector sequences in continuous space mathbb R^d and autoregressively generating these sequences using a decoder-only diffusion transformer (ARDiT). Our findings indicate that ARDiT excels in zero-shot text-to-speech and exhibits performance that compares to or even surpasses that of state-of-the-art models. High-bitrate continuous speech representation enables almost flawless reconstruction, allowing our model to achieve nearly perfect speech editing. Our experiments reveal that employing Integral Kullback-Leibler (IKL) divergence for distillation at each autoregressive step significantly boosts the perceived quality of the samples. Simultaneously, it condenses the iterative sampling process of the diffusion model into a single step. Furthermore, ARDiT can be trained to predict several continuous vectors in one step, significantly reducing latency during sampling. Impressively, one of our models can generate 170 ms of 24 kHz speech per evaluation step with minimal degradation in performance. Audio samples are available at http://ardit-tts.github.io/ .

AI for partial differential equations (PDEs) has garnered significant attention, particularly with the emergence of Physics-informed neural networks (PINNs). The recent advent of Kolmogorov-Arnold Network (KAN) indicates that there is potential to revisit and enhance the previously MLP-based PINNs. Compared to MLPs, KANs offer interpretability and require fewer parameters. PDEs can be described in various forms, such as strong form, energy form, and inverse form. While mathematically equivalent, these forms are not computationally equivalent, making the exploration of different PDE formulations significant in computational physics. Thus, we propose different PDE forms based on KAN instead of MLP, termed Kolmogorov-Arnold-Informed Neural Network (KINN). We systematically compare MLP and KAN in various numerical examples of PDEs, including multi-scale, singularity, stress concentration, nonlinear hyperelasticity, heterogeneous, and complex geometry problems. Our results demonstrate that KINN significantly outperforms MLP in terms of accuracy and convergence speed for numerous PDEs in computational solid mechanics, except for the complex geometry problem. This highlights KINN's potential for more efficient and accurate PDE solutions in AI for PDEs.

Multi-task learning (MTL) is often achieved by merging datasets before fine-tuning, but the growing availability of fine-tuned models has led to new approaches such as model merging via task arithmetic. A major challenge in this setting is task interference, which worsens as the number of tasks increases. We propose a method that merges models trained on different tasks into a single model, maintaining strong performance across all tasks. Our approach leverages Jensen-Shannon divergence to guide the merging process without requiring additional labelled data, and automatically balances task importance. Unlike existing methods, our approach remains robust as the number of tasks grows and consistently outperforms prior work.

At initialization, artificial neural networks (ANNs) are equivalent to Gaussian processes in the infinite-width limit, thus connecting them to kernel methods. We prove that the evolution of an ANN during training can also be described by a kernel: during gradient descent on the parameters of an ANN, the network function f_theta (which maps input vectors to output vectors) follows the kernel gradient of the functional cost (which is convex, in contrast to the parameter cost) w.r.t. a new kernel: the Neural Tangent Kernel (NTK). This kernel is central to describe the generalization features of ANNs. While the NTK is random at initialization and varies during training, in the infinite-width limit it converges to an explicit limiting kernel and it stays constant during training. This makes it possible to study the training of ANNs in function space instead of parameter space. Convergence of the training can then be related to the positive-definiteness of the limiting NTK. We prove the positive-definiteness of the limiting NTK when the data is supported on the sphere and the non-linearity is non-polynomial. We then focus on the setting of least-squares regression and show that in the infinite-width limit, the network function f_theta follows a linear differential equation during training. The convergence is fastest along the largest kernel principal components of the input data with respect to the NTK, hence suggesting a theoretical motivation for early stopping. Finally we study the NTK numerically, observe its behavior for wide networks, and compare it to the infinite-width limit.

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 introduce KALL-E, a novel autoregressive (AR) language modeling approach with next-distribution prediction for text-to-speech (TTS) synthesis. Unlike existing methods, KALL-E directly models and predicts the continuous speech distribution conditioned on text without relying on VAE- or diffusion-based components. Specifically, we use WaveVAE to extract continuous speech distributions from waveforms instead of using discrete speech tokens. A single AR language model predicts these continuous speech distributions from text, with a Kullback-Leibler divergence loss as the constraint. Experimental results show that KALL-E outperforms open-source implementations of YourTTS, VALL-E, NaturalSpeech 2, and CosyVoice in terms of naturalness and speaker similarity in zero-shot TTS scenarios. Moreover, KALL-E demonstrates exceptional zero-shot capabilities in emotion and accent cloning. Importantly, KALL-E presents a more straightforward and effective paradigm for using continuous speech representations in TTS. Audio samples are available at: https://zxf-icpc.github.io/kalle/.

Masked diffusion models have demonstrated competitive results on various tasks including language generation. However, due to its iterative refinement process, the inference is often bottlenecked by slow and static sampling speed. To overcome this problem, we introduce `KL-Adaptive Stability Sampling' (KLASS), a fast yet effective sampling method that exploits token-level KL divergence to identify stable, high-confidence predictions. By unmasking multiple tokens in each iteration without any additional model training, our approach speeds up generation significantly while maintaining sample quality. On reasoning benchmarks, KLASS achieves up to 2.78times wall-clock speedups while improving performance over standard greedy decoding, attaining state-of-the-art results among diffusion-based samplers. We further validate KLASS across diverse domains, including text, image, and molecular generation, showing its effectiveness as a broadly applicable sampler across different models.

Knowledge distillation plays a key role in compressing the Large Language Models (LLMs), which boosts a small-size student model under large teacher models' guidance. However, existing LLM distillation methods overly rely on student-generated outputs, which may introduce generation errors and misguide the distillation process. Moreover, the distillation loss functions introduced in previous art struggle to align the most informative part due to the complex distribution of LLMs' outputs. To address these problems, we propose a multi-granularity semantic revision method for LLM distillation. At the sequence level, we propose a sequence correction and re-generation (SCRG) strategy. SCRG first calculates the semantic cognitive difference between the teacher and student to detect the error token, then corrects it with the teacher-generated one, and re-generates the sequence to reduce generation errors and enhance generation diversity. At the token level, we design a distribution adaptive clipping Kullback-Leibler (DAC-KL) loss as the distillation objective function. DAC-KL loss exploits a learnable sub-network to adaptively extract semantically dense areas from the teacher's output, avoiding the interference of redundant information in the distillation process. Finally, at the span level, we leverage the span priors of a sequence to compute the probability correlations within spans, and constrain the teacher and student's probability correlations to be consistent, further enhancing the transfer of semantic information. Extensive experiments across different model families with parameters ranging from 0.1B to 13B demonstrate the superiority of our method compared to existing methods.

Post-training processes are essential phases in grounding pre-trained language models to real-world tasks, with learning from demonstrations or preference signals playing a crucial role in this adaptation. We present a unified theoretical framework bridging Supervised Fine-Tuning (SFT) and preference learning in Large Language Model (LLM) post-training. Through rigorous mathematical derivation, we demonstrate that both SFT and preference learning methods like Direct Preference Optimization (DPO) operate within the same optimal policy-reward subspace, with SFT representing a special case of implicit reward learning. Our analysis reveals a critical limitation in conventional SFT: the KL divergence term in distribution matching becomes constant with respect to the policy during optimization, failing to constrain model updates. To address this, we propose a simple yet effective learning rate reduction approach that yields significant performance improvements (up to 25\% relative gain and 6\% absolute win rate increase in instruction following tasks. Additionally, we derive alternative SFT objectives from various f-divergence functions that preserve the KL term during optimization, further enhancing post-DPO model performance. Finally, we extend the theoretical relationship between LLM logits and Q-functions from preference learning to the SFT context, providing mathematical derivations and experimental validation.

