Daily Papers

Chain-of-Thought (CoT) prompting has emerged as a powerful approach to enhancing the reasoning capabilities of Large Language Models (LLMs). However, existing implementations, such as in-context learning and fine-tuning, remain costly and inefficient. To improve CoT reasoning at a lower cost, and inspired by the task vector paradigm, we introduce CoT Vectors, compact representations that encode task-general, multi-step reasoning knowledge. Through experiments with Extracted CoT Vectors, we observe pronounced layer-wise instability, manifesting as a U-shaped performance curve that reflects a systematic three-stage reasoning process in LLMs. To address this limitation, we propose Learnable CoT Vectors, optimized under a teacher-student framework to provide more stable and robust guidance. Extensive evaluations across diverse benchmarks and models demonstrate that CoT Vectors not only outperform existing baselines but also achieve performance comparable to parameter-efficient fine-tuning methods, while requiring fewer trainable parameters. Moreover, by treating CoT Vectors as a probe, we uncover how their effectiveness varies due to latent space structure, information density, acquisition mechanisms, and pre-training differences, offering new insights into the functional organization of multi-step reasoning in LLMs. The source code will be released.

We present Submatrix-wise Vector Embedding Learner (Swivel), a method for generating low-dimensional feature embeddings from a feature co-occurrence matrix. Swivel performs approximate factorization of the point-wise mutual information matrix via stochastic gradient descent. It uses a piecewise loss with special handling for unobserved co-occurrences, and thus makes use of all the information in the matrix. While this requires computation proportional to the size of the entire matrix, we make use of vectorized multiplication to process thousands of rows and columns at once to compute millions of predicted values. Furthermore, we partition the matrix into shards in order to parallelize the computation across many nodes. This approach results in more accurate embeddings than can be achieved with methods that consider only observed co-occurrences, and can scale to much larger corpora than can be handled with sampling methods.

Large language models (LLMs) have shown impressive capabilities in various tasks, yet they still struggle with math reasoning. Despite efforts to optimize Chain-of-Thoughts (CoT) prompts and fine-tune LLMs, the potential of few-shot learning remains unexplored. In this work, we propose CoT-Max, a novel approach pushing the boundaries of few-shot CoT learning to improve LLM math reasoning capabilities. CoT-Max addresses the challenges of the selection of useful examples and limited number of examples due to restricted context window length. Inspired by our observation that natural language inputs contain many redundancy, we propose a coarse-to-fine pruner as a plug-and-play module for LLMs, which first identifies crucial CoT examples from a large batch and then further prunes unimportant tokens. To train the pruner, we collect a math reasoning dataset with diverse difficulty and steps, introduce a reward to measure both the input's effectiveness for math reasoning and token length constraints, and propose a novel training approach with reinforcement learning. As a result, CoT-Max significantly outperforms CoT and few-shot prompting baselines across various LLMs (LLaMA2-7B, 13B, 70B) and 5 mathematical datasets, achieving up to 4.55% absolute improvements. Remarkably, without any fine-tuning, LLaMA2-70B with CoT-Max surpasses GPT-3.5 and a wide range of larger LLMs (PaLM, Minerva, etc.) on the GSM8K.

Dense passage retrieval aims to retrieve the relevant passages of a query from a large corpus based on dense representations (i.e., vectors) of the query and the passages. Recent studies have explored improving pre-trained language models to boost dense retrieval performance. This paper proposes CoT-MAE (ConTextual Masked Auto-Encoder), a simple yet effective generative pre-training method for dense passage retrieval. CoT-MAE employs an asymmetric encoder-decoder architecture that learns to compress the sentence semantics into a dense vector through self-supervised and context-supervised masked auto-encoding. Precisely, self-supervised masked auto-encoding learns to model the semantics of the tokens inside a text span, and context-supervised masked auto-encoding learns to model the semantical correlation between the text spans. We conduct experiments on large-scale passage retrieval benchmarks and show considerable improvements over strong baselines, demonstrating the high efficiency of CoT-MAE. Our code is available at https://github.com/caskcsg/ir/tree/main/cotmae.

Large language models (LLMs) have demonstrated remarkable capabilities in tasks requiring reasoning and multi-step problem-solving through the use of chain-of-thought (CoT) prompting. However, generating the full CoT process results in significantly longer output sequences, leading to increased computational costs and latency during inference. To address this challenge, we propose a novel approach to compress the CoT process through semantic alignment, enabling more efficient decoding while preserving the benefits of CoT reasoning. Our method introduces an auxiliary CoT model that learns to generate and compress the full thought process into a compact special token representation semantically aligned with the original CoT output. This compressed representation is then integrated into the input of the Hidden Chain-of-Thought (HCoT) model. The training process follows a two-stage procedure: First, the CoT model is optimized to generate the compressed token representations aligned with the ground-truth CoT outputs using a contrastive loss. Subsequently, with the CoT model parameters frozen, the HCoT model is fine-tuned to generate accurate subsequent predictions conditioned on the prefix instruction and the compressed CoT representations from the CoT model. Extensive experiments across three challenging domains - mathematical reasoning, agent invocation, and question answering - demonstrate that our semantic compression approach achieves competitive or improved performance compared to the full CoT baseline, while providing significant speedups of at least 1.5x in decoding time. Moreover, incorporating contrastive learning objectives further enhances the quality of the compressed representations, leading to better CoT prompting and improved task accuracy. Our work paves the way for more efficient exploitation of multi-step reasoning capabilities in LLMs across a wide range of applications.

Generating Chain-of-Thought (CoT) before deriving the answer can effectively improve the reasoning capabilities of large language models (LLMs) and significantly improve the accuracy of the generated answer. However, in most cases, the length of the generated CoT is much longer than the desired final answer, which results in additional decoding costs. Furthermore, existing research has discovered that shortening the reasoning steps in CoT, even while preserving the key information, diminishes LLMs' abilities. These phenomena make it difficult to use LLMs and CoT in many real-world applications that only require the final answer and are sensitive to latency, such as search and recommendation. To reduce the costs of model decoding and shorten the length of the generated CoT, this paper presents Conditioned Compressed Chain-of-Thought (C3oT), a CoT compression framework that involves a compressor to compress an original longer CoT into a shorter CoT while maintaining key information and interpretability, a conditioned training method to train LLMs with both longer CoT and shorter CoT simultaneously to learn the corresponding relationships between them, and a conditioned inference method to gain the reasoning ability learned from longer CoT by generating shorter CoT. We conduct experiments over four datasets from arithmetic and commonsense scenarios, showing that the proposed method is capable of compressing the length of generated CoT by up to more than 50% without compromising its effectiveness.

Prevalent text-to-video retrieval systems mainly adopt embedding models for feature extraction and compute cosine similarities for ranking. However, this design presents two limitations. Low-quality text-video data pairs could compromise the retrieval, yet are hard to identify and examine. Cosine similarity alone provides no explanation for the ranking results, limiting the interpretability. We ask that can we interpret the ranking results, so as to assess the retrieval models and examine the text-video data? This work proposes X-CoT, an explainable retrieval framework upon LLM CoT reasoning in place of the embedding model-based similarity ranking. We first expand the existing benchmarks with additional video annotations to support semantic understanding and reduce data bias. We also devise a retrieval CoT consisting of pairwise comparison steps, yielding detailed reasoning and complete ranking. X-CoT empirically improves the retrieval performance and produces detailed rationales. It also facilitates the model behavior and data quality analysis. Code and data are available at: https://github.com/PrasannaPulakurthi/X-CoT.

Chain-of-Thought significantly enhances a model's reasoning capability, but it also comes with a considerable increase in inference costs due to long chains. With the observation that the reasoning path can be easily compressed under easy tasks but struggle on hard tasks, we explore the feasibility of elastically controlling the length of reasoning paths with only one model, thereby reducing the inference overhead of reasoning models dynamically based on task difficulty. We introduce a new tuning and inference strategy named CoT-Valve, designed to allow models to generate reasoning chains of varying lengths. To achieve this, we propose to identify a direction in the parameter space that, when manipulated, can effectively control the length of generated CoT. Moreover, we show that this property is valuable for compressing the reasoning chain. We construct datasets with chains from long to short for the same questions and explore two enhanced strategies for CoT-Valve: (1) a precise length-compressible CoT tuning method, and (2) a progressive chain length compression approach. Our experiments show that CoT-Valve successfully enables controllability and compressibility of the chain and shows better performance than the prompt-based control. We applied this method to QwQ-32B-Preview, reducing reasoning chains on GSM8K from 741 to 225 tokens with a minor performance drop (95.07% to 94.92%) and on AIME from 6827 to 4629 tokens, with only one additional incorrect answer.

Popular PEFT methods achieve parameter efficiency by assuming that incremental weight updates are inherently low-rank, which often leads to a performance gap compared to full fine-tuning. While recent methods have attempted to address this limitation, they typically lack sufficient parameter and memory efficiency. We propose VectorFit, an effective and easily deployable approach that adaptively trains the singular vectors and biases of pre-trained weight matrices. We demonstrate that the utilization of structural and transformational characteristics of pre-trained weights enables high-rank updates comparable to those of full fine-tuning. As a result, VectorFit achieves superior performance with 9X less trainable parameters compared to state-of-the-art PEFT methods. Through extensive experiments over 17 datasets spanning diverse language and vision tasks such as natural language understanding and generation, question answering, image classification, and image generation, we exhibit that VectorFit consistently outperforms baselines, even in extremely low-budget scenarios.

Current language models generate chain-of-thought traces by autoregressively sampling tokens from a finite vocabulary. While this discrete sampling has achieved remarkable success, conducting chain-of-thought with continuously-valued tokens (CoT2) offers a richer and more expressive alternative. Our work examines the benefits of CoT2 through logical reasoning tasks that inherently require search capabilities and provide optimization and exploration methods for CoT2. Theoretically, we show that CoT2 allows the model to track multiple traces in parallel and quantify its benefits for inference efficiency. Notably, one layer transformer equipped with CoT2 can provably solve the combinatorial "subset sum problem" given sufficient embedding dimension. These insights lead to a novel and effective supervision strategy where we match the softmax outputs to the empirical token distributions of a set of target traces. Complementing this, we introduce sampling strategies that unlock policy optimization and self-improvement for CoT2. Our first strategy samples and composes K discrete tokens at each decoding step to control the level of parallelism, and reduces to standard CoT when K=1. Our second strategy relies on continuous exploration over the probability simplex. Experiments confirm that policy optimization with CoT2 indeed improves the performance of the model beyond its initial discrete or continuous supervision.

Implicit Chain-of-Thought (CoT) methods present a promising, token-efficient alternative to explicit CoT reasoning in Large Language Models (LLMs), but a persistent performance gap has limited the application of implicit CoT. We identify a core latent instability issue by scaling the computational budget of implicit CoT approaches: as we increase the number of implicit reasoning tokens to enhance performance, the training process often becomes unstable and collapses. Our analysis reveals that this instability arises from the latent representations becoming homogeneous and losing their semantic diversity, a failure caused by insufficient step-level supervision in existing implicit CoT approaches. To address this issue, we propose SIM-CoT, a plug-and-play training module that introduces step-level supervision to stabilize and enrich the latent reasoning space. Specifically, SIM-CoT employs an auxiliary decoder during training to align each implicit token with its corresponding explicit reasoning step, ensuring that latent states capture distinct and meaningful information. The proposed auxiliary decoder is removed during inference, preserving the computational efficiency of implicit CoT methods with no added overhead. In addition, the auxiliary decoder affords interpretability of implicit reasoning by projecting each latent token onto an explicit reasoning vocabulary, enabling per-step visualization of semantic roles and diagnosis. SIM-CoT significantly enhances both the in-domain accuracy and out-of-domain stability of various implicit CoT methods, boosting baselines like Coconut by +8.2% on GPT-2 and CODI by +3.0% on LLaMA-3.1 8B. Demonstrating strong scalability, SIM-CoT also surpasses the explicit CoT baseline on GPT-2 by 2.1% with 2.3\times greater token efficiency, while substantially closing the performance gap on larger models like LLaMA-3.1 8B.

We propose Equiangular Basis Vectors (EBVs) for classification tasks. In deep neural networks, models usually end with a k-way fully connected layer with softmax to handle different classification tasks. The learning objective of these methods can be summarized as mapping the learned feature representations to the samples' label space. While in metric learning approaches, the main objective is to learn a transformation function that maps training data points from the original space to a new space where similar points are closer while dissimilar points become farther apart. Different from previous methods, our EBVs generate normalized vector embeddings as "predefined classifiers" which are required to not only be with the equal status between each other, but also be as orthogonal as possible. By minimizing the spherical distance of the embedding of an input between its categorical EBV in training, the predictions can be obtained by identifying the categorical EBV with the smallest distance during inference. Various experiments on the ImageNet-1K dataset and other downstream tasks demonstrate that our method outperforms the general fully connected classifier while it does not introduce huge additional computation compared with classical metric learning methods. Our EBVs won the first place in the 2022 DIGIX Global AI Challenge, and our code is open-source and available at https://github.com/NJUST-VIPGroup/Equiangular-Basis-Vectors.