The upscaling of Large Language Models (LLMs) has yielded impressive advances in natural language processing, yet it also poses significant deployment challenges. Weight quantization has emerged as a widely embraced solution to reduce memory and computational demands. This paper introduces BitDistiller, a framework that synergizes Quantization-Aware Training (QAT) with Knowledge Distillation (KD) to boost the performance of LLMs at ultra-low precisions (sub-4-bit). Specifically, BitDistiller first incorporates a tailored asymmetric quantization and clipping technique to maximally preserve the fidelity of quantized weights, and then proposes a novel Confidence-Aware Kullback-Leibler Divergence (CAKLD) objective, which is employed in a self-distillation manner to enable faster convergence and superior model performance. Empirical evaluations demonstrate that BitDistiller significantly surpasses existing methods in both 3-bit and 2-bit configurations on general language understanding and complex reasoning benchmarks. Notably, BitDistiller is shown to be more cost-effective, demanding fewer data and training resources. The code is available at https://github.com/DD-DuDa/BitDistiller.

Intelligent tutoring systems optimize the selection and timing of learning materials to enhance understanding and long-term retention. This requires estimates of both the learner's progress (''knowledge tracing''; KT), and the prerequisite structure of the learning domain (''knowledge mapping''). While recent deep learning models achieve high KT accuracy, they do so at the expense of the interpretability of psychologically-inspired models. In this work, we present a solution to this trade-off. PSI-KT is a hierarchical generative approach that explicitly models how both individual cognitive traits and the prerequisite structure of knowledge influence learning dynamics, thus achieving interpretability by design. Moreover, by using scalable Bayesian inference, PSI-KT targets the real-world need for efficient personalization even with a growing body of learners and learning histories. Evaluated on three datasets from online learning platforms, PSI-KT achieves superior multi-step predictive accuracy and scalable inference in continual-learning settings, all while providing interpretable representations of learner-specific traits and the prerequisite structure of knowledge that causally supports learning. In sum, predictive, scalable and interpretable knowledge tracing with solid knowledge mapping lays a key foundation for effective personalized learning to make education accessible to a broad, global audience.

We present a faster algorithm to generate a warm start for sampling an arbitrary logconcave density specified by an evaluation oracle, leading to the first sub-cubic sampling algorithms for inputs in (near-)isotropic position. A long line of prior work incurred a warm-start penalty of at least linear in the dimension, hitting a cubic barrier, even for the special case of uniform sampling from convex bodies. Our improvement relies on two key ingredients of independent interest. (1) We show how to sample given a warm start in weaker notions of distance, in particular q-R\'enyi divergence for q=mathcal{O}(1), whereas previous analyses required stringent infty-R\'enyi divergence (with the exception of Hit-and-Run, whose known mixing time is higher). This marks the first improvement in the required warmness since Lov\'asz and Simonovits (1991). (2) We refine and generalize the log-Sobolev inequality of Lee and Vempala (2018), originally established for isotropic logconcave distributions in terms of the diameter of the support, to logconcave distributions in terms of a geometric average of the support diameter and the largest eigenvalue of the covariance matrix.

Policy-based optimizations are widely adopted today for the training and alignment of language models, where one of the most recent and effective approaches is Group-relative Policy Optimization (GRPO). In this paper, we reveals and analyze two major limitations of GRPO: (i) tokens frequently appear in completions with both positive and negative rewards, leading to conflicting gradient updates that can reduce their output probability, even though can be essential for maintaining proper structure; (ii) negatively rewarded completions may penalize confident responses and shift model decisions toward unlikely tokens, progressively flattening the output distribution and degrading learning. To address these issues and provide a more stable and effective policy optimization strategy, we introduce GTPO (Group-relative Trajectory-based Policy Optimization), which identifies conflict tokens, tokens appearing in the same position across completions with opposite rewards, protects them by skipping negative updates, while amplifying positive ones. To further prevent policy collapse, GTPO filters out completions whose entropy exceeds a provable threshold. Unlike GRPO, GTPO does not rely on KL-divergence regularization, eliminating the need for a reference model during training, while still ensuring greater training stability and improved performance, validated through multiple experiments on GSM8K, MATH and AIME 2024 benchmarks.

Comparison of fine-tuning models with reinforcement learning (RL) and supervised fine-tuning (SFT) reveals that, despite similar performance at a new task, RL preserves prior knowledge and capabilities significantly better. We find that the degree of forgetting is determined by the distributional shift, measured as the KL-divergence between the fine-tuned and base policy evaluated on the new task. Our analysis reveals that on-policy RL is implicitly biased towards KL-minimal solutions among the many that solve the new task, whereas SFT can converge to distributions arbitrarily far from the base model. We validate these findings through experiments with large language models and robotic foundation models and further provide theoretical justification for why on-policy RL updates lead to a smaller KL change. We term this principle RL's Razor: among all ways to solve a new task, RL prefers those closest in KL to the original model.

As an adaptive method, Shampoo employs a structured second-moment estimation, and its effectiveness has attracted growing attention. Prior work has primarily analyzed its estimation scheme through the Frobenius norm. Motivated by the natural connection between the second moment and a covariance matrix, we propose studying Shampoo's estimation as covariance estimation through the lens of Kullback-Leibler (KL) minimization. This alternative perspective reveals a previously hidden limitation, motivating improvements to Shampoo's design. Building on this insight, we develop a practical estimation scheme, termed KL-Shampoo, that eliminates Shampoo's reliance on Adam for stabilization, thereby removing the additional memory overhead introduced by Adam. Preliminary results show that KL-Shampoo improves Shampoo's performance, enabling it to stabilize without Adam and even outperform its Adam-stabilized variant, SOAP, in neural network pretraining.

Knowledge Distillation (KD) trains a smaller student model using a large, pre-trained teacher model, with temperature as a key hyperparameter controlling the softness of output probabilities. Traditional methods use a fixed temperature throughout training, which is suboptimal. Moreover, architectural differences between teacher and student often result in mismatched logit magnitudes. We demonstrate that students benefit from softer probabilities early in training but require sharper probabilities in later stages. We introduce Dynamic Temperature Scheduler (DTS), which adjusts temperature dynamically based on the cross-entropy loss gap between teacher and student. To our knowledge, this is the first temperature scheduling method that adapts based on the divergence between teacher and student distributions. Our method integrates seamlessly with existing KD frameworks. We validate DTS across multiple KD strategies on vision (CIFAR-100, Tiny-ImageNet) and NLP tasks (GLUE, Dolly, SelfIns, UnNI, S-NI), consistently outperforming static-temperature baselines. Code is available at https://github.com/Sibgat-Ul/DTS.