Vector embeddings have been tasked with an ever-increasing set of retrieval tasks over the years, with a nascent rise in using them for reasoning, instruction-following, coding, and more. These new benchmarks push embeddings to work for any query and any notion of relevance that could be given. While prior works have pointed out theoretical limitations of vector embeddings, there is a common assumption that these difficulties are exclusively due to unrealistic queries, and those that are not can be overcome with better training data and larger models. In this work, we demonstrate that we may encounter these theoretical limitations in realistic settings with extremely simple queries. We connect known results in learning theory, showing that the number of top-k subsets of documents capable of being returned as the result of some query is limited by the dimension of the embedding. We empirically show that this holds true even if we restrict to k=2, and directly optimize on the test set with free parameterized embeddings. We then create a realistic dataset called LIMIT that stress tests models based on these theoretical results, and observe that even state-of-the-art models fail on this dataset despite the simple nature of the task. Our work shows the limits of embedding models under the existing single vector paradigm and calls for future research to develop methods that can resolve this fundamental limitation.

Pre-trained models produce strong generic representations that can be adapted via fine-tuning. The learned weight difference relative to the pre-trained model, known as a task vector, characterises the direction and stride of fine-tuning. The significance of task vectors is such that simple arithmetic operations on them can be used to combine diverse representations from different domains. This paper builds on these properties of task vectors and aims to answer (1) whether components of task vectors, particularly parameter blocks, exhibit similar characteristics, and (2) how such blocks can be used to enhance knowledge composition and transfer. To this end, we introduce aTLAS, an algorithm that linearly combines parameter blocks with different learned coefficients, resulting in anisotropic scaling at the task vector level. We show that such linear combinations explicitly exploit the low intrinsic dimensionality of pre-trained models, with only a few coefficients being the learnable parameters. Furthermore, composition of parameter blocks leverages the already learned representations, thereby reducing the dependency on large amounts of data. We demonstrate the effectiveness of our method in task arithmetic, few-shot recognition and test-time adaptation, with supervised or unsupervised objectives. In particular, we show that (1) learned anisotropic scaling allows task vectors to be more disentangled, causing less interference in composition; (2) task vector composition excels with scarce or no labeled data and is less prone to domain shift, thus leading to better generalisability; (3) mixing the most informative parameter blocks across different task vectors prior to training can reduce the memory footprint and improve the flexibility of knowledge transfer. Moreover, we show the potential of aTLAS as a PEFT method, particularly with less data, and demonstrate that its scalibility.

We introduce CCI4.0, a large-scale bilingual pre-training dataset engineered for superior data quality and diverse human-like reasoning trajectory. CCI4.0 occupies roughly 35 TB of disk space and comprises two sub-datasets: CCI4.0-M2-Base and CCI4.0-M2-CoT. CCI4.0-M2-Base combines a 5.2 TB carefully curated Chinese web corpus, a 22.5 TB English subset from Nemotron-CC, and diverse sources from math, wiki, arxiv, and code. Although these data are mostly sourced from well-processed datasets, the quality standards of various domains are dynamic and require extensive expert experience and labor to process. So, we propose a novel pipeline justifying data quality mainly based on models through two-stage deduplication, multiclassifier quality scoring, and domain-aware fluency filtering. We extract 4.5 billion pieces of CoT(Chain-of-Thought) templates, named CCI4.0-M2-CoT. Differing from the distillation of CoT from larger models, our proposed staged CoT extraction exemplifies diverse reasoning patterns and significantly decreases the possibility of hallucination. Empirical evaluations demonstrate that LLMs pre-trained in CCI4.0 benefit from cleaner, more reliable training signals, yielding consistent improvements in downstream tasks, especially in math and code reflection tasks. Our results underscore the critical role of rigorous data curation and human thinking templates in advancing LLM performance, shedding some light on automatically processing pretraining corpora.

Multi-vector models, such as ColBERT, are a significant advancement in neural information retrieval (IR), delivering state-of-the-art performance by representing queries and documents by multiple contextualized token-level embeddings. However, this increased representation size introduces considerable storage and computational overheads which have hindered widespread adoption in practice. A common approach to mitigate this overhead is to cluster the model's frozen vectors, but this strategy's effectiveness is fundamentally limited by the intrinsic clusterability of these embeddings. In this work, we introduce CRISP (Clustered Representations with Intrinsic Structure Pruning), a novel multi-vector training method which learns inherently clusterable representations directly within the end-to-end training process. By integrating clustering into the training phase rather than imposing it post-hoc, CRISP significantly outperforms post-hoc clustering at all representation sizes, as well as other token pruning methods. On the BEIR retrieval benchmarks, CRISP achieves a significant rate of ~3x reduction in the number of vectors while outperforming the original unpruned model. This indicates that learned clustering effectively denoises the model by filtering irrelevant information, thereby generating more robust multi-vector representations. With more aggressive clustering, CRISP achieves an 11x reduction in the number of vectors with only a 3.6% quality loss.

A common approach for teaching large language models (LLMs) to reason is to train on chain-of-thought (CoT) traces of in-distribution reasoning problems, but such annotated data is costly to obtain for every problem of interest. We want reasoning models to generalize beyond their training distribution, and ideally to generalize compositionally: combine atomic reasoning skills to solve harder, unseen reasoning tasks. We take a step towards compositional generalization of reasoning skills when addressing a target compositional task that has no labeled CoT data. We find that simply training models on CoT data of atomic tasks leads to limited generalization, but minimally modifying CoT formats of constituent atomic tasks to be composable can lead to improvements. We can train "atomic CoT" models on the atomic tasks with Composable CoT data and combine them with multitask learning or model merging for better zero-shot performance on the target compositional task. Such a combined model can be further bootstrapped on a small amount of compositional data using rejection sampling fine-tuning (RFT). Results on string operations and natural language skill compositions show that training LLMs on Composable CoT outperforms multitask learning and continued fine-tuning baselines within a given training data budget.

Large Language Models (LLMs) using Chain-of-Thought (CoT) prompting excel at complex reasoning but generate verbose thought processes with considerable redundancy, leading to increased inference costs and reduced efficiency. We introduce a novel CoT compression framework based on step entropy, a metric that quantifies the informational contribution of individual reasoning steps to identify redundancy. Through theoretical analysis and extensive empirical validation on mathematical reasoning benchmarks, we demonstrate that steps with low entropy are indeed highly redundant. Our experiments reveal that an astonishing 80\% of low-entropy intermediate steps can be pruned with minor degradation in the final answer accuracy across DeepSeek-R1-7B, 14B and Qwen3-8B. This finding sharply contrasts with random or high-entropy pruning, which severely impairs reasoning performance. Building on this, we propose a novel two-stage training strategy combining Supervised Fine-Tuning (SFT) and Group Relative Policy Optimization (GRPO) reinforcement learning. This approach enables LLMs to autonomously learn to generate compressed COTs during inference by strategically incorporating [SKIP] tokens. Our method significantly enhances LLM inference efficiency while rigorously preserving accuracy, offering profound implications for practical LLM deployment and a deeper understanding of reasoning structures.

We propose a game-based formulation for learning dimensionality-reducing representations of feature vectors, when only a prior knowledge on future prediction tasks is available. In this game, the first player chooses a representation, and then the second player adversarially chooses a prediction task from a given class, representing the prior knowledge. The first player aims is to minimize, and the second player to maximize, the regret: The minimal prediction loss using the representation, compared to the same loss using the original features. For the canonical setting in which the representation, the response to predict and the predictors are all linear functions, and under the mean squared error loss function, we derive the theoretically optimal representation in pure strategies, which shows the effectiveness of the prior knowledge, and the optimal regret in mixed strategies, which shows the usefulness of randomizing the representation. For general representations and loss functions, we propose an efficient algorithm to optimize a randomized representation. The algorithm only requires the gradients of the loss function, and is based on incrementally adding a representation rule to a mixture of such rules.

We introduce the first method for translating text embeddings from one vector space to another without any paired data, encoders, or predefined sets of matches. Our unsupervised approach translates any embedding to and from a universal latent representation (i.e., a universal semantic structure conjectured by the Platonic Representation Hypothesis). Our translations achieve high cosine similarity across model pairs with different architectures, parameter counts, and training datasets. The ability to translate unknown embeddings into a different space while preserving their geometry has serious implications for the security of vector databases. An adversary with access only to embedding vectors can extract sensitive information about the underlying documents, sufficient for classification and attribute inference.

Compositional zero-shot learning (CZSL) aims to recognize unseen compositions with prior knowledge of known primitives (attribute and object). Previous works for CZSL often suffer from grasping the contextuality between attribute and object, as well as the discriminability of visual features, and the long-tailed distribution of real-world compositional data. We propose a simple and scalable framework called Composition Transformer (CoT) to address these issues. CoT employs object and attribute experts in distinctive manners to generate representative embeddings, using the visual network hierarchically. The object expert extracts representative object embeddings from the final layer in a bottom-up manner, while the attribute expert makes attribute embeddings in a top-down manner with a proposed object-guided attention module that models contextuality explicitly. To remedy biased prediction caused by imbalanced data distribution, we develop a simple minority attribute augmentation (MAA) that synthesizes virtual samples by mixing two images and oversampling minority attribute classes. Our method achieves SoTA performance on several benchmarks, including MIT-States, C-GQA, and VAW-CZSL. We also demonstrate the effectiveness of CoT in improving visual discrimination and addressing the model bias from the imbalanced data distribution. The code is available at https://github.com/HanjaeKim98/CoT.

We introduce CoT-ICL Lab, a framework and methodology to generate synthetic tokenized datasets and systematically study chain-of-thought (CoT) in-context learning (ICL) in language models. CoT-ICL Lab allows fine grained control over the complexity of in-context examples by decoupling (1) the causal structure involved in chain token generation from (2) the underlying token processing functions. We train decoder-only transformers (up to 700M parameters) on these datasets and show that CoT accelerates the accuracy transition to higher values across model sizes. In particular, we find that model depth is crucial for leveraging CoT with limited in-context examples, while more examples help shallow models match deeper model performance. Additionally, limiting the diversity of token processing functions throughout training improves causal structure learning via ICL. We also interpret these transitions by analyzing transformer embeddings and attention maps. Overall, CoT-ICL Lab serves as a simple yet powerful testbed for theoretical and empirical insights into ICL and CoT in language models.

We present Simplex Random Features (SimRFs), a new random feature (RF) mechanism for unbiased approximation of the softmax and Gaussian kernels by geometrical correlation of random projection vectors. We prove that SimRFs provide the smallest possible mean square error (MSE) on unbiased estimates of these kernels among the class of weight-independent geometrically-coupled positive random feature (PRF) mechanisms, substantially outperforming the previously most accurate Orthogonal Random Features at no observable extra cost. We present a more computationally expensive SimRFs+ variant, which we prove is asymptotically optimal in the broader family of weight-dependent geometrical coupling schemes (which permit correlations between random vector directions and norms). In extensive empirical studies, we show consistent gains provided by SimRFs in settings including pointwise kernel estimation, nonparametric classification and scalable Transformers.

Invariance learning algorithms that conditionally filter out domain-specific random variables as distractors, do so based only on the data semantics, and not the target domain under evaluation. We show that a provably optimal and sample-efficient way of learning conditional invariances is by relaxing the invariance criterion to be non-commutatively directed towards the target domain. Under domain asymmetry, i.e., when the target domain contains semantically relevant information absent in the source, the risk of the encoder varphi^* that is optimal on average across domains is strictly lower-bounded by the risk of the target-specific optimal encoder Phi^*_tau. We prove that non-commutativity steers the optimization towards Phi^*_tau instead of varphi^*, bringing the H-divergence between domains down to zero, leading to a stricter bound on the target risk. Both our theory and experiments demonstrate that non-commutative invariance (NCI) can leverage source domain samples to meet the sample complexity needs of learning Phi^*_tau, surpassing SOTA invariance learning algorithms for domain adaptation, at times by over 2%, approaching the performance of an oracle. Implementation is available at https://github.com/abhrac/nci.

Low-rank adaptation (LoRA) has emerged as a leading parameter-efficient fine-tuning technique for adapting large foundation models, yet it often locks adapters into suboptimal minima near their initialization. This hampers model generalization and limits downstream operators such as adapter merging and pruning. Here, we propose CoTo, a progressive training strategy that gradually increases adapters' activation probability over the course of fine-tuning. By stochastically deactivating adapters, CoTo encourages more balanced optimization and broader exploration of the loss landscape. We provide a theoretical analysis showing that CoTo promotes layer-wise dropout stability and linear mode connectivity, and we adopt a cooperative-game approach to quantify each adapter's marginal contribution. Extensive experiments demonstrate that CoTo consistently boosts single-task performance, enhances multi-task merging accuracy, improves pruning robustness, and reduces training overhead, all while remaining compatible with diverse LoRA variants. Code is available at https://github.com/zwebzone/coto.