Knowledge distillation (KD) is a common approach to compress a teacher model to reduce its inference cost and memory footprint, by training a smaller student model. However, in the context of autoregressive language models (LMs), we empirically find that larger teacher LMs might dramatically result in a poorer student. In response to this problem, we conduct a series of analyses and reveal that different tokens have different teaching modes, neglecting which will lead to performance degradation. Motivated by this, we propose a simple yet effective adaptive teaching approach (ATKD) to improve the KD. The core of ATKD is to reduce rote learning and make teaching more diverse and flexible. Extensive experiments on 8 LM tasks show that, with the help of ATKD, various baseline KD methods can achieve consistent and significant performance gains (up to +3.04% average score) across all model types and sizes. More encouragingly, ATKD can improve the student model generalization effectively.

Knowledge distillation (KD) is a general neural network training approach that uses a teacher model to guide the student model. Existing works mainly study KD from the network output side (e.g., trying to design a better KD loss function), while few have attempted to understand it from the input side. Especially, its interplay with data augmentation (DA) has not been well understood. In this paper, we ask: Why do some DA schemes (e.g., CutMix) inherently perform much better than others in KD? What makes a "good" DA in KD? Our investigation from a statistical perspective suggests that a good DA scheme should reduce the covariance of the teacher-student cross-entropy. A practical metric, the stddev of teacher's mean probability (T. stddev), is further presented and well justified empirically. Besides the theoretical understanding, we also introduce a new entropy-based data-mixing DA scheme, CutMixPick, to further enhance CutMix. Extensive empirical studies support our claims and demonstrate how we can harvest considerable performance gains simply by using a better DA scheme in knowledge distillation.

Consider a random uniform sample of n points in a compact region A of Euclidean d-space, d geq 2, with a smooth or (when d=2) polygonal boundary. Fix k bf N. Let T_{n,k} be the threshold r at which the geometric graph on these n vertices with distance parameter r becomes k-connected. We show that if d=2 then n (pi/|A|) T_{n,1}^2 - log n is asymptotically standard Gumbel. For (d,k) neq (2,1), it is n (theta_d/|A|) T_{n,k}^d - (2-2/d) log n - (4-2k-2/d) log log n that converges in distribution to a nondegenerate limit, where theta_d is the volume of the unit ball. The limit is Gumbel with scale parameter 2 except when (d,k)=(2,2) where the limit is two component extreme value distributed. The different cases reflect the fact that boundary effects are more more important in some cases than others. We also give similar results for the largest k-nearest neighbour link U_{n,k} in the sample, and show T_{n,k}=U_{n,k} with high probability. We provide estimates on rates of convergence and give similar results for Poisson samples in A. Finally, we give similar results even for non-uniform samples, with a less explicit sequence of centring constants.

Gradient methods have become mainstream techniques for Bi-Level Optimization (BLO) in learning fields. The validity of existing works heavily rely on either a restrictive Lower- Level Strong Convexity (LLSC) condition or on solving a series of approximation subproblems with high accuracy or both. In this work, by averaging the upper and lower level objectives, we propose a single loop Bi-level Averaged Method of Multipliers (sl-BAMM) for BLO that is simple yet efficient for large-scale BLO and gets rid of the limited LLSC restriction. We further provide non-asymptotic convergence analysis of sl-BAMM towards KKT stationary points, and the comparative advantage of our analysis lies in the absence of strong gradient boundedness assumption, which is always required by others. Thus our theory safely captures a wider variety of applications in deep learning, especially where the upper-level objective is quadratic w.r.t. the lower-level variable. Experimental results demonstrate the superiority of our method.

Appropriately designing the proposal kernel of particle filters is an issue of significant importance, since a bad choice may lead to deterioration of the particle sample and, consequently, waste of computational power. In this paper we introduce a novel algorithm adaptively approximating the so-called optimal proposal kernel by a mixture of integrated curved exponential distributions with logistic weights. This family of distributions, referred to as mixtures of experts, is broad enough to be used in the presence of multi-modality or strongly skewed distributions. The mixtures are fitted, via online-EM methods, to the optimal kernel through minimisation of the Kullback-Leibler divergence between the auxiliary target and instrumental distributions of the particle filter. At each iteration of the particle filter, the algorithm is required to solve only a single optimisation problem for the whole particle sample, yielding an algorithm with only linear complexity. In addition, we illustrate in a simulation study how the method can be successfully applied to optimal filtering in nonlinear state-space models.

Post-training quantization (PTQ) has emerged as a widely adopted technique for compressing and accelerating Large Language Models (LLMs). The major challenge in LLM quantization is that uneven and heavy-tailed data distributions can expand the quantization range, thereby reducing bit precision for most values. Recent methods attempt to eliminate outliers and balance inter-channel differences by employing linear transformations; however, they remain heuristic and are often overlook optimizing the data distribution across the entire quantization space.In this paper, we introduce Quantization Space Utilization Rate (QSUR), a novel metric that effectively assesses the quantizability of transformed data by measuring the space utilization of the data in the quantization space. We complement QSUR with mathematical derivations that examine the effects and limitations of various transformations, guiding our development of Orthogonal and Scaling Transformation-based Quantization (OSTQuant). OSQuant employs a learnable equivalent transformation, consisting of an orthogonal transformation and a scaling transformation, to optimize the distributions of weights and activations across the entire quantization space. Futhermore, we propose the KL-Top loss function, designed to mitigate noise during optimization while retaining richer semantic information within the limited calibration data imposed by PTQ. OSTQuant outperforms existing work on various LLMs and benchmarks. In the W4-only setting, it retains 99.5\% of the floating-point accuracy. In the more challenging W4A4KV4 configuration, OSTQuant reduces the performance gap by 32\% on the LLaMA-3-8B model compared to state-of-the-art methods. https://github.com/BrotherHappy/OSTQuant{https://github.com/BrotherHappy/OSTQuant}.

We propose (G)I-DLE, a new approach to constrained decoding that leverages KL divergence minimization to preserve the intrinsic conditional probability distribution of autoregressive language models while excluding undesirable tokens. Unlike conventional methods that naively set banned tokens' logits to -infty, which can distort the conversion from raw logits to posterior probabilities and increase output variance, (G)I-DLE re-normalizes the allowed token probabilities to minimize such distortion. We validate our method on the K2-Eval dataset, specifically designed to assess Korean language fluency, logical reasoning, and cultural appropriateness. Experimental results on Qwen2.5 models (ranging from 1.5B to 14B) demonstrate that G-IDLE not only boosts mean evaluation scores but also substantially reduces the variance of output quality.

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.

Inspired by the Kolmogorov-Arnold representation theorem, we propose Kolmogorov-Arnold Networks (KANs) as promising alternatives to Multi-Layer Perceptrons (MLPs). While MLPs have fixed activation functions on nodes ("neurons"), KANs have learnable activation functions on edges ("weights"). KANs have no linear weights at all -- every weight parameter is replaced by a univariate function parametrized as a spline. We show that this seemingly simple change makes KANs outperform MLPs in terms of accuracy and interpretability. For accuracy, much smaller KANs can achieve comparable or better accuracy than much larger MLPs in data fitting and PDE solving. Theoretically and empirically, KANs possess faster neural scaling laws than MLPs. For interpretability, KANs can be intuitively visualized and can easily interact with human users. Through two examples in mathematics and physics, KANs are shown to be useful collaborators helping scientists (re)discover mathematical and physical laws. In summary, KANs are promising alternatives for MLPs, opening opportunities for further improving today's deep learning models which rely heavily on MLPs.

The asymptotically precise estimation of the generalization of kernel methods has recently received attention due to the parallels between neural networks and their associated kernels. However, prior works derive such estimates for training by kernel ridge regression (KRR), whereas neural networks are typically trained with gradient descent (GD). In the present work, we consider the training of kernels with a family of spectral algorithms specified by profile h(lambda), and including KRR and GD as special cases. Then, we derive the generalization error as a functional of learning profile h(lambda) for two data models: high-dimensional Gaussian and low-dimensional translation-invariant model. Under power-law assumptions on the spectrum of the kernel and target, we use our framework to (i) give full loss asymptotics for both noisy and noiseless observations (ii) show that the loss localizes on certain spectral scales, giving a new perspective on the KRR saturation phenomenon (iii) conjecture, and demonstrate for the considered data models, the universality of the loss w.r.t. non-spectral details of the problem, but only in case of noisy observation.

Reinforcement learning from human feedback (RLHF) aligns large language models (LLMs) by encouraging their generations to have high rewards, using a reward model trained on human preferences. To prevent the forgetting of pre-trained knowledge, RLHF usually incorporates a KL regularization; this forces the policy to remain close to its supervised fine-tuned initialization, though it hinders the reward optimization. To tackle the trade-off between KL and reward, in this paper we introduce a novel alignment strategy named Weight Averaged Rewarded Policies (WARP). WARP merges policies in the weight space at three distinct stages. First, it uses the exponential moving average of the policy as a dynamic anchor in the KL regularization. Second, it applies spherical interpolation to merge independently fine-tuned policies into a new enhanced one. Third, it linearly interpolates between this merged model and the initialization, to recover features from pre-training. This procedure is then applied iteratively, with each iteration's final model used as an advanced initialization for the next, progressively refining the KL-reward Pareto front, achieving superior rewards at fixed KL. Experiments with GEMMA policies validate that WARP improves their quality and alignment, outperforming other open-source LLMs.

Risk-averse reinforcement learning finds application in various high-stakes fields. Unlike classical reinforcement learning, which aims to maximize expected returns, risk-averse agents choose policies that minimize risk, occasionally sacrificing expected value. These preferences can be framed through utility theory. We focus on the specific case of the exponential utility function, where we can derive the Bellman equations and employ various reinforcement learning algorithms with few modifications. However, these methods suffer from numerical instability due to the need for exponent computation throughout the process. To address this, we introduce a numerically stable and mathematically sound loss function based on the Itakura-Saito divergence for learning state-value and action-value functions. We evaluate our proposed loss function against established alternatives, both theoretically and empirically. In the experimental section, we explore multiple financial scenarios, some with known analytical solutions, and show that our loss function outperforms the alternatives.

In the misspecified kernel ridge regression problem, researchers usually assume the underground true function f_{rho}^{*} in [H]^{s}, a less-smooth interpolation space of a reproducing kernel Hilbert space (RKHS) H for some sin (0,1). The existing minimax optimal results require |f_{rho}^{*}|_{L^{infty}}<infty which implicitly requires s > alpha_{0} where alpha_{0}in (0,1) is the embedding index, a constant depending on H. Whether the KRR is optimal for all sin (0,1) is an outstanding problem lasting for years. In this paper, we show that KRR is minimax optimal for any sin (0,1) when the H is a Sobolev RKHS.

As the scale of training corpora for large language models (LLMs) grows, model developers become increasingly reluctant to disclose details on their data. This lack of transparency poses challenges to scientific evaluation and ethical deployment. Recently, pretraining data detection approaches, which infer whether a given text was part of an LLM's training data through black-box access, have been explored. The Min-K\% Prob method, which has achieved state-of-the-art results, assumes that a non-training example tends to contain a few outlier words with low token probabilities. However, the effectiveness may be limited as it tends to misclassify non-training texts that contain many common words with high probabilities predicted by LLMs. To address this issue, we introduce a divergence-based calibration method, inspired by the divergence-from-randomness concept, to calibrate token probabilities for pretraining data detection. We compute the cross-entropy (i.e., the divergence) between the token probability distribution and the token frequency distribution to derive a detection score. We have developed a Chinese-language benchmark, PatentMIA, to assess the performance of detection approaches for LLMs on Chinese text. Experimental results on English-language benchmarks and PatentMIA demonstrate that our proposed method significantly outperforms existing methods. Our code and PatentMIA benchmark are available at https://github.com/zhang-wei-chao/DC-PDD.

Compression is at the heart of intelligence. A theoretically optimal way to compress any sequence of data is to find the shortest program that outputs that sequence and then halts. However, such 'Kolmogorov compression' is uncomputable, and code generating LLMs struggle to approximate this theoretical ideal, as it requires reasoning, planning and search capabilities beyond those of current models. In this work, we introduce the KoLMogorov-Test (KT), a compression-as-intelligence test for code generating LLMs. In KT a model is presented with a sequence of data at inference time, and asked to generate the shortest program that produces the sequence. We identify several benefits of KT for both evaluation and training: an essentially infinite number of problem instances of varying difficulty is readily available, strong baselines already exist, the evaluation metric (compression) cannot be gamed, and pretraining data contamination is highly unlikely. To evaluate current models, we use audio, text, and DNA data, as well as sequences produced by random synthetic programs. Current flagship models perform poorly - both GPT4-o and Llama-3.1-405B struggle on our natural and synthetic sequences. On our synthetic distribution, we are able to train code generation models with lower compression rates than previous approaches. Moreover, we show that gains on synthetic data generalize poorly to real data, suggesting that new innovations are necessary for additional gains on KT.

We present a generalization of the proximal operator defined through a convex combination of convex objectives, where the coefficients are updated in a minimax fashion. We prove that this new operator is Bregman firmly nonexpansive with respect to a Bregman divergence that combines Euclidean and information geometries.

It is believed that in knowledge distillation (KD), the role of the teacher is to provide an estimate for the unknown Bayes conditional probability distribution (BCPD) to be used in the student training process. Conventionally, this estimate is obtained by training the teacher using maximum log-likelihood (MLL) method. To improve this estimate for KD, in this paper we introduce the concept of conditional mutual information (CMI) into the estimation of BCPD and propose a novel estimator called the maximum CMI (MCMI) method. Specifically, in MCMI estimation, both the log-likelihood and CMI of the teacher are simultaneously maximized when the teacher is trained. Through Eigen-CAM, it is further shown that maximizing the teacher's CMI value allows the teacher to capture more contextual information in an image cluster. Via conducting a thorough set of experiments, we show that by employing a teacher trained via MCMI estimation rather than one trained via MLL estimation in various state-of-the-art KD frameworks, the student's classification accuracy consistently increases, with the gain of up to 3.32\%. This suggests that the teacher's BCPD estimate provided by MCMI method is more accurate than that provided by MLL method. In addition, we show that such improvements in the student's accuracy are more drastic in zero-shot and few-shot settings. Notably, the student's accuracy increases with the gain of up to 5.72\% when 5\% of the training samples are available to the student (few-shot), and increases from 0\% to as high as 84\% for an omitted class (zero-shot). The code is available at https://github.com/iclr2024mcmi/ICLRMCMI.