Many machine learning algorithms require the input to be represented as a fixed-length feature vector. When it comes to texts, one of the most common fixed-length features is bag-of-words. Despite their popularity, bag-of-words features have two major weaknesses: they lose the ordering of the words and they also ignore semantics of the words. For example, "powerful," "strong" and "Paris" are equally distant. In this paper, we propose Paragraph Vector, an unsupervised algorithm that learns fixed-length feature representations from variable-length pieces of texts, such as sentences, paragraphs, and documents. Our algorithm represents each document by a dense vector which is trained to predict words in the document. Its construction gives our algorithm the potential to overcome the weaknesses of bag-of-words models. Empirical results show that Paragraph Vectors outperform bag-of-words models as well as other techniques for text representations. Finally, we achieve new state-of-the-art results on several text classification and sentiment analysis tasks.

In this dissertation we report results of our research on dense distributed representations of text data. We propose two novel neural models for learning such representations. The first model learns representations at the document level, while the second model learns word-level representations. For document-level representations we propose Binary Paragraph Vector: a neural network models for learning binary representations of text documents, which can be used for fast document retrieval. We provide a thorough evaluation of these models and demonstrate that they outperform the seminal method in the field in the information retrieval task. We also report strong results in transfer learning settings, where our models are trained on a generic text corpus and then used to infer codes for documents from a domain-specific dataset. In contrast to previously proposed approaches, Binary Paragraph Vector models learn embeddings directly from raw text data. For word-level representations we propose Disambiguated Skip-gram: a neural network model for learning multi-sense word embeddings. Representations learned by this model can be used in downstream tasks, like part-of-speech tagging or identification of semantic relations. In the word sense induction task Disambiguated Skip-gram outperforms state-of-the-art models on three out of four benchmarks datasets. Our model has an elegant probabilistic interpretation. Furthermore, unlike previous models of this kind, it is differentiable with respect to all its parameters and can be trained with backpropagation. In addition to quantitative results, we present qualitative evaluation of Disambiguated Skip-gram, including two-dimensional visualisations of selected word-sense embeddings.

Learnable embedding vector is one of the most important applications in machine learning, and is widely used in various database-related domains. However, the high dimensionality of sparse data in recommendation tasks and the huge volume of corpus in retrieval-related tasks lead to a large memory consumption of the embedding table, which poses a great challenge to the training and deployment of models. Recent research has proposed various methods to compress the embeddings at the cost of a slight decrease in model quality or the introduction of other overheads. Nevertheless, the relative performance of these methods remains unclear. Existing experimental comparisons only cover a subset of these methods and focus on limited metrics. In this paper, we perform a comprehensive comparative analysis and experimental evaluation of embedding compression. We introduce a new taxonomy that categorizes these techniques based on their characteristics and methodologies, and further develop a modular benchmarking framework that integrates 14 representative methods. Under a uniform test environment, our benchmark fairly evaluates each approach, presents their strengths and weaknesses under different memory budgets, and recommends the best method based on the use case. In addition to providing useful guidelines, our study also uncovers the limitations of current methods and suggests potential directions for future research.

Large language models (LLMs) excel at complex reasoning when they include intermediate steps, known as "chains of thought" (CoTs). However, these rationales are often overly verbose, even for simple problems, leading to wasted context, increased latency, and higher energy consumption. We observe that verbose, English-heavy CoTs and concise, math-centric CoTs occupy distinct regions in the model's residual-stream activation space. By extracting and injecting a "steering vector" to transition between these modes, we can reliably shift generation toward more concise reasoning, effectively compressing CoTs without retraining. We formalize this approach as Activation-Steered Compression (ASC), an inference-time technique that shortens reasoning traces by directly modifying hidden representations. In addition, we provide a theoretical analysis of the impact of ASC on the output distribution, derived from a closed-form KL-divergence-bounded constraint to regulate steering strength. Using only 100 paired verbose and concise examples, ASC achieves up to 67.43% reduction in CoT length on MATH500 and GSM8K datasets, while maintaining accuracy across 7B, 8B, and 32B parameter models. As a training-free method, ASC introduces negligible runtime overhead and, on MATH500, delivers an average 2.73x speedup in end-to-end reasoning wall-clock time on an 8B model. This makes ASC a practical and efficient tool for streamlining the deployment of reasoning-capable LLMs in latency- or cost-sensitive settings. The code is available at: https://github.com/ArminAzizi98/ASC

Learning compact discrete representations of data is a key task on its own or for facilitating subsequent processing of data. In this paper we present a model that produces Discrete InfoMax Codes (DIMCO); we learn a probabilistic encoder that yields k-way d-dimensional codes associated with input data. Our model's learning objective is to maximize the mutual information between codes and labels with a regularization, which enforces entries of a codeword to be as independent as possible. We show that the infomax principle also justifies previous loss functions (e.g., cross-entropy) as its special cases. Our analysis also shows that using shorter codes, as DIMCO does, reduces overfitting in the context of few-shot classification. Through experiments in various domains, we observe this implicit meta-regularization effect of DIMCO. Furthermore, we show that the codes learned by DIMCO are efficient in terms of both memory and retrieval time compared to previous methods.

Recent empirical works have successfully used unlabeled data to learn feature representations that are broadly useful in downstream classification tasks. Several of these methods are reminiscent of the well-known word2vec embedding algorithm: leveraging availability of pairs of semantically "similar" data points and "negative samples," the learner forces the inner product of representations of similar pairs with each other to be higher on average than with negative samples. The current paper uses the term contrastive learning for such algorithms and presents a theoretical framework for analyzing them by introducing latent classes and hypothesizing that semantically similar points are sampled from the same latent class. This framework allows us to show provable guarantees on the performance of the learned representations on the average classification task that is comprised of a subset of the same set of latent classes. Our generalization bound also shows that learned representations can reduce (labeled) sample complexity on downstream tasks. We conduct controlled experiments in both the text and image domains to support the theory.

The task of "unlearning" certain concepts in large language models (LLMs) has attracted immense attention recently, due to its importance for mitigating undesirable model behaviours, such as the generation of harmful, private, or incorrect information. Current protocols to evaluate unlearning methods largely rely on behavioral tests, without monitoring the presence of unlearned knowledge within the model's parameters. This residual knowledge can be adversarially exploited to recover the erased information post-unlearning. We argue that unlearning should also be evaluated internally, by considering changes in the parametric knowledge traces of the unlearned concepts. To this end, we propose a general methodology for eliciting directions in the parameter space (termed "concept vectors") that encode concrete concepts, and construct ConceptVectors, a benchmark dataset containing hundreds of common concepts and their parametric knowledge traces within two open-source LLMs. Evaluation on ConceptVectors shows that existing unlearning methods minimally impact concept vectors, while directly ablating these vectors demonstrably removes the associated knowledge from the LLMs and significantly reduces their susceptibility to adversarial manipulation. Our results highlight limitations in behavioral-based unlearning evaluations and call for future work to include parametric-based evaluations. To support this, we release our code and benchmark at https://github.com/yihuaihong/ConceptVectors.

Recent advancements in Long Chain-of-Thought (CoT) reasoning models have improved performance on complex tasks, but they suffer from overthinking, which generates redundant reasoning steps, especially for simple questions. This paper revisits the reasoning patterns of Long and Short CoT models, observing that the Short CoT patterns offer concise reasoning efficiently, while the Long CoT patterns excel in challenging scenarios where the Short CoT patterns struggle. To enable models to leverage both patterns, we propose Question-Free Fine-Tuning (QFFT), a fine-tuning approach that removes the input question during training and learns exclusively from Long CoT responses. This approach enables the model to adaptively employ both reasoning patterns: it prioritizes the Short CoT patterns and activates the Long CoT patterns only when necessary. Experiments on various mathematical datasets demonstrate that QFFT reduces average response length by more than 50\%, while achieving performance comparable to Supervised Fine-Tuning (SFT). Additionally, QFFT exhibits superior performance compared to SFT in noisy, out-of-domain, and low-resource scenarios.

Reasoning-capable language models achieve state-of-the-art performance in diverse complex tasks by generating long, explicit Chain-of-Thought (CoT) traces. While recent works show that base models can acquire such reasoning traces via reinforcement learning or distillation from stronger models like DeepSeek-R1, previous works demonstrate that even short CoT prompting without fine-tuning is able to improve reasoning. We ask whether long CoT can be induced in a base model using only prompting or minimal tuning. Using just 20 long CoT examples from the reasoning model QwQ-32B-Preview, we lightly fine-tune the base model Qwen2.5-32B. The resulting model outperforms the much larger Qwen2.5-Math-72B-Instruct, showing that a handful of high-quality examples can unlock strong reasoning capabilities. We further explore using CoT data from non-reasoning models and human annotators, enhanced with prompt engineering, multi-pass editing, and structural guidance. However, neither matches the performance of reasoning model traces, suggesting that certain latent qualities of expert CoT are difficult to replicate. We analyze key properties of reasoning data, such as problem difficulty, diversity, and answer length, that influence reasoning distillation. While challenges remain, we are optimistic that carefully curated human-written CoT, even in small quantities, can activate reasoning behaviors in base models. We release our human-authored dataset across refinement stages and invite further investigation into what makes small-scale reasoning supervision so effective.

We study the problem of teaching multiple learners simultaneously in the nonparametric iterative teaching setting, where the teacher iteratively provides examples to the learner for accelerating the acquisition of a target concept. This problem is motivated by the gap between current single-learner teaching setting and the real-world scenario of human instruction where a teacher typically imparts knowledge to multiple students. Under the new problem formulation, we introduce a novel framework -- Multi-learner Nonparametric Teaching (MINT). In MINT, the teacher aims to instruct multiple learners, with each learner focusing on learning a scalar-valued target model. To achieve this, we frame the problem as teaching a vector-valued target model and extend the target model space from a scalar-valued reproducing kernel Hilbert space used in single-learner scenarios to a vector-valued space. Furthermore, we demonstrate that MINT offers significant teaching speed-up over repeated single-learner teaching, particularly when the multiple learners can communicate with each other. Lastly, we conduct extensive experiments to validate the practicality and efficiency of MINT.

Generalization to novel compound tasks under distribution shift is important for deploying transformer-based language models (LMs). This work investigates Chain-of-Thought (CoT) reasoning as a means to enhance OOD generalization. Through controlled experiments across several compound tasks, we reveal three key insights: (1) While QA-trained models achieve near-perfect in-distribution accuracy, their OOD performance degrades catastrophically, even with 10000k+ training examples; (2) the granularity of CoT data strongly correlates with generalization performance; finer-grained CoT data leads to better generalization; (3) CoT exhibits remarkable sample efficiency, matching QA performance with much less (even 80%) data. Theoretically, we demonstrate that compound tasks inherently permit shortcuts in Q-A data that misalign with true reasoning principles, while CoT forces internalization of valid dependency structures, and thus can achieve better generalization. Further, we show that transformer positional embeddings can amplify generalization by emphasizing subtask condition recurrence in long CoT sequences. Our combined theoretical and empirical analysis provides compelling evidence for CoT reasoning as a crucial training paradigm for enabling LM generalization under real-world distributional shifts for compound tasks.

Chain-of-thought (CoT) prompting with large language models has proven effective in numerous natural language processing tasks, but designing prompts that generalize well to diverse problem types can be challenging, especially in the context of math word problem (MWP) solving. Additionally, it is common to have a large amount of training data that have a better diversity coverage but CoT annotations are not available, which limits the use of supervised learning techniques. To address these issues, we investigate two approaches to leverage the training data in a few-shot prompting scenario: dynamic program prompting and program distillation. Our approach is largely inspired by Gao et al., (2022), where they proposed to replace the CoT with the programs as the intermediate reasoning step. Such a prompting strategy allows us to accurately verify the answer correctness through program execution in MWP solving. Our dynamic program prompting involves annotating the training data by sampling correct programs from a large language model, while program distillation involves adapting a smaller model to the program-annotated training data. Our experiments on three standard MWP datasets demonstrate the effectiveness of these approaches, yielding significant improvements over previous baselines for prompting and fine-tuning. Our results suggest that leveraging a large amount of training data can improve the generalization ability of prompts and boost the performance of fine-tuned small models in MWP solving.

Instructing the model to generate a sequence of intermediate steps, a.k.a., a chain of thought (CoT), is a highly effective method to improve the accuracy of large language models (LLMs) on arithmetics and symbolic reasoning tasks. However, the mechanism behind CoT remains unclear. This work provides a theoretical understanding of the power of CoT for decoder-only transformers through the lens of expressiveness. Conceptually, CoT empowers the model with the ability to perform inherently serial computation, which is otherwise lacking in transformers, especially when depth is low. Given input length n, previous works have shown that constant-depth transformers with finite precision poly(n) embedding size can only solve problems in TC^0 without CoT. We first show an even tighter expressiveness upper bound for constant-depth transformers with constant-bit precision, which can only solve problems in AC^0, a proper subset of TC^0. However, with T steps of CoT, constant-depth transformers using constant-bit precision and O(log n) embedding size can solve any problem solvable by boolean circuits of size T. Empirically, enabling CoT dramatically improves the accuracy for tasks that are hard for parallel computation, including the composition of permutation groups, iterated squaring, and circuit value problems, especially for low-depth transformers.