Multimodal Large Language Models (MLLMs) have gained significant traction for their ability to process diverse input data types and generate coherent, contextually relevant outputs across various applications. While supervised fine-tuning (SFT) has been the predominant approach to enhance MLLM capabilities in task-specific optimization, it often falls short in fostering crucial generalized reasoning abilities. Although reinforcement learning (RL) holds great promise in overcoming these limitations, it encounters two significant challenges: (1) its generalized capacities in multimodal tasks remain largely unexplored, and (2) its training constraints, including the constant Kullback-Leibler divergence or the clamp strategy, often result in suboptimal bottlenecks. To address these challenges, we propose OThink-MR1, an advanced MLLM equipped with profound comprehension and reasoning capabilities across multimodal tasks. Specifically, we introduce Group Relative Policy Optimization with a dynamic Kullback-Leibler strategy (GRPO-D), which markedly enhances reinforcement learning (RL) performance. For Qwen2-VL-2B-Instruct, GRPO-D achieves a relative improvement of more than 5.72% over SFT and more than 13.59% over GRPO in same-task evaluation on two adapted datasets. Furthermore, GRPO-D demonstrates remarkable cross-task generalization capabilities, with an average relative improvement of more than 61.63% over SFT in cross-task evaluation. These results highlight that the MLLM trained with GRPO-D on one multimodal task can be effectively transferred to another task, underscoring the superior generalized reasoning capabilities of our proposed OThink-MR1 model.

While multilingual neural machine translation has achieved great success, it suffers from the off-target issue, where the translation is in the wrong language. This problem is more pronounced on zero-shot translation tasks. In this work, we find that failing in encoding discriminative target language signal will lead to off-target and a closer lexical distance (i.e., KL-divergence) between two languages' vocabularies is related with a higher off-target rate. We also find that solely isolating the vocab of different languages in the decoder can alleviate the problem. Motivated by the findings, we propose Language Aware Vocabulary Sharing (LAVS), a simple and effective algorithm to construct the multilingual vocabulary, that greatly alleviates the off-target problem of the translation model by increasing the KL-divergence between languages. We conduct experiments on a multilingual machine translation benchmark in 11 languages. Experiments show that the off-target rate for 90 translation tasks is reduced from 29\% to 8\%, while the overall BLEU score is improved by an average of 1.9 points without extra training cost or sacrificing the supervised directions' performance. We release the code at https://github.com/PKUnlp-icler/Off-Target-MNMT for reproduction.

Optimal transport (OT) theory has been been used in machine learning to study and characterize maps that can push-forward efficiently a probability measure onto another. Recent works have drawn inspiration from Brenier's theorem, which states that when the ground cost is the squared-Euclidean distance, the ``best'' map to morph a continuous measure in P(Rd) into another must be the gradient of a convex function. To exploit that result, [Makkuva+ 2020, Korotin+2020] consider maps T=nabla f_theta, where f_theta is an input convex neural network (ICNN), as defined by Amos+2017, and fit theta with SGD using samples. Despite their mathematical elegance, fitting OT maps with ICNNs raises many challenges, due notably to the many constraints imposed on theta; the need to approximate the conjugate of f_theta; or the limitation that they only work for the squared-Euclidean cost. More generally, we question the relevance of using Brenier's result, which only applies to densities, to constrain the architecture of candidate maps fitted on samples. Motivated by these limitations, we propose a radically different approach to estimating OT maps: Given a cost c and a reference measure rho, we introduce a regularizer, the Monge gap M^c_{rho}(T) of a map T. That gap quantifies how far a map T deviates from the ideal properties we expect from a c-OT map. In practice, we drop all architecture requirements for T and simply minimize a distance (e.g., the Sinkhorn divergence) between Tsharpmu and nu, regularized by M^c_rho(T). We study M^c_{rho}, and show how our simple pipeline outperforms significantly other baselines in practice.

Modeling the time-dependent evolution of electron density is essential for understanding quantum mechanical behaviors of condensed matter and enabling predictive simulations in spectroscopy, photochemistry, and ultrafast science. Yet, while machine learning methods have advanced static density prediction, modeling its spatiotemporal dynamics remains largely unexplored. In this work, we introduce a generative framework that combines a 3D convolutional autoencoder with a latent diffusion model (LDM) to learn electron density trajectories from ab-initio molecular dynamics (AIMD) simulations. Our method encodes electron densities into a compact latent space and predicts their future states by sampling from the learned conditional distribution, enabling stable long-horizon rollouts without drift or collapse. To preserve statistical fidelity, we incorporate a scaled Jensen-Shannon divergence regularization that aligns generated and reference density distributions. On AIMD trajectories of liquid lithium at 800 K, our model accurately captures both the spatial correlations and the log-normal-like statistical structure of the density. The proposed framework has the potential to accelerate the simulation of quantum dynamics and overcome key challenges faced by current spatiotemporal machine learning methods as surrogates of quantum mechanical simulators.

Speech produced by human vocal apparatus conveys substantial non-semantic information including the gender of the speaker, voice quality, affective state, abnormalities in the vocal apparatus etc. Such information is attributed to the properties of the voice source signal, which is usually estimated from the speech signal. However, most of the source estimation techniques depend heavily on the goodness of the model assumptions and are prone to noise. A popular alternative is to indirectly obtain the source information through the Electroglottographic (EGG) signal that measures the electrical admittance around the vocal folds using dedicated hardware. In this paper, we address the problem of estimating the EGG signal directly from the speech signal, devoid of any hardware. Sampling from the intractable conditional distribution of the EGG signal given the speech signal is accomplished through optimization of an evidence lower bound. This is constructed via minimization of the KL-divergence between the true and the approximated posteriors of a latent variable learned using a deep neural auto-encoder that serves an informative prior. We demonstrate the efficacy of the method at generating the EGG signal by conducting several experiments on datasets comprising multiple speakers, voice qualities, noise settings and speech pathologies. The proposed method is evaluated on many benchmark metrics and is found to agree with the gold standard while proving better than the state-of-the-art algorithms on a few tasks such as epoch extraction.

Knowledge distillation (KD) exploits a large well-trained model (i.e., teacher) to train a small student model on the same dataset for the same task. Treating teacher features as knowledge, prevailing methods of knowledge distillation train student by aligning its features with the teacher's, e.g., by minimizing the KL-divergence between their logits or L2 distance between their intermediate features. While it is natural to believe that better alignment of student features to the teacher better distills teacher knowledge, simply forcing this alignment does not directly contribute to the student's performance, e.g., classification accuracy. In this work, we propose to align student features with class-mean of teacher features, where class-mean naturally serves as a strong classifier. To this end, we explore baseline techniques such as adopting the cosine distance based loss to encourage the similarity between student features and their corresponding class-means of the teacher. Moreover, we train the student to produce large-norm features, inspired by other lines of work (e.g., model pruning and domain adaptation), which find the large-norm features to be more significant. Finally, we propose a rather simple loss term (dubbed ND loss) to simultaneously (1) encourage student to produce large-norm features, and (2) align the direction of student features and teacher class-means. Experiments on standard benchmarks demonstrate that our explored techniques help existing KD methods achieve better performance, i.e., higher classification accuracy on ImageNet and CIFAR100 datasets, and higher detection precision on COCO dataset. Importantly, our proposed ND loss helps the most, leading to the state-of-the-art performance on these benchmarks. The source code is available at https://github.com/WangYZ1608/Knowledge-Distillation-via-ND.

Black box optimisation of an unknown function from expensive and noisy evaluations is a ubiquitous problem in machine learning, academic research and industrial production. An abstraction of the problem can be formulated as a kernel based bandit problem (also known as Bayesian optimisation), where a learner aims at optimising a kernelized function through sequential noisy observations. The existing work predominantly assumes feedback is immediately available; an assumption which fails in many real world situations, including recommendation systems, clinical trials and hyperparameter tuning. We consider a kernel bandit problem under stochastically delayed feedback, and propose an algorithm with mathcal{O}(Gamma_k(T)T+E[tau]) regret, where T is the number of time steps, Gamma_k(T) is the maximum information gain of the kernel with T observations, and tau is the delay random variable. This represents a significant improvement over the state of the art regret bound of mathcal{O}(Gamma_k(T)T+E[tau]Gamma_k(T)) reported in Verma et al. (2022). In particular, for very non-smooth kernels, the information gain grows almost linearly in time, trivializing the existing results. We also validate our theoretical results with simulations.

For many evaluation metrics commonly used as benchmarks for unconditional image generation, trivially memorizing the training set attains a better score than models which are considered state-of-the-art; we consider this problematic. We clarify a necessary condition for an evaluation metric not to behave this way: estimating the function must require a large sample from the model. In search of such a metric, we turn to neural network divergences (NNDs), which are defined in terms of a neural network trained to distinguish between distributions. The resulting benchmarks cannot be "won" by training set memorization, while still being perceptually correlated and computable only from samples. We survey past work on using NNDs for evaluation and implement an example black-box metric based on these ideas. Through experimental validation we show that it can effectively measure diversity, sample quality, and generalization.

Knowledge Distillation (KD) is a model-agnostic technique to improve model quality while having a fixed capacity budget. It is a commonly used technique for model compression, where a larger capacity teacher model with better quality is used to train a more compact student model with better inference efficiency. Through distillation, one hopes to benefit from student's compactness, without sacrificing too much on model quality. Despite the large success of knowledge distillation, better understanding of how it benefits student model's training dynamics remains under-explored. In this paper, we categorize teacher's knowledge into three hierarchical levels and study its effects on knowledge distillation: (1) knowledge of the `universe', where KD brings a regularization effect through label smoothing; (2) domain knowledge, where teacher injects class relationships prior to student's logit layer geometry; and (3) instance specific knowledge, where teacher rescales student model's per-instance gradients based on its measurement on the event difficulty. Using systematic analyses and extensive empirical studies on both synthetic and real-world datasets, we confirm that the aforementioned three factors play a major role in knowledge distillation. Furthermore, based on our findings, we diagnose some of the failure cases of applying KD from recent studies.

Post-hoc out-of-distribution (OOD) detection has garnered intensive attention in reliable machine learning. Many efforts have been dedicated to deriving score functions based on logits, distances, or rigorous data distribution assumptions to identify low-scoring OOD samples. Nevertheless, these estimate scores may fail to accurately reflect the true data density or impose impractical constraints. To provide a unified perspective on density-based score design, we propose a novel theoretical framework grounded in Bregman divergence, which extends distribution considerations to encompass an exponential family of distributions. Leveraging the conjugation constraint revealed in our theorem, we introduce a ConjNorm method, reframing density function design as a search for the optimal norm coefficient p against the given dataset. In light of the computational challenges of normalization, we devise an unbiased and analytically tractable estimator of the partition function using the Monte Carlo-based importance sampling technique. Extensive experiments across OOD detection benchmarks empirically demonstrate that our proposed ConjNorm has established a new state-of-the-art in a variety of OOD detection setups, outperforming the current best method by up to 13.25% and 28.19% (FPR95) on CIFAR-100 and ImageNet-1K, respectively.

Mixture of experts (MoE) has a well-principled finite mixture model construction for prediction, allowing the gating network (mixture weights) to learn from the predictors (explanatory variables) together with the experts' network (mixture component densities). We investigate the estimation properties of MoEs in a high-dimensional setting, where the number of predictors is much larger than the sample size, for which the literature lacks computational and especially theoretical results. We consider the class of finite MoE models with softmax gating functions and Gaussian regression experts, and focus on the theoretical properties of their l_1-regularized estimation via the Lasso. We provide a lower bound on the regularization parameter of the Lasso penalty that ensures an l_1-oracle inequality is satisfied by the Lasso estimator according to the Kullback--Leibler loss. We further state an l_1-ball oracle inequality for the l_1-penalized maximum likelihood estimator from the model selection.

Effective cross-modal retrieval requires robust alignment of heterogeneous data types. Most existing methods focus on bi-modal retrieval tasks and rely on distributional alignment techniques such as Kullback-Leibler divergence, Maximum Mean Discrepancy, and correlation alignment. However, these methods often suffer from critical limitations, including numerical instability, sensitivity to hyperparameters, and their inability to capture the full structure of the underlying distributions. In this paper, we introduce the Cauchy-Schwarz (CS) divergence, a hyperparameter-free measure that improves both training stability and retrieval performance. We further propose a novel Generalized CS (GCS) divergence inspired by H\"older's inequality. This extension enables direct alignment of three or more modalities within a unified mathematical framework through a bidirectional circular comparison scheme, eliminating the need for exhaustive pairwise comparisons. Extensive experiments on six benchmark datasets demonstrate the effectiveness of our method in both bi-modal and tri-modal retrieval tasks. The code of our CS/GCS divergence is publicly available at https://github.com/JiahaoZhang666/CSD.

We propose a Distributional Approach for addressing Controlled Text Generation from pre-trained Language Models (LMs). This approach permits to specify, in a single formal framework, both "pointwise" and "distributional" constraints over the target LM -- to our knowledge, the first model with such generality -- while minimizing KL divergence from the initial LM distribution. The optimal target distribution is then uniquely determined as an explicit EBM (Energy-Based Model) representation. From that optimal representation we then train a target controlled Autoregressive LM through an adaptive distributional variant of Policy Gradient. We conduct a first set of experiments over pointwise constraints showing the advantages of our approach over a set of baselines, in terms of obtaining a controlled LM balancing constraint satisfaction with divergence from the initial LM. We then perform experiments over distributional constraints, a unique feature of our approach, demonstrating its potential as a remedy to the problem of Bias in Language Models. Through an ablation study, we show the effectiveness of our adaptive technique for obtaining faster convergence. (Code available at https://github.com/naver/gdc)

The modern paradigm in machine learning involves pre-training on diverse data, followed by task-specific fine-tuning. In reinforcement learning (RL), this translates to learning via offline RL on a diverse historical dataset, followed by rapid online RL fine-tuning using interaction data. Most RL fine-tuning methods require continued training on offline data for stability and performance. However, this is undesirable because training on diverse offline data is slow and expensive for large datasets, and in principle, also limit the performance improvement possible because of constraints or pessimism on offline data. In this paper, we show that retaining offline data is unnecessary as long as we use a properly-designed online RL approach for fine-tuning offline RL initializations. To build this approach, we start by analyzing the role of retaining offline data in online fine-tuning. We find that continued training on offline data is mostly useful for preventing a sudden divergence in the value function at the onset of fine-tuning, caused by a distribution mismatch between the offline data and online rollouts. This divergence typically results in unlearning and forgetting the benefits of offline pre-training. Our approach, Warm-start RL (WSRL), mitigates the catastrophic forgetting of pre-trained initializations using a very simple idea. WSRL employs a warmup phase that seeds the online RL run with a very small number of rollouts from the pre-trained policy to do fast online RL. The data collected during warmup helps ``recalibrate'' the offline Q-function to the online distribution, allowing us to completely discard offline data without destabilizing the online RL fine-tuning. We show that WSRL is able to fine-tune without retaining any offline data, and is able to learn faster and attains higher performance than existing algorithms irrespective of whether they retain offline data or not.