Self-supervised learning excels at learning representations from large amounts of data. At the same time, generative models offer the complementary property of learning information about the underlying data generation process. In this study, we aim at establishing a principled connection between these two paradigms and highlight the benefits of their complementarity. In particular, we perform an analysis of self-supervised learning objectives, elucidating the underlying probabilistic graphical models and presenting a standardized methodology for their derivation from first principles. The analysis suggests a natural means of integrating self-supervised learning with likelihood-based generative models. We instantiate this concept within the realm of cluster-based self-supervised learning and energy models, introducing a lower bound proven to reliably penalize the most important failure modes and unlocking full unification. Our theoretical findings are substantiated through experiments on synthetic and real-world data, including SVHN, CIFAR10, and CIFAR100, demonstrating that our objective function allows to jointly train a backbone network in a discriminative and generative fashion, consequently outperforming existing self-supervised learning strategies in terms of clustering, generation and out-of-distribution detection performance by a wide margin. We also demonstrate that the solution can be integrated into a neuro-symbolic framework to tackle a simple yet non-trivial instantiation of the symbol grounding problem. The code is publicly available at https://github.com/emsansone/GEDI.

Representation learning has become an invaluable approach for learning from symbolic data such as text and graphs. However, while complex symbolic datasets often exhibit a latent hierarchical structure, state-of-the-art methods typically learn embeddings in Euclidean vector spaces, which do not account for this property. For this purpose, we introduce a new approach for learning hierarchical representations of symbolic data by embedding them into hyperbolic space -- or more precisely into an n-dimensional Poincar\'e ball. Due to the underlying hyperbolic geometry, this allows us to learn parsimonious representations of symbolic data by simultaneously capturing hierarchy and similarity. We introduce an efficient algorithm to learn the embeddings based on Riemannian optimization and show experimentally that Poincar\'e embeddings outperform Euclidean embeddings significantly on data with latent hierarchies, both in terms of representation capacity and in terms of generalization ability.

A range of recent works addresses the problem of compression of sequence of tokens into a shorter sequence of real-valued vectors to be used as inputs instead of token embeddings or key-value cache. These approaches allow to reduce the amount of compute in existing language models. Despite relying on powerful models as encoders, the maximum attainable lossless compression ratio is typically not higher than x10. This fact is highly intriguing because, in theory, the maximum information capacity of large real-valued vectors is far beyond the presented rates even for 16-bit precision and a modest vector size. In this work, we explore the limits of compression by replacing the encoder with a per-sample optimization procedure. We show that vectors with compression ratios up to x1500 exist, which highlights two orders of magnitude gap between existing and practically attainable solutions. Furthermore, we empirically show that the compression limits are determined not by the length of the input but by the amount of uncertainty to be reduced, namely, the cross-entropy loss on this sequence without any conditioning. The obtained limits highlight the substantial gap between the theoretical capacity of input embeddings and their practical utilization, suggesting significant room for optimization in model design.

We propose two novel model architectures for computing continuous vector representations of words from very large data sets. The quality of these representations is measured in a word similarity task, and the results are compared to the previously best performing techniques based on different types of neural networks. We observe large improvements in accuracy at much lower computational cost, i.e. it takes less than a day to learn high quality word vectors from a 1.6 billion words data set. Furthermore, we show that these vectors provide state-of-the-art performance on our test set for measuring syntactic and semantic word similarities.

Attention mechanisms, particularly softmax attention, have been instrumental in the success of transformer-based models such as GPT. However, the quadratic memory complexity of softmax attention with respect to sequence length poses significant challenges for processing longer sequences. We introduce Cottention, a novel attention mechanism that replaces the softmax operation with cosine similarity. By leveraging the properties of cosine similarity and rearranging the attention equation, Cottention achieves native linear memory complexity with respect to sequence length, making it inherently more memory-efficient than softmax attention. We demonstrate that Cottention can be reformulated as a recurrent neural network (RNN) with a finite hidden state, allowing for constant memory usage during inference. We evaluate Cottention on both the bidirectional BERT and causal GPT tasks, demonstrating comparable performance to softmax attention while significantly reducing memory requirements. To ensure efficient computation, we develop a custom CUDA kernel for Cottention. Our results show that Cottention is a promising alternative to softmax attention, enabling the processing of longer sequences without sacrificing performance, due to its native linear memory complexity and ability to maintain a constant memory footprint during inference.

Chain-of-Thought (CoT) has widely enhanced mathematical reasoning in Large Language Models (LLMs), but it still remains challenging for extending it to multimodal domains. Existing works either adopt a similar textual reasoning for image input, or seek to interleave visual signals into mathematical CoT. However, they face three key limitations for math problem-solving: reliance on coarse-grained box-shaped image regions, limited perception of vision encoders on math content, and dependence on external capabilities for visual modification. In this paper, we propose MINT-CoT, introducing Mathematical INterleaved Tokens for Chain-of-Thought visual reasoning. MINT-CoT adaptively interleaves relevant visual tokens into textual reasoning steps via an Interleave Token, which dynamically selects visual regions of any shapes within math figures. To empower this capability, we construct the MINT-CoT dataset, containing 54K mathematical problems aligning each reasoning step with visual regions at the token level, accompanied by a rigorous data generation pipeline. We further present a three-stage MINT-CoT training strategy, progressively combining text-only CoT SFT, interleaved CoT SFT, and interleaved CoT RL, which derives our MINT-CoT-7B model. Extensive experiments demonstrate the effectiveness of our method for effective visual interleaved reasoning in mathematical domains, where MINT-CoT-7B outperforms the baseline model by +34.08% on MathVista, +28.78% on GeoQA, and +23.2% on MMStar, respectively. Our code and data are available at https://github.com/xinyan-cxy/MINT-CoT

Continuous Bag of Words (CBOW) is a powerful text embedding method. Due to its strong capabilities to encode word content, CBOW embeddings perform well on a wide range of downstream tasks while being efficient to compute. However, CBOW is not capable of capturing the word order. The reason is that the computation of CBOW's word embeddings is commutative, i.e., embeddings of XYZ and ZYX are the same. In order to address this shortcoming, we propose a learning algorithm for the Continuous Matrix Space Model, which we call Continual Multiplication of Words (CMOW). Our algorithm is an adaptation of word2vec, so that it can be trained on large quantities of unlabeled text. We empirically show that CMOW better captures linguistic properties, but it is inferior to CBOW in memorizing word content. Motivated by these findings, we propose a hybrid model that combines the strengths of CBOW and CMOW. Our results show that the hybrid CBOW-CMOW-model retains CBOW's strong ability to memorize word content while at the same time substantially improving its ability to encode other linguistic information by 8%. As a result, the hybrid also performs better on 8 out of 11 supervised downstream tasks with an average improvement of 1.2%.

Class-Incremental Learning (CIL) requires a learning system to continually learn new classes without forgetting. Despite the strong performance of Pre-Trained Models (PTMs) in CIL, a critical issue persists: learning new classes often results in the overwriting of old ones. Excessive modification of the network causes forgetting, while minimal adjustments lead to an inadequate fit for new classes. As a result, it is desired to figure out a way of efficient model updating without harming former knowledge. In this paper, we propose ExpAndable Subspace Ensemble (EASE) for PTM-based CIL. To enable model updating without conflict, we train a distinct lightweight adapter module for each new task, aiming to create task-specific subspaces. These adapters span a high-dimensional feature space, enabling joint decision-making across multiple subspaces. As data evolves, the expanding subspaces render the old class classifiers incompatible with new-stage spaces. Correspondingly, we design a semantic-guided prototype complement strategy that synthesizes old classes' new features without using any old class instance. Extensive experiments on seven benchmark datasets verify EASE's state-of-the-art performance. Code is available at: https://github.com/sun-hailong/CVPR24-Ease

Neural embedding models have become a fundamental component of modern information retrieval (IR) pipelines. These models produce a single embedding x in R^d per data-point, allowing for fast retrieval via highly optimized maximum inner product search (MIPS) algorithms. Recently, beginning with the landmark ColBERT paper, multi-vector models, which produce a set of embedding per data point, have achieved markedly superior performance for IR tasks. Unfortunately, using these models for IR is computationally expensive due to the increased complexity of multi-vector retrieval and scoring. In this paper, we introduce MUVERA (MUlti-VEctor Retrieval Algorithm), a retrieval mechanism which reduces multi-vector similarity search to single-vector similarity search. This enables the usage of off-the-shelf MIPS solvers for multi-vector retrieval. MUVERA asymmetrically generates Fixed Dimensional Encodings (FDEs) of queries and documents, which are vectors whose inner product approximates multi-vector similarity. We prove that FDEs give high-quality epsilon-approximations, thus providing the first single-vector proxy for multi-vector similarity with theoretical guarantees. Empirically, we find that FDEs achieve the same recall as prior state-of-the-art heuristics while retrieving 2-5times fewer candidates. Compared to prior state of the art implementations, MUVERA achieves consistently good end-to-end recall and latency across a diverse set of the BEIR retrieval datasets, achieving an average of 10% improved recall with 90% lower latency.

We investigate compositional structures in data embeddings from pre-trained vision-language models (VLMs). Traditionally, compositionality has been associated with algebraic operations on embeddings of words from a pre-existing vocabulary. In contrast, we seek to approximate representations from an encoder as combinations of a smaller set of vectors in the embedding space. These vectors can be seen as "ideal words" for generating concepts directly within the embedding space of the model. We first present a framework for understanding compositional structures from a geometric perspective. We then explain what these compositional structures entail probabilistically in the case of VLM embeddings, providing intuitions for why they arise in practice. Finally, we empirically explore these structures in CLIP's embeddings and we evaluate their usefulness for solving different vision-language tasks such as classification, debiasing, and retrieval. Our results show that simple linear algebraic operations on embedding vectors can be used as compositional and interpretable methods for regulating the behavior of VLMs.

We develop a rigorous mathematical analysis of zero-shot learning with attributes. In this setting, the goal is to label novel classes with no training data, only detectors for attributes and a description of how those attributes are correlated with the target classes, called the class-attribute matrix. We develop the first non-trivial lower bound on the worst-case error of the best map from attributes to classes for this setting, even with perfect attribute detectors. The lower bound characterizes the theoretical intrinsic difficulty of the zero-shot problem based on the available information -- the class-attribute matrix -- and the bound is practically computable from it. Our lower bound is tight, as we show that we can always find a randomized map from attributes to classes whose expected error is upper bounded by the value of the lower bound. We show that our analysis can be predictive of how standard zero-shot methods behave in practice, including which classes will likely be confused with others.

This work examines the challenges of training neural networks using vector quantization using straight-through estimation. We find that a primary cause of training instability is the discrepancy between the model embedding and the code-vector distribution. We identify the factors that contribute to this issue, including the codebook gradient sparsity and the asymmetric nature of the commitment loss, which leads to misaligned code-vector assignments. We propose to address this issue via affine re-parameterization of the code vectors. Additionally, we introduce an alternating optimization to reduce the gradient error introduced by the straight-through estimation. Moreover, we propose an improvement to the commitment loss to ensure better alignment between the codebook representation and the model embedding. These optimization methods improve the mathematical approximation of the straight-through estimation and, ultimately, the model performance. We demonstrate the effectiveness of our methods on several common model architectures, such as AlexNet, ResNet, and ViT, across various tasks, including image classification and generative modeling.

Recent studies have discovered that Chain-of-Thought prompting (CoT) can dramatically improve the performance of Large Language Models (LLMs), particularly when dealing with complex tasks involving mathematics or reasoning. Despite the enormous empirical success, the underlying mechanisms behind CoT and how it unlocks the potential of LLMs remain elusive. In this paper, we take a first step towards theoretically answering these questions. Specifically, we examine the expressivity of LLMs with CoT in solving fundamental mathematical and decision-making problems. By using circuit complexity theory, we first give impossibility results showing that bounded-depth Transformers are unable to directly produce correct answers for basic arithmetic/equation tasks unless the model size grows super-polynomially with respect to the input length. In contrast, we then prove by construction that autoregressive Transformers of constant size suffice to solve both tasks by generating CoT derivations using a commonly used math language format. Moreover, we show LLMs with CoT can handle a general class of decision-making problems known as Dynamic Programming, thus justifying its power in tackling complex real-world tasks. Finally, an extensive set of experiments show that, while Transformers always fail to directly predict the answers, they can consistently learn to generate correct solutions step-by-step given sufficient CoT demonstrations.

When leveraging language models for reasoning tasks, generating explicit chain-of-thought (CoT) steps often proves essential for achieving high accuracy in final outputs. In this paper, we investigate if models can be taught to internalize these CoT steps. To this end, we propose a simple yet effective method for internalizing CoT steps: starting with a model trained for explicit CoT reasoning, we gradually remove the intermediate steps and finetune the model. This process allows the model to internalize the intermediate reasoning steps, thus simplifying the reasoning process while maintaining high performance. Our approach enables a GPT-2 Small model to solve 9-by-9 multiplication with up to 99% accuracy, whereas standard training cannot solve beyond 4-by-4 multiplication. Furthermore, our method proves effective on larger language models, such as Mistral 7B, achieving over 50% accuracy on GSM8K without producing any intermediate steps.

Recently, Large Reasoning Models (LRMs) have demonstrated remarkable capabilities in code reasoning by scaling up the length of Chain-of-Thought (CoT). However, excessively long reasoning traces introduce substantial challenges in terms of training cost, inference latency, and deployment feasibility. While various CoT compression approaches have emerged to address this challenge, they face inherent trade-offs: token-level methods often disrupt syntactic and logical coherence, while step-level methods based on perplexity fail to reliably capture the logically critical reasoning steps. In this paper, we propose ASAP (Anchor-guided, Surprisal-based Pruning), a novel coarse-to-fine framework for CoT compression. ASAP first performs anchor-guided pruning to preserve the core reasoning structure, which efficiently reduces the search space for subsequent processing. It then enables a logic-aware pruning by selecting logically essential reasoning steps based on a novel first-token surprisal metric. Finally, ASAP teaches models to autonomously generate and leverage these concise CoTs at inference time, enabling efficient reasoning in coding tasks. Experiments show that ASAP achieves state-of-the-art accuracy across multiple code generation benchmarks while substantially reducing training and inference costs. On the challenging LiveCodeBench v4_v5 benchmark, our approach reduces token generation by 23.5% and inference latency by 43.5% compared to the strongest baseline, while achieving a competitive accuracy of 36.19% in Pass@1. Our results highlight a promising direction for building powerful and efficient LRMs.

Dense retrieval models use bi-encoder network architectures for learning query and document representations. These representations are often in the form of a vector representation and their similarities are often computed using the dot product function. In this paper, we propose a new representation learning framework for dense retrieval. Instead of learning a vector for each query and document, our framework learns a multivariate distribution and uses negative multivariate KL divergence to compute the similarity between distributions. For simplicity and efficiency reasons, we assume that the distributions are multivariate normals and then train large language models to produce mean and variance vectors for these distributions. We provide a theoretical foundation for the proposed framework and show that it can be seamlessly integrated into the existing approximate nearest neighbor algorithms to perform retrieval efficiently. We conduct an extensive suite of experiments on a wide range of datasets, and demonstrate significant improvements compared to competitive dense retrieval models.

We report on a series of experiments with convolutional neural networks (CNN) trained on top of pre-trained word vectors for sentence-level classification tasks. We show that a simple CNN with little hyperparameter tuning and static vectors achieves excellent results on multiple benchmarks. Learning task-specific vectors through fine-tuning offers further gains in performance. We additionally propose a simple modification to the architecture to allow for the use of both task-specific and static vectors. The CNN models discussed herein improve upon the state of the art on 4 out of 7 tasks, which include sentiment analysis and question classification.

Step-by-step reasoning approaches like chain of thought (CoT) have proved to be very effective in inducing reasoning capabilities in large language models. However, the success of the CoT approach is fundamentally tied to the model size, and billion parameter-scale models are often needed to get CoT to work. In this paper, we propose a knowledge distillation approach that leverages the step-by-step CoT reasoning capabilities of larger models and distills these abilities into smaller models. In this work, we propose an alternative reasoning scheme, Socratic CoT, that learns a decomposition of the original problem into a sequence of subproblems and uses it to guide the intermediate reasoning steps. We use Socratic CoT to train a combination of two small distilled models: a problem decomposer and a subproblem solver. In practice, given a new problem, the two distilled models work in sync to decompose and solve complex problems. On multiple reasoning datasets (GSM8K, StrategyQA, and SVAMP), our proposed distillation strategies boosts the performance of smaller models over 70% compared to the baselines. Finally, we investigate when Socratic CoT is an effective alternative to CoT, demonstrating cases where a much smaller model (GPT-2 large) can outperform a 10X larger model (GPT-3 6B). Our code is available here: https://github.com/kumar-shridhar/Distiiling-LM

A major challenge in structured prediction is to represent the interdependencies within output structures. When outputs are structured as sequences, linear-chain conditional random fields (CRFs) are a widely used model class which can learn local dependencies in the output. However, the CRF's Markov assumption makes it impossible for CRFs to represent distributions with nonlocal dependencies, and standard CRFs are unable to respect nonlocal constraints of the data (such as global arity constraints on output labels). We present a generalization of CRFs that can enforce a broad class of constraints, including nonlocal ones, by specifying the space of possible output structures as a regular language L. The resulting regular-constrained CRF (RegCCRF) has the same formal properties as a standard CRF, but assigns zero probability to all label sequences not in L. Notably, RegCCRFs can incorporate their constraints during training, while related models only enforce constraints during decoding. We prove that constrained training is never worse than constrained decoding, and show empirically that it can be substantially better in practice. Additionally, we demonstrate a practical benefit on downstream tasks by incorporating a RegCCRF into a deep neural model for semantic role labeling, exceeding state-of-the-art results on a standard dataset.

Unsupervised pretraining, which learns a useful representation using a large amount of unlabeled data to facilitate the learning of downstream tasks, is a critical component of modern large-scale machine learning systems. Despite its tremendous empirical success, the rigorous theoretical understanding of why unsupervised pretraining generally helps remains rather limited -- most existing results are restricted to particular methods or approaches for unsupervised pretraining with specialized structural assumptions. This paper studies a generic framework, where the unsupervised representation learning task is specified by an abstract class of latent variable models Phi and the downstream task is specified by a class of prediction functions Psi. We consider a natural approach of using Maximum Likelihood Estimation (MLE) for unsupervised pretraining and Empirical Risk Minimization (ERM) for learning downstream tasks. We prove that, under a mild ''informative'' condition, our algorithm achieves an excess risk of mathcal{O}(mathcal{C_Phi/m} + mathcal{C_Psi/n}) for downstream tasks, where C_Phi, C_Psi are complexity measures of function classes Phi, Psi, and m, n are the number of unlabeled and labeled data respectively. Comparing to the baseline of mathcal{O}(mathcal{C_{Phi circ Psi}/n}) achieved by performing supervised learning using only the labeled data, our result rigorously shows the benefit of unsupervised pretraining when m gg n and C_{Phicirc Psi} > C_Psi. This paper further shows that our generic framework covers a wide range of approaches for unsupervised pretraining, including factor models, Gaussian mixture models, and contrastive learning.

Prediction tasks over nodes and edges in networks require careful effort in engineering features used by learning algorithms. Recent research in the broader field of representation learning has led to significant progress in automating prediction by learning the features themselves. However, present feature learning approaches are not expressive enough to capture the diversity of connectivity patterns observed in networks. Here we propose node2vec, an algorithmic framework for learning continuous feature representations for nodes in networks. In node2vec, we learn a mapping of nodes to a low-dimensional space of features that maximizes the likelihood of preserving network neighborhoods of nodes. We define a flexible notion of a node's network neighborhood and design a biased random walk procedure, which efficiently explores diverse neighborhoods. Our algorithm generalizes prior work which is based on rigid notions of network neighborhoods, and we argue that the added flexibility in exploring neighborhoods is the key to learning richer representations. We demonstrate the efficacy of node2vec over existing state-of-the-art techniques on multi-label classification and link prediction in several real-world networks from diverse domains. Taken together, our work represents a new way for efficiently learning state-of-the-art task-independent representations in complex networks.

Training large language models (LLMs) with high-quality Chain-of-Thought (CoT) annotations has become a widely adopted strategy due to its significant enhancement of reasoning capabilities. To fully comprehend this approach, two questions naturally arise: (Q1) What advantages does training with CoT offer compared to training without CoT? (Q2) If there are advantages, what are the underlying mechanisms of explicit CoT training? Analyzing the advantages and mechanisms of CoT training is challenging due to the many factors involved. To address this, we conduct a detailed analysis using clear and controllable data distributions and, for the first time, reveal that CoT training offers the following advantages: (1) Training with CoT markedly improves reasoning generalization, extending it from in-distribution (ID) to both ID and out-of-distribution (OOD) scenarios, while also speeding up convergence; (2) Even when training with CoT includes a certain range of erroneous reasoning steps, it still enables the model to learn reasoning patterns, leading to systematic generalization. We further explore the underlying mechanisms from a circuit perspective: (1) The data distribution (e.g., ratio lambda and pattern) plays a crucial role in influencing the model's systematic generalization; (2) CoT training (with two-hop facts) internalizes reasoning into a two-stage generalizing circuit, where the number of stages corresponds to the explicit reasoning steps during training. Our findings elucidate the mechanisms underlying explicit CoT training and offer critical insights into tuning strategies for LLMs to achieve robust generalization.

This paper presents our work on the Light-R1 series, with models, data, and code all released. We first focus on training long COT models from scratch, specifically starting from models initially lacking long COT capabilities. Using a curriculum training recipe consisting of two-stage SFT and semi-on-policy DPO, we train our model Light-R1-32B from Qwen2.5-32B-Instruct, resulting in superior math performance compared to DeepSeek-R1-Distill-Qwen-32B. Despite being trained exclusively on math data, Light-R1-32B shows strong generalization across other domains. In the subsequent phase of this work, we highlight the significant benefit of the 3k dataset constructed for the second SFT stage on enhancing other models. By fine-tuning DeepSeek-R1-Distilled models using this dataset, we obtain new SOTA models in 7B and 14B, while the 32B model, Light-R1-32B-DS performed comparably to QwQ-32B and DeepSeek-R1. Furthermore, we extend our work by applying reinforcement learning, specifically GRPO, on long-COT models to further improve reasoning performance. We successfully train our final Light-R1-14B-DS with RL, achieving SOTA performance among 14B parameter models in math. With AIME24 & 25 scores of 74.0 and 60.2 respectively, Light-R1-14B-DS surpasses even many 32B models and DeepSeek-R1-Distill-Llama-70B. Its RL training also exhibits well expected behavior, showing simultaneous increase in response length and reward score. The Light-R1 series of work validates training long-COT models from scratch, showcases the art in SFT data and releases SOTA models from RL.

We introduce a new neural architecture to learn the conditional probability of an output sequence with elements that are discrete tokens corresponding to positions in an input sequence. Such problems cannot be trivially addressed by existent approaches such as sequence-to-sequence and Neural Turing Machines, because the number of target classes in each step of the output depends on the length of the input, which is variable. Problems such as sorting variable sized sequences, and various combinatorial optimization problems belong to this class. Our model solves the problem of variable size output dictionaries using a recently proposed mechanism of neural attention. It differs from the previous attention attempts in that, instead of using attention to blend hidden units of an encoder to a context vector at each decoder step, it uses attention as a pointer to select a member of the input sequence as the output. We call this architecture a Pointer Net (Ptr-Net). We show Ptr-Nets can be used to learn approximate solutions to three challenging geometric problems -- finding planar convex hulls, computing Delaunay triangulations, and the planar Travelling Salesman Problem -- using training examples alone. Ptr-Nets not only improve over sequence-to-sequence with input attention, but also allow us to generalize to variable size output dictionaries. We show that the learnt models generalize beyond the maximum lengths they were trained on. We hope our results on these tasks will encourage a broader exploration of neural learning for discrete problems.

Chain-of-Thought (CoT) enhances Large Language Models (LLMs) by enabling step-by-step reasoning in natural language. However, the language space may be suboptimal for reasoning. While implicit CoT methods attempt to enable reasoning without explicit CoT tokens, they have consistently lagged behind explicit CoT method in task performance. We propose CODI (Continuous Chain-of-Thought via Self-Distillation), a novel framework that distills CoT into a continuous space, where a shared model acts as both teacher and student, jointly learning explicit and implicit CoT while aligning their hidden activation on the token generating the final answer. CODI is the first implicit CoT method to match explicit CoT's performance on GSM8k while achieving 3.1x compression, surpassing the previous state-of-the-art by 28.2% in accuracy. Furthermore, CODI demonstrates scalability, robustness, and generalizability to more complex CoT datasets. Additionally, CODI retains interpretability by decoding its continuous thoughts, making its reasoning process transparent. Our findings establish implicit CoT as not only a more efficient but a powerful alternative to explicit CoT.

Recent advances in Chain-of-Thought (CoT) reasoning have improved complex video understanding, but existing methods often struggle to adapt to domain-specific skills (e.g., event detection, spatial relation understanding, emotion understanding) over various video content. To address this, we propose Video-Skill-CoT (a.k.a. Video-SKoT), a framework that automatically constructs and leverages skill-aware CoT supervisions for domain-adaptive video reasoning. First, we construct skill-based CoT annotations: we extract domain-relevant reasoning skills from training questions, cluster them into a shared skill taxonomy, and create detailed multi-step CoT rationale tailored to each video-question pair for training. Second, we introduce a skill-specific expert learning framework. Each expert module specializes in a subset of reasoning skills and is trained with lightweight adapters using the collected CoT supervision. We demonstrate the effectiveness of the proposed approach on three video understanding benchmarks, where Video-SKoT consistently outperforms strong baselines. We also provide in-depth analyses on comparing different CoT annotation pipelines and learned skills over multiple video domains.