Knowledge Distillation (KD) consists of transferring “knowledge” from one machine learning model (the teacher) to another (the student). Commonly, the teacher is a high-capacity model with formidable performance, while the student is more compact. By transferring knowledge, one hopes to benefit from the student’s compactness, without sacrificing too much performance. We study KD from a new perspective: rather than compressing models, we train students parameterized identically to their teachers. Surprisingly, these Born-Again Networks (BANs), outperform their teachers significantly, both on computer vision and language modeling tasks. Our experiments with BANs based on DenseNets demonstrate state-of-the-art performance on the CIFAR-10 (3.5%) and CIFAR-100 (15.5%) datasets, by validation error. Additional experiments explore two distillation objectives: (i) Confidence-Weighted by Teacher Max (CWTM) and (ii) Dark Knowledge with Permuted Predictions (DKPP). Both methods elucidate the essential components of KD, demonstrating the effect of the teacher outputs on both predicted and non-predicted classes.

Knowledge Distillation (KD) uses the teacher's prediction logits as soft labels to guide the student, while self-KD does not need a real teacher to require the soft labels. This work unifies the formulations of the two tasks by decomposing and reorganizing the generic KD loss into a Normalized KD (NKD) loss and customized soft labels for both target class (image's category) and non-target classes named Universal Self-Knowledge Distillation (USKD). We decompose the KD loss and find the non-target loss from it forces the student's non-target logits to match the teacher's, but the sum of the two non-target logits is different, preventing them from being identical. NKD normalizes the non-target logits to equalize their sum. It can be generally used for KD and self-KD to better use the soft labels for distillation loss. USKD generates customized soft labels for both target and non-target classes without a teacher. It smooths the target logit of the student as the soft target label and uses the rank of the intermediate feature to generate the soft non-target labels with Zipf's law. For KD with teachers, our NKD achieves state-of-the-art performance on CIFAR-100 and ImageNet datasets, boosting the ImageNet Top-1 accuracy of ResNet18 from 69.90% to 71.96% with a ResNet-34 teacher. For self-KD without teachers, USKD is the first self-KD method that can be effectively applied to both CNN and ViT models with negligible additional time and memory cost, resulting in new state-of-the-art results, such as 1.17% and 0.55% accuracy gains on ImageNet for MobileNet and DeiT-Tiny, respectively. Our codes are available at https://github.com/yzd-v/cls_KD.

Recent advances in knowledge distillation (KD) have enabled smaller student models to approach the performance of larger teacher models. However, popular methods such as supervised KD and on-policy KD, are adversely impacted by the knowledge gaps between teacher-student in practical scenarios. Supervised KD suffers from a distribution mismatch between training with a static dataset and inference over final student-generated outputs. Conversely, on-policy KD, which uses student-generated samples for training, can suffer from low-quality training examples with which teacher models are not familiar, resulting in inaccurate teacher feedback. To address these limitations, we introduce Speculative Knowledge Distillation (SKD), a novel approach that leverages cooperation between student and teacher models to generate high-quality training data on-the-fly while aligning with the student's inference-time distribution. In SKD, the student proposes tokens, and the teacher replaces poorly ranked ones based on its own distribution, transferring high-quality knowledge adaptively. We evaluate SKD on various text generation tasks, including translation, summarization, math, and instruction following, and show that SKD consistently outperforms existing KD methods across different domains, data sizes, and model initialization strategies.

We study a family of loss functions named label-distributionally robust (LDR) losses for multi-class classification that are formulated from distributionally robust optimization (DRO) perspective, where the uncertainty in the given label information are modeled and captured by taking the worse case of distributional weights. The benefits of this perspective are several fold: (i) it provides a unified framework to explain the classical cross-entropy (CE) loss and SVM loss and their variants, (ii) it includes a special family corresponding to the temperature-scaled CE loss, which is widely adopted but poorly understood; (iii) it allows us to achieve adaptivity to the uncertainty degree of label information at an instance level. Our contributions include: (1) we study both consistency and robustness by establishing top-k (forall kgeq 1) consistency of LDR losses for multi-class classification, and a negative result that a top-1 consistent and symmetric robust loss cannot achieve top-k consistency simultaneously for all kgeq 2; (2) we propose a new adaptive LDR loss that automatically adapts the individualized temperature parameter to the noise degree of class label of each instance; (3) we demonstrate stable and competitive performance for the proposed adaptive LDR loss on 7 benchmark datasets under 6 noisy label and 1 clean settings against 13 loss functions, and on one real-world noisy dataset. The code is open-sourced at https://github.com/Optimization-AI/ICML2023_LDR.

Neural network training is inherently sensitive to initialization and the randomness induced by stochastic gradient descent. However, it is unclear to what extent such effects lead to meaningfully different networks, either in terms of the models' weights or the underlying functions that were learned. In this work, we show that during the initial "chaotic" phase of training, even extremely small perturbations reliably causes otherwise identical training trajectories to diverge-an effect that diminishes rapidly over training time. We quantify this divergence through (i) L^2 distance between parameters, (ii) the loss barrier when interpolating between networks, (iii) L^2 and barrier between parameters after permutation alignment, and (iv) representational similarity between intermediate activations; revealing how perturbations across different hyperparameter or fine-tuning settings drive training trajectories toward distinct loss minima. Our findings provide insights into neural network training stability, with practical implications for fine-tuning, model merging, and diversity of model ensembles.

Lack of large expert annotated MR datasets makes training deep learning models difficult. Therefore, a cross-modality (MR-CT) deep learning segmentation approach that augments training data using pseudo MR images produced by transforming expert-segmented CT images was developed. Eighty-One T2-weighted MRI scans from 28 patients with non-small cell lung cancers were analyzed. Cross-modality prior encoding the transformation of CT to pseudo MR images resembling T2w MRI was learned as a generative adversarial deep learning model. This model augmented training data arising from 6 expert-segmented T2w MR patient scans with 377 pseudo MRI from non-small cell lung cancer CT patient scans with obtained from the Cancer Imaging Archive. A two-dimensional Unet implemented with batch normalization was trained to segment the tumors from T2w MRI. This method was benchmarked against (a) standard data augmentation and two state-of-the art cross-modality pseudo MR-based augmentation and (b) two segmentation networks. Segmentation accuracy was computed using Dice similarity coefficient (DSC), Hausdroff distance metrics, and volume ratio. The proposed approach produced the lowest statistical variability in the intensity distribution between pseudo and T2w MR images measured as Kullback-Leibler divergence of 0.069. This method produced the highest segmentation accuracy with a DSC of 0.75 and the lowest Hausdroff distance on the test dataset. This approach produced highly similar estimations of tumor growth as an expert (P = 0.37). A novel deep learning MR segmentation was developed that overcomes the limitation of learning robust models from small datasets by leveraging learned cross-modality priors to augment training. The results show the feasibility of the approach and the corresponding improvement over the state-of-the-art methods.