Word embeddings are an essential component in a wide range of natural language processing applications. However, distributional semantic models are known to struggle when only a small number of context sentences are available. Several methods have been proposed to obtain higher-quality vectors for these words, leveraging both this context information and sometimes the word forms themselves through a hybrid approach. We show that the current tasks do not suffice to evaluate models that use word-form information, as such models can easily leverage word forms in the training data that are related to word forms in the test data. We introduce 3 new tasks, allowing for a more balanced comparison between models. Furthermore, we show that hyperparameters that have largely been ignored in previous work can consistently improve the performance of both baseline and advanced models, achieving a new state of the art on 4 out of 6 tasks.

Current parameter-efficient fine-tuning (PEFT) methods build adapters without considering the context of downstream task to learn, or the context of important knowledge to maintain. As a result, there is often a performance gap compared to full-parameter finetuning, and meanwhile the finetuned model suffers from catastrophic forgetting of the pre-trained world knowledge. In this paper, we propose CorDA, a Context-oriented Decomposition Adaptation method that builds learnable adapters from weight decomposition oriented by the context of downstream task or world knowledge. Concretely, we collect a few data samples, and perform singular value decomposition for each linear layer of a pre-trained LLM multiplied by the covariance matrix of the input activation using these samples. By doing so, the context of the representative samples is captured through deciding the factorizing orientation. Our method enables two options, the knowledge-preserved adaptation and the instruction-previewed adaptation. For the former, we use question-answering samples to obtain the covariance matrices, and use the decomposed components with the smallest r singular values to initialize a learnable adapter, with the others frozen such that the world knowledge is better preserved. For the latter, we use the instruction data from the finetuning task, such as math or coding, to orientate the decomposition and train the largest r components that capture the main characteristics of the task to learn. We conduct extensive experiments on Math, Code, and Instruction Following tasks. Our knowledge-preserved adaptation not only achieves better performance than LoRA on finetuning tasks, but also mitigates the forgetting of world knowledge. Our instruction-previewed adaptation is able to further enhance the finetuning performance, surpassing full-parameter finetuning and the state-of-the-art PEFT methods.

Contrastive learning, a prominent approach to representation learning, traditionally assumes positive pairs are closely related samples (the same image or class) and negative pairs are distinct samples. We challenge this assumption by proposing to learn from arbitrary pairs, allowing any pair of samples to be positive within our framework.The primary challenge of the proposed approach lies in applying contrastive learning to disparate pairs which are semantically distant. Motivated by the discovery that SimCLR can separate given arbitrary pairs (e.g., garter snake and table lamp) in a subspace, we propose a feature filter in the condition of class pairs that creates the requisite subspaces by gate vectors selectively activating or deactivating dimensions. This filter can be optimized through gradient descent within a conventional contrastive learning mechanism. We present Hydra, a universal contrastive learning framework for visual representations that extends conventional contrastive learning to accommodate arbitrary pairs. Our approach is validated using IN1K, where 1K diverse classes compose 500,500 pairs, most of them being distinct. Surprisingly, Hydra achieves superior performance in this challenging setting. Additional benefits include the prevention of dimensional collapse and the discovery of class relationships. Our work highlights the value of learning common features of arbitrary pairs and potentially broadens the applicability of contrastive learning techniques on the sample pairs with weak relationships.

Dimensionality reduction methods are unsupervised approaches which learn low-dimensional spaces where some properties of the initial space, typically the notion of "neighborhood", are preserved. Such methods usually require propagation on large k-NN graphs or complicated optimization solvers. On the other hand, self-supervised learning approaches, typically used to learn representations from scratch, rely on simple and more scalable frameworks for learning. In this paper, we propose TLDR, a dimensionality reduction method for generic input spaces that is porting the recent self-supervised learning framework of Zbontar et al. (2021) to the specific task of dimensionality reduction, over arbitrary representations. We propose to use nearest neighbors to build pairs from a training set and a redundancy reduction loss to learn an encoder that produces representations invariant across such pairs. TLDR is a method that is simple, easy to train, and of broad applicability; it consists of an offline nearest neighbor computation step that can be highly approximated, and a straightforward learning process. Aiming for scalability, we focus on improving linear dimensionality reduction, and show consistent gains on image and document retrieval tasks, e.g. gaining +4% mAP over PCA on ROxford for GeM- AP, improving the performance of DINO on ImageNet or retaining it with a 10x compression.

Chain-of-Thought (CoT) reasoning has been extensively explored in large models to tackle complex understanding tasks. However, it still remains an open question whether such strategies can be applied to verifying and reinforcing image generation scenarios. In this paper, we provide the first comprehensive investigation of the potential of CoT reasoning to enhance autoregressive image generation. We focus on three techniques: scaling test-time computation for verification, aligning model preferences with Direct Preference Optimization (DPO), and integrating these techniques for complementary effects. Our results demonstrate that these approaches can be effectively adapted and combined to significantly improve image generation performance. Furthermore, given the pivotal role of reward models in our findings, we propose the Potential Assessment Reward Model (PARM) and PARM++, specialized for autoregressive image generation. PARM adaptively assesses each generation step through a potential assessment approach, merging the strengths of existing reward models, and PARM++ further introduces a reflection mechanism to self-correct the generated unsatisfactory image. Using our investigated reasoning strategies, we enhance a baseline model, Show-o, to achieve superior results, with a significant +24% improvement on the GenEval benchmark, surpassing Stable Diffusion 3 by +15%. We hope our study provides unique insights and paves a new path for integrating CoT reasoning with autoregressive image generation. Code and models are released at https://github.com/ZiyuGuo99/Image-Generation-CoT

Learning unified text embeddings that excel across diverse downstream tasks is a central goal in representation learning, yet negative transfer remains a persistent obstacle. This challenge is particularly pronounced when jointly training a single encoder for Information Retrieval (IR) and Semantic Textual Similarity (STS), two essential but fundamentally disparate tasks for which naive co-training typically yields steep performance trade-offs. We argue that resolving this conflict requires systematically decoupling task-specific learning signals throughout the training pipeline. To this end, we introduce CoDiEmb, a unified framework that reconciles the divergent requirements of IR and STS in a collaborative yet distinct manner. CoDiEmb integrates three key innovations for effective joint optimization: (1) Task-specialized objectives paired with a dynamic sampler that forms single-task batches and balances per-task updates, thereby preventing gradient interference. For IR, we employ a contrastive loss with multiple positives and hard negatives, augmented by cross-device sampling. For STS, we adopt order-aware objectives that directly optimize correlation and ranking consistency. (2) A delta-guided model fusion strategy that computes fine-grained merging weights for checkpoints by analyzing each parameter's deviation from its pre-trained initialization, proving more effective than traditional Model Soups. (3) An efficient, single-stage training pipeline that is simple to implement and converges stably. Extensive experiments on 15 standard IR and STS benchmarks across three base encoders validate CoDiEmb. Our results and analysis demonstrate that the framework not only mitigates cross-task trade-offs but also measurably improves the geometric properties of the embedding space.

Coordinate-based neural representations have shown significant promise as an alternative to discrete, array-based representations for complex low dimensional signals. However, optimizing a coordinate-based network from randomly initialized weights for each new signal is inefficient. We propose applying standard meta-learning algorithms to learn the initial weight parameters for these fully-connected networks based on the underlying class of signals being represented (e.g., images of faces or 3D models of chairs). Despite requiring only a minor change in implementation, using these learned initial weights enables faster convergence during optimization and can serve as a strong prior over the signal class being modeled, resulting in better generalization when only partial observations of a given signal are available. We explore these benefits across a variety of tasks, including representing 2D images, reconstructing CT scans, and recovering 3D shapes and scenes from 2D image observations.

General-purpose multilingual vector representations, used in retrieval, regression and classification, are traditionally obtained from bidirectional encoder models. Despite their wide applicability, encoders have been recently overshadowed by advances in generative decoder-only models. However, many innovations driving this progress are not inherently tied to decoders. In this paper, we revisit the development of multilingual encoders through the lens of these advances, and introduce EuroBERT, a family of multilingual encoders covering European and widely spoken global languages. Our models outperform existing alternatives across a diverse range of tasks, spanning multilingual capabilities, mathematics, and coding, and natively supporting sequences of up to 8,192 tokens. We also examine the design decisions behind EuroBERT, offering insights into our dataset composition and training pipeline. We publicly release the EuroBERT models, including intermediate training checkpoints, together with our training framework.

We introduce Reprompting, an iterative sampling algorithm that searches for the Chain-of-Thought (CoT) recipes for a given task without human intervention. Through Gibbs sampling, we infer CoT recipes that work consistently well for a set of training samples. Our method iteratively samples new recipes using previously sampled solutions as parent prompts to solve other training problems. On five Big-Bench Hard tasks that require multi-step reasoning, Reprompting achieves consistently better performance than the zero-shot, few-shot, and human-written CoT baselines. Reprompting can also facilitate transfer of knowledge from a stronger model to a weaker model leading to substantially improved performance of the weaker model. Overall, Reprompting brings up to +17 point improvements over the previous state-of-the-art method that uses human-written CoT prompts.

Self-supervised learning (SSL) has emerged as a powerful paradigm for learning representations without labeled data, often by enforcing invariance to input transformations such as rotations or blurring. Recent studies have highlighted two pivotal properties for effective representations: (i) avoiding dimensional collapse-where the learned features occupy only a low-dimensional subspace, and (ii) enhancing uniformity of the induced distribution. In this work, we introduce T-REGS, a simple regularization framework for SSL based on the length of the Minimum Spanning Tree (MST) over the learned representation. We provide theoretical analysis demonstrating that T-REGS simultaneously mitigates dimensional collapse and promotes distribution uniformity on arbitrary compact Riemannian manifolds. Several experiments on synthetic data and on classical SSL benchmarks validate the effectiveness of our approach at enhancing representation quality.

In this work, we demonstrate that a simple two-layer neural network with standard activation functions can learn an arbitrary word operation in any finite group, provided sufficient width is available and exhibits grokking while doing so. To explain the mechanism by which this is achieved, we reframe the problem as that of learning a particular 3-tensor, which we show is typically of low rank. A key insight is that low-rank implementations of this tensor can be obtained by decomposing it along triplets of basic self-conjugate representations of the group and leveraging the fusion structure to rule out many components. Focusing on a phenomenologically similar but more tractable surrogate model, we show that the network is able to find such low-rank implementations (or approximations thereof), thereby using limited width to approximate the word-tensor in a generalizable way. In the case of the simple multiplication word, we further elucidate the form of these low-rank implementations, showing that the network effectively implements efficient matrix multiplication in the sense of Strassen. Our work also sheds light on the mechanism by which a network reaches such a solution under gradient descent.

The recently introduced continuous Skip-gram model is an efficient method for learning high-quality distributed vector representations that capture a large number of precise syntactic and semantic word relationships. In this paper we present several extensions that improve both the quality of the vectors and the training speed. By subsampling of the frequent words we obtain significant speedup and also learn more regular word representations. We also describe a simple alternative to the hierarchical softmax called negative sampling. An inherent limitation of word representations is their indifference to word order and their inability to represent idiomatic phrases. For example, the meanings of "Canada" and "Air" cannot be easily combined to obtain "Air Canada". Motivated by this example, we present a simple method for finding phrases in text, and show that learning good vector representations for millions of phrases is possible.

In this work, we study the problem of Text-to-Image In-Context Learning (T2I-ICL). While Unified Multimodal LLMs (MLLMs) have advanced rapidly in recent years, they struggle with contextual reasoning in T2I-ICL scenarios. To address this limitation, we propose a novel framework that incorporates a thought process called ImageGen-CoT prior to image generation. To avoid generating unstructured ineffective reasoning steps, we develop an automatic pipeline to curate a high-quality ImageGen-CoT dataset. We then fine-tune MLLMs using this dataset to enhance their contextual reasoning capabilities. To further enhance performance, we explore test-time scale-up strategies and propose a novel hybrid scaling approach. This approach first generates multiple ImageGen-CoT chains and then produces multiple images for each chain via sampling. Extensive experiments demonstrate the effectiveness of our proposed method. Notably, fine-tuning with the ImageGen-CoT dataset leads to a substantial 80\% performance gain for SEED-X on T2I-ICL tasks. See our project page at https://ImageGen-CoT.github.io/. Code and model weights will be open-sourced.

Recently, DeepSeek-R1 (671B) (DeepSeek-AIet al., 2025) has demonstrated its excellent reasoning ability in complex tasks and has publiclyshared its methodology. This provides potentially high-quality chain-of-thought (CoT) data for stimulating the reasoning abilities of small-sized large language models (LLMs). To generate high-quality CoT data for different LLMs, we seek an efficient method for generating high-quality CoT data with LLM-Adaptive questiondifficulty levels. First, we grade the difficulty of the questions according to the reasoning ability of the LLMs themselves and construct a LLM-Adaptive question database. Second, we sample the problem database based on a distribution of difficulty levels of the questions and then use DeepSeek-R1 (671B) (DeepSeek-AI et al., 2025) to generate the corresponding high-quality CoT data with correct answers. Thanks to the construction of CoT data with LLM-Adaptive difficulty levels, we have significantly reduced the cost of data generation and enhanced the efficiency of model supervised fine-tuning (SFT). Finally, we have validated the effectiveness and generalizability of the proposed method in the fields of complex mathematical competitions and code generation tasks. Notably, with only 2k high-quality mathematical CoT data, our ZMath-32B surpasses DeepSeek-Distill-32B in math reasoning task. Similarly, with only 2k high-quality code CoT data, our ZCode-32B surpasses DeepSeek-Distill-32B in code reasoning tasks.

We present a simple picture of the training process of joint embedding self-supervised learning methods. We find that these methods learn their high-dimensional embeddings one dimension at a time in a sequence of discrete, well-separated steps. We arrive at this conclusion via the study of a linearized model of Barlow Twins applicable to the case in which the trained network is infinitely wide. We solve the training dynamics of this model from small initialization, finding that the model learns the top eigenmodes of a certain contrastive kernel in a stepwise fashion, and obtain a closed-form expression for the final learned representations. Remarkably, we then see the same stepwise learning phenomenon when training deep ResNets using the Barlow Twins, SimCLR, and VICReg losses. Our theory suggests that, just as kernel regression can be thought of as a model of supervised learning, kernel PCA may serve as a useful model of self-supervised learning.

The use of continuous instead of discrete tokens during the Chain-of-Thought (CoT) phase of reasoning LLMs has garnered attention recently, based on the intuition that a continuous mixture of discrete tokens could simulate a superposition of several reasoning paths simultaneously. Theoretical results have formally proven that continuous tokens have much greater expressivity and can solve specific problems more efficiently. However, practical use of continuous tokens has been limited by strong training difficulties: previous works either just use continuous tokens at inference time on a pre-trained discrete-token model, or must distill the continuous CoT from ground-truth discrete CoTs and face computational costs that limit the CoT to very few tokens. This is the first work introducing a scalable method to learn continuous CoTs via reinforcement learning (RL), without distilling from reference discrete CoTs. We use "soft" tokens: mixtures of tokens together with noise on the input embedding to provide RL exploration. Computational overhead is minimal, enabling us to learn continuous CoTs with hundreds of tokens. On math reasoning benchmarks with Llama and Qwen models up to 8B, training with continuous CoTs match discrete-token CoTs for pass@1 and surpass them for pass@32, showing greater CoT diversity. In systematic comparisons, the best-performing scenario is to train with continuous CoT tokens then use discrete tokens for inference, meaning the "soft" models can be deployed in a standard way. Finally, we show continuous CoT RL training better preserves the predictions of the base model on out-of-domain tasks, thus providing a softer touch to the base model.

Recent self-supervised methods for image representation learning are based on maximizing the agreement between embedding vectors from different views of the same image. A trivial solution is obtained when the encoder outputs constant vectors. This collapse problem is often avoided through implicit biases in the learning architecture, that often lack a clear justification or interpretation. In this paper, we introduce VICReg (Variance-Invariance-Covariance Regularization), a method that explicitly avoids the collapse problem with a simple regularization term on the variance of the embeddings along each dimension individually. VICReg combines the variance term with a decorrelation mechanism based on redundancy reduction and covariance regularization, and achieves results on par with the state of the art on several downstream tasks. In addition, we show that incorporating our new variance term into other methods helps stabilize the training and leads to performance improvements.

Recent works have demonstrated reasonable success of representation learning in hypercomplex space. Specifically, "fully-connected layers with Quaternions" (4D hypercomplex numbers), which replace real-valued matrix multiplications in fully-connected layers with Hamilton products of Quaternions, both enjoy parameter savings with only 1/4 learnable parameters and achieve comparable performance in various applications. However, one key caveat is that hypercomplex space only exists at very few predefined dimensions (4D, 8D, and 16D). This restricts the flexibility of models that leverage hypercomplex multiplications. To this end, we propose parameterizing hypercomplex multiplications, allowing models to learn multiplication rules from data regardless of whether such rules are predefined. As a result, our method not only subsumes the Hamilton product, but also learns to operate on any arbitrary nD hypercomplex space, providing more architectural flexibility using arbitrarily 1/n learnable parameters compared with the fully-connected layer counterpart. Experiments of applications to the LSTM and Transformer models on natural language inference, machine translation, text style transfer, and subject verb agreement demonstrate architectural flexibility and effectiveness of the proposed approach.

We study whether visual embedding models capture continuous, ordinal attributes along linear directions, which we term _rank axes_. We define a model as _rankable_ for an attribute if projecting embeddings onto such an axis preserves the attribute's order. Across 7 popular encoders and 9 datasets with attributes like age, crowd count, head pose, aesthetics, and recency, we find that many embeddings are inherently rankable. Surprisingly, a small number of samples, or even just two extreme examples, often suffice to recover meaningful rank axes, without full-scale supervision. These findings open up new use cases for image ranking in vector databases and motivate further study into the structure and learning of rankable embeddings. Our code is available at https://github.com/aktsonthalia/rankable-vision-embeddings.

Vector Quantization (VQ) is a widely used method for converting continuous representations into discrete codes, which has become fundamental in unsupervised representation learning and latent generative models. However, VQ models are often hindered by the problem of representation collapse in the latent space, which leads to low codebook utilization and limits the scalability of the codebook for large-scale training. Existing methods designed to mitigate representation collapse typically reduce the dimensionality of latent space at the expense of model capacity, which do not fully resolve the core issue. In this study, we conduct a theoretical analysis of representation collapse in VQ models and identify its primary cause as the disjoint optimization of the codebook, where only a small subset of code vectors are updated through gradient descent. To address this issue, we propose SimVQ, a novel method which reparameterizes the code vectors through a linear transformation layer based on a learnable latent basis. This transformation optimizes the entire linear space spanned by the codebook, rather than merely updating the code vector selected by the nearest-neighbor search in vanilla VQ models. Although it is commonly understood that the multiplication of two linear matrices is equivalent to applying a single linear layer, our approach works surprisingly well in resolving the collapse issue in VQ models with just one linear layer. We validate the efficacy of SimVQ through extensive experiments across various modalities, including image and audio data with different model architectures. Our code is available at https://github.com/youngsheen/SimVQ.

Learning high-quality feature embeddings efficiently and effectively is critical for the performance of web-scale machine learning systems. A typical model ingests hundreds of features with vocabularies on the order of millions to billions of tokens. The standard approach is to represent each feature value as a d-dimensional embedding, introducing hundreds of billions of parameters for extremely high-cardinality features. This bottleneck has led to substantial progress in alternative embedding algorithms. Many of these methods, however, make the assumption that each feature uses an independent embedding table. This work introduces a simple yet highly effective framework, Feature Multiplexing, where one single representation space is used across many different categorical features. Our theoretical and empirical analysis reveals that multiplexed embeddings can be decomposed into components from each constituent feature, allowing models to distinguish between features. We show that multiplexed representations lead to Pareto-optimal parameter-accuracy tradeoffs for three public benchmark datasets. Further, we propose a highly practical approach called Unified Embedding with three major benefits: simplified feature configuration, strong adaptation to dynamic data distributions, and compatibility with modern hardware. Unified embedding gives significant improvements in offline and online metrics compared to highly competitive baselines across five web-scale search, ads, and recommender systems, where it serves billions of users across the world in industry-leading products.

In this work, we present a variety of novel information-theoretic generalization bounds for learning algorithms, from the supersample setting of Steinke & Zakynthinou (2020)-the setting of the "conditional mutual information" framework. Our development exploits projecting the loss pair (obtained from a training instance and a testing instance) down to a single number and correlating loss values with a Rademacher sequence (and its shifted variants). The presented bounds include square-root bounds, fast-rate bounds, including those based on variance and sharpness, and bounds for interpolating algorithms etc. We show theoretically or empirically that these bounds are tighter than all information-theoretic bounds known to date on the same supersample setting.

Transformers are widely used to extract semantic meanings from input tokens, yet they usually operate as black-box models. In this paper, we present a simple yet informative decomposition of hidden states (or embeddings) of trained transformers into interpretable components. For any layer, embedding vectors of input sequence samples are represented by a tensor h in R^{C times T times d}. Given embedding vector h_{c,t} in R^d at sequence position t le T in a sequence (or context) c le C, extracting the mean effects yields the decomposition \[ h_{c,t} = \mu + pos_t + ctx_c + resid_{c,t} \] where mu is the global mean vector, pos_t and ctx_c are the mean vectors across contexts and across positions respectively, and resid_{c,t} is the residual vector. For popular transformer architectures and diverse text datasets, empirically we find pervasive mathematical structure: (1) (pos_t)_{t} forms a low-dimensional, continuous, and often spiral shape across layers, (2) (ctx_c)_c shows clear cluster structure that falls into context topics, and (3) (pos_t)_{t} and (ctx_c)_c are mutually nearly orthogonal. We argue that smoothness is pervasive and beneficial to transformers trained on languages, and our decomposition leads to improved model interpretability.

Recent works have shown that chain-of-thought (CoT) prompting can elicit language models to solve complex reasoning tasks, step-by-step. However, prompt-based CoT methods are dependent on very large models such as GPT-3 175B which are prohibitive to deploy at scale. In this paper, we use these large models as reasoning teachers to enable complex reasoning in smaller models and reduce model size requirements by several orders of magnitude. We propose Fine-tune-CoT, a method that generates reasoning samples from very large teacher models to fine-tune smaller models. We evaluate our method on a wide range of public models and complex tasks. We find that Fine-tune-CoT enables substantial reasoning capability in small models, far outperforming prompt-based baselines and even the teacher model in many tasks. Additionally, we extend our method by leveraging the teacher model's ability to generate multiple distinct rationales for each original sample. Enriching the fine-tuning data with such diverse reasoning results in a substantial performance boost across datasets, even for very small models. We conduct ablations and sample studies to understand the emergence of reasoning capabilities of student models. Our code implementation and data are available at https://github.com/itsnamgyu/reasoning-teacher.

Mixture models arise in many regression problems, but most methods have seen limited adoption partly due to these algorithms' highly-tailored and model-specific nature. On the other hand, transformers are flexible, neural sequence models that present the intriguing possibility of providing general-purpose prediction methods, even in this mixture setting. In this work, we investigate the hypothesis that transformers can learn an optimal predictor for mixtures of regressions. We construct a generative process for a mixture of linear regressions for which the decision-theoretic optimal procedure is given by data-driven exponential weights on a finite set of parameters. We observe that transformers achieve low mean-squared error on data generated via this process. By probing the transformer's output at inference time, we also show that transformers typically make predictions that are close to the optimal predictor. Our experiments also demonstrate that transformers can learn mixtures of regressions in a sample-efficient fashion and are somewhat robust to distribution shifts. We complement our experimental observations by proving constructively that the decision-theoretic optimal procedure is indeed implementable by a transformer.

An emerging solution for explaining Transformer-based models is to use vector-based analysis on how the representations are formed. However, providing a faithful vector-based explanation for a multi-layer model could be challenging in three aspects: (1) Incorporating all components into the analysis, (2) Aggregating the layer dynamics to determine the information flow and mixture throughout the entire model, and (3) Identifying the connection between the vector-based analysis and the model's predictions. In this paper, we present DecompX to tackle these challenges. DecompX is based on the construction of decomposed token representations and their successive propagation throughout the model without mixing them in between layers. Additionally, our proposal provides multiple advantages over existing solutions for its inclusion of all encoder components (especially nonlinear feed-forward networks) and the classification head. The former allows acquiring precise vectors while the latter transforms the decomposition into meaningful prediction-based values, eliminating the need for norm- or summation-based vector aggregation. According to the standard faithfulness evaluations, DecompX consistently outperforms existing gradient-based and vector-based approaches on various datasets. Our code is available at https://github.com/mohsenfayyaz/DecompX.

Machine learning models -- including prominent zero-shot models -- are often trained on datasets whose labels are only a small proportion of a larger label space. Such spaces are commonly equipped with a metric that relates the labels via distances between them. We propose a simple approach to exploit this information to adapt the trained model to reliably predict new classes -- or, in the case of zero-shot prediction, to improve its performance -- without any additional training. Our technique is a drop-in replacement of the standard prediction rule, swapping argmax with the Fr\'echet mean. We provide a comprehensive theoretical analysis for this approach, studying (i) learning-theoretic results trading off label space diameter, sample complexity, and model dimension, (ii) characterizations of the full range of scenarios in which it is possible to predict any unobserved class, and (iii) an optimal active learning-like next class selection procedure to obtain optimal training classes for when it is not possible to predict the entire range of unobserved classes. Empirically, using easily-available external metrics, our proposed approach, Loki, gains up to 29.7% relative improvement over SimCLR on ImageNet and scales to hundreds of thousands of classes. When no such metric is available, Loki can use self-derived metrics from class embeddings and obtains a 10.5% improvement on pretrained zero-shot models such as CLIP.