Recent work in scientific machine learning (SciML) has focused on incorporating partial differential equation (PDE) information into the learning process. Much of this work has focused on relatively ``easy'' PDE operators (e.g., elliptic and parabolic), with less emphasis on relatively ``hard'' PDE operators (e.g., hyperbolic). Within numerical PDEs, the latter problem class requires control of a type of volume element or conservation constraint, which is known to be challenging. Delivering on the promise of SciML requires seamlessly incorporating both types of problems into the learning process. To address this issue, we propose ProbConserv, a framework for incorporating conservation constraints into a generic SciML architecture. To do so, ProbConserv combines the integral form of a conservation law with a Bayesian update. We provide a detailed analysis of ProbConserv on learning with the Generalized Porous Medium Equation (GPME), a widely-applicable parameterized family of PDEs that illustrates the qualitative properties of both easier and harder PDEs. ProbConserv is effective for easy GPME variants, performing well with state-of-the-art competitors; and for harder GPME variants it outperforms other approaches that do not guarantee volume conservation. ProbConserv seamlessly enforces physical conservation constraints, maintains probabilistic uncertainty quantification (UQ), and deals well with shocks and heteroscedasticities. In each case, it achieves superior predictive performance on downstream tasks.

Knowledge distillation (KD) is a powerful model compression technique broadly used in practical deep learning applications. It is focused on training a small student network to mimic a larger teacher network. While it is widely known that KD can offer an improvement to student generalization in i.i.d setting, its performance under domain shift, i.e. the performance of student networks on data from domains unseen during training, has received little attention in the literature. In this paper we make a step towards bridging the research fields of knowledge distillation and domain generalization. We show that weight averaging techniques proposed in domain generalization literature, such as SWAD and SMA, also improve the performance of knowledge distillation under domain shift. In addition, we propose a simplistic weight averaging strategy that does not require evaluation on validation data during training and show that it performs on par with SWAD and SMA when applied to KD. We name our final distillation approach Weight-Averaged Knowledge Distillation (WAKD).

Neural scaling laws (NSL) refer to the phenomenon where model performance improves with scale. Sharma & Kaplan analyzed NSL using approximation theory and predict that MSE losses decay as N^{-alpha}, alpha=4/d, where N is the number of model parameters, and d is the intrinsic input dimension. Although their theory works well for some cases (e.g., ReLU networks), we surprisingly find that a simple 1D problem y=x^2 manifests a different scaling law (alpha=1) from their predictions (alpha=4). We opened the neural networks and found that the new scaling law originates from lottery ticket ensembling: a wider network on average has more "lottery tickets", which are ensembled to reduce the variance of outputs. We support the ensembling mechanism by mechanistically interpreting single neural networks, as well as studying them statistically. We attribute the N^{-1} scaling law to the "central limit theorem" of lottery tickets. Finally, we discuss its potential implications for large language models and statistical physics-type theories of learning.

Markov categories are a novel framework to describe and treat problems in probability and information theory. In this work we combine the categorical formalism with the traditional quantitative notions of entropy, mutual information, and data processing inequalities. We show that several quantitative aspects of information theory can be captured by an enriched version of Markov categories, where the spaces of morphisms are equipped with a divergence or even a metric. As it is customary in information theory, mutual information can be defined as a measure of how far a joint source is from displaying independence of its components. More strikingly, Markov categories give a notion of determinism for sources and channels, and we can define entropy exactly by measuring how far a source or channel is from being deterministic. This recovers Shannon and R\'enyi entropies, as well as the Gini-Simpson index used in ecology to quantify diversity, and it can be used to give a conceptual definition of generalized entropy.

Kahneman & Tversky's prospect theory tells us that humans perceive random variables in a biased but well-defined manner; for example, humans are famously loss-averse. We show that objectives for aligning LLMs with human feedback implicitly incorporate many of these biases -- the success of these objectives (e.g., DPO) over cross-entropy minimization can partly be ascribed to them being human-aware loss functions (HALOs). However, the utility functions these methods attribute to humans still differ from those in the prospect theory literature. Using a Kahneman-Tversky model of human utility, we propose a HALO that directly maximizes the utility of generations instead of maximizing the log-likelihood of preferences, as current methods do. We call this approach Kahneman-Tversky Optimization (KTO), and it matches or exceeds the performance of preference-based methods at scales from 1B to 30B. Crucially, KTO does not need preferences -- only a binary signal of whether an output is desirable or undesirable for a given input. This makes it far easier to use in the real world, where preference data is scarce and expensive.

In this paper, we introduce the Diff-Instruct* (DI*), an image data-free approach for building one-step text-to-image generative models that align with human preference while maintaining the ability to generate highly realistic images. We frame human preference alignment as online reinforcement learning using human feedback (RLHF), where the goal is to maximize the reward function while regularizing the generator distribution to remain close to a reference diffusion process. Unlike traditional RLHF approaches, which rely on the KL divergence for regularization, we introduce a novel score-based divergence regularization, which leads to significantly better performances. Although the direct calculation of this preference alignment objective remains intractable, we demonstrate that we can efficiently compute its gradient by deriving an equivalent yet tractable loss function. Remarkably, we used Diff-Instruct* to train a Stable Diffusion-XL-based 1-step model, the 2.6B DI*-SDXL-1step text-to-image model, which can generate images of a resolution of 1024x1024 with only 1 generation step. DI*-SDXL-1step model uses only 1.88% inference time and 29.30% GPU memory cost to outperform 12B FLUX-dev-50step significantly in PickScore, ImageReward, and CLIPScore on Parti prompt benchmark and HPSv2.1 on Human Preference Score benchmark, establishing a new state-of-the-art benchmark of human-preferred 1-step text-to-image generative models. Besides the strong quantitative performances, extensive qualitative comparisons also confirm the advantages of DI* in terms of maintaining diversity, improving image layouts, and enhancing aesthetic colors. We have released our industry-ready model on the homepage: https://github.com/pkulwj1994/diff_instruct_star.

A standard Variational Autoencoder, with a Euclidean latent space, is structurally incapable of capturing topological properties of certain datasets. To remove topological obstructions, we introduce Diffusion Variational Autoencoders with arbitrary manifolds as a latent space. A Diffusion Variational Autoencoder uses transition kernels of Brownian motion on the manifold. In particular, it uses properties of the Brownian motion to implement the reparametrization trick and fast approximations to the KL divergence. We show that the Diffusion Variational Autoencoder is capable of capturing topological properties of synthetic datasets. Additionally, we train MNIST on spheres, tori, projective spaces, SO(3), and a torus embedded in R3. Although a natural dataset like MNIST does not have latent variables with a clear-cut topological structure, training it on a manifold can still highlight topological and geometrical properties.

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.