Compatible features enable the direct comparison of old and new learned features allowing to use them interchangeably over time. In visual search systems, this eliminates the need to extract new features from the gallery-set when the representation model is upgraded with novel data. This has a big value in real applications as re-indexing the gallery-set can be computationally expensive when the gallery-set is large, or even infeasible due to privacy or other concerns of the application. In this paper, we propose CoReS, a new training procedure to learn representations that are compatible with those previously learned, grounding on the stationarity of the features as provided by fixed classifiers based on polytopes. With this solution, classes are maximally separated in the representation space and maintain their spatial configuration stationary as new classes are added, so that there is no need to learn any mappings between representations nor to impose pairwise training with the previously learned model. We demonstrate that our training procedure largely outperforms the current state of the art and is particularly effective in the case of multiple upgrades of the training-set, which is the typical case in real applications.

Chain-of-thought (CoT) reasoning has been widely applied in the mathematical reasoning of Large Language Models (LLMs). Recently, the introduction of derivative process supervision on CoT trajectories has sparked discussions on enhancing scaling capabilities during test time, thereby boosting the potential of these models. However, in multimodal mathematical reasoning, the scarcity of high-quality CoT training data has hindered existing models from achieving high-precision CoT reasoning and has limited the realization of reasoning potential during test time. In this work, we propose a three-module synthesis strategy that integrates CoT distillation, trajectory-format rewriting, and format unification. It results in a high-quality CoT reasoning instruction fine-tuning dataset in multimodal mathematics, MMathCoT-1M. We comprehensively validate the state-of-the-art (SOTA) performance of the trained URSA-7B model on multiple multimodal mathematical benchmarks. For test-time scaling, we introduce a data synthesis strategy that automatically generates process annotation datasets, known as DualMath-1.1M, focusing on both interpretation and logic. By further training URSA-7B on DualMath-1.1M, we transition from CoT reasoning capabilities to robust supervision abilities. The trained URSA-RM-7B acts as a verifier, effectively enhancing the performance of URSA-7B at test time. URSA-RM-7B also demonstrates excellent out-of-distribution (OOD) verifying capabilities, showcasing its generalization. Model weights, training data and code will be open-sourced.

Prompt-based text embedding models, which generate task-specific embeddings upon receiving tailored prompts, have recently demonstrated remarkable performance. However, their resulting embeddings often have thousands of dimensions, leading to high storage costs and increased computational costs of embedding-based operations. In this paper, we investigate how post-hoc dimensionality reduction applied to the embeddings affects the performance of various tasks that leverage these embeddings, specifically classification, clustering, retrieval, and semantic textual similarity (STS) tasks. Our experiments show that even a naive dimensionality reduction, which keeps only the first 25% of the dimensions of the embeddings, results in a very slight performance degradation, indicating that these embeddings are highly redundant. Notably, for classification and clustering, even when embeddings are reduced to less than 0.5% of the original dimensionality the performance degradation is very small. To quantitatively analyze this redundancy, we perform an analysis based on the intrinsic dimensionality and isotropy of the embeddings. Our analysis reveals that embeddings for classification and clustering, which are considered to have very high dimensional redundancy, exhibit lower intrinsic dimensionality and less isotropy compared with those for retrieval and STS.

Loss functions serve as the foundation of supervised learning and are often chosen prior to model development. To avoid potentially ad hoc choices of losses, statistical decision theory describes a desirable property for losses known as properness, which asserts that Bayes' rule is optimal. Recent works have sought to learn losses and models jointly. Existing methods do this by fitting an inverse canonical link function which monotonically maps R to [0,1] to estimate probabilities for binary problems. In this paper, we extend monotonicity to maps between R^{C-1} and the projected probability simplex Delta^{C-1} by using monotonicity of gradients of convex functions. We present {\sc LegendreTron} as a novel and practical method that jointly learns proper canonical losses and probabilities for multiclass problems. Tested on a benchmark of domains with up to 1,000 classes, our experimental results show that our method consistently outperforms the natural multiclass baseline under a t-test at 99% significance on all datasets with greater than 10 classes.

Large language models (LLMs) have shown remarkable in-context learning (ICL) capabilities on textual data. We explore whether these capabilities can be extended to continuous vectors from diverse domains, obtained from black-box pretrained encoders. By aligning input data with an LLM's embedding space through lightweight projectors, we observe that LLMs can effectively process and learn from these projected vectors, which we term Vector-ICL. In particular, we find that pretraining projectors with general language modeling objectives enables Vector-ICL, while task-specific finetuning further enhances performance. In our experiments across various tasks and modalities, including text reconstruction, numerical function regression, text classification, summarization, molecule captioning, time-series classification, graph classification, and fMRI decoding, Vector-ICL often surpasses both few-shot ICL and domain-specific model or tuning. We further conduct analyses and case studies, indicating the potential of LLMs to process vector representations beyond traditional token-based paradigms.

Large Language Models (LLMs) have demonstrated remarkable performance across diverse tasks and exhibited impressive reasoning abilities by applying zero-shot Chain-of-Thought (CoT) prompting. However, due to the evolving nature of sentence prefixes during the pre-training phase, existing zero-shot CoT prompting methods that employ identical CoT prompting across all task instances may not be optimal. In this paper, we introduce a novel zero-shot prompting method that leverages evolutionary algorithms to generate diverse promptings for LLMs dynamically. Our approach involves initializing two CoT promptings, performing evolutionary operations based on LLMs to create a varied set, and utilizing the LLMs to select a suitable CoT prompting for a given problem. Additionally, a rewriting operation, guided by the selected CoT prompting, enhances the understanding of the LLMs about the problem. Extensive experiments conducted across ten reasoning datasets demonstrate the superior performance of our proposed method compared to current zero-shot CoT prompting methods on GPT-3.5-turbo and GPT-4. Moreover, in-depth analytical experiments underscore the adaptability and effectiveness of our method in various reasoning tasks.

One of the main methods for computational interpretation of a text is mapping it into a vector in some embedding space. Such vectors can then be used for a variety of textual processing tasks. Recently, most embedding spaces are a product of training large language models (LLMs). One major drawback of this type of representation is their incomprehensibility to humans. Understanding the embedding space is crucial for several important needs, including the need to debug the embedding method and compare it to alternatives, and the need to detect biases hidden in the model. In this paper, we present a novel method of understanding embeddings by transforming a latent embedding space into a comprehensible conceptual space. We present an algorithm for deriving a conceptual space with dynamic on-demand granularity. We devise a new evaluation method, using either human rater or LLM-based raters, to show that the conceptualized vectors indeed represent the semantics of the original latent ones. We show the use of our method for various tasks, including comparing the semantics of alternative models and tracing the layers of the LLM. The code is available online https://github.com/adiSimhi/Interpreting-Embedding-Spaces-by-Conceptualization.

Distributed dense word vectors have been shown to be effective at capturing token-level semantic and syntactic regularities in language, while topic models can form interpretable representations over documents. In this work, we describe lda2vec, a model that learns dense word vectors jointly with Dirichlet-distributed latent document-level mixtures of topic vectors. In contrast to continuous dense document representations, this formulation produces sparse, interpretable document mixtures through a non-negative simplex constraint. Our method is simple to incorporate into existing automatic differentiation frameworks and allows for unsupervised document representations geared for use by scientists while simultaneously learning word vectors and the linear relationships between them.

Decoder-only Transformers often struggle with complex reasoning tasks, particularly arithmetic reasoning requiring multiple sequential operations. In this work, we identify representation collapse in the model's intermediate layers as a key factor limiting their reasoning capabilities. To address this, we propose Sequential Variance-Covariance Regularization (Seq-VCR), which enhances the entropy of intermediate representations and prevents collapse. Combined with dummy pause tokens as substitutes for chain-of-thought (CoT) tokens, our method significantly improves performance in arithmetic reasoning problems. In the challenging 5 times 5 integer multiplication task, our approach achieves 99.5% exact match accuracy, outperforming models of the same size (which yield 0% accuracy) and GPT-4 with five-shot CoT prompting (44%). We also demonstrate superior results on arithmetic expression and longest increasing subsequence (LIS) datasets. Our findings highlight the importance of preventing intermediate layer representation collapse to enhance the reasoning capabilities of Transformers and show that Seq-VCR offers an effective solution without requiring explicit CoT supervision.

In this work, we introduce Boolformer, the first Transformer architecture trained to perform end-to-end symbolic regression of Boolean functions. First, we show that it can predict compact formulas for complex functions which were not seen during training, when provided a clean truth table. Then, we demonstrate its ability to find approximate expressions when provided incomplete and noisy observations. We evaluate the Boolformer on a broad set of real-world binary classification datasets, demonstrating its potential as an interpretable alternative to classic machine learning methods. Finally, we apply it to the widespread task of modelling the dynamics of gene regulatory networks. Using a recent benchmark, we show that Boolformer is competitive with state-of-the art genetic algorithms with a speedup of several orders of magnitude. Our code and models are available publicly.

Conformal prediction has shown spurring performance in constructing statistically rigorous prediction sets for arbitrary black-box machine learning models, assuming the data is exchangeable. However, even small adversarial perturbations during the inference can violate the exchangeability assumption, challenge the coverage guarantees, and result in a subsequent decline in empirical coverage. In this work, we propose a certifiably robust learning-reasoning conformal prediction framework (COLEP) via probabilistic circuits, which comprise a data-driven learning component that trains statistical models to learn different semantic concepts, and a reasoning component that encodes knowledge and characterizes the relationships among the trained models for logic reasoning. To achieve exact and efficient reasoning, we employ probabilistic circuits (PCs) within the reasoning component. Theoretically, we provide end-to-end certification of prediction coverage for COLEP in the presence of bounded adversarial perturbations. We also provide certified coverage considering the finite size of the calibration set. Furthermore, we prove that COLEP achieves higher prediction coverage and accuracy over a single model as long as the utilities of knowledge models are non-trivial. Empirically, we show the validity and tightness of our certified coverage, demonstrating the robust conformal prediction of COLEP on various datasets, including GTSRB, CIFAR10, and AwA2. We show that COLEP achieves up to 12% improvement in certified coverage on GTSRB, 9% on CIFAR-10, and 14% on AwA2.

While dense retrieval has been shown effective and efficient across tasks and languages, it remains difficult to create effective fully zero-shot dense retrieval systems when no relevance label is available. In this paper, we recognize the difficulty of zero-shot learning and encoding relevance. Instead, we propose to pivot through Hypothetical Document Embeddings~(HyDE). Given a query, HyDE first zero-shot instructs an instruction-following language model (e.g. InstructGPT) to generate a hypothetical document. The document captures relevance patterns but is unreal and may contain false details. Then, an unsupervised contrastively learned encoder~(e.g. Contriever) encodes the document into an embedding vector. This vector identifies a neighborhood in the corpus embedding space, where similar real documents are retrieved based on vector similarity. This second step ground the generated document to the actual corpus, with the encoder's dense bottleneck filtering out the incorrect details. Our experiments show that HyDE significantly outperforms the state-of-the-art unsupervised dense retriever Contriever and shows strong performance comparable to fine-tuned retrievers, across various tasks (e.g. web search, QA, fact verification) and languages~(e.g. sw, ko, ja).

Word embedding is a key component in many downstream applications in processing natural languages. Existing approaches often assume the existence of a large collection of text for learning effective word embedding. However, such a corpus may not be available for some low-resource languages. In this paper, we study how to effectively learn a word embedding model on a corpus with only a few million tokens. In such a situation, the co-occurrence matrix is sparse as the co-occurrences of many word pairs are unobserved. In contrast to existing approaches often only sample a few unobserved word pairs as negative samples, we argue that the zero entries in the co-occurrence matrix also provide valuable information. We then design a Positive-Unlabeled Learning (PU-Learning) approach to factorize the co-occurrence matrix and validate the proposed approaches in four different languages.

Vectors are universal mathematical objects that can represent text, images, speech, or a mix of these data modalities. That happens regardless of whether data is represented by hand-crafted features or learnt embeddings. Collect a large enough quantity of such vectors and the question of retrieval becomes urgently relevant: Finding vectors that are more similar to a query vector. This monograph is concerned with the question above and covers fundamental concepts along with advanced data structures and algorithms for vector retrieval. In doing so, it recaps this fascinating topic and lowers barriers of entry into this rich area of research.

Cosine-similarity is the cosine of the angle between two vectors, or equivalently the dot product between their normalizations. A popular application is to quantify semantic similarity between high-dimensional objects by applying cosine-similarity to a learned low-dimensional feature embedding. This can work better but sometimes also worse than the unnormalized dot-product between embedded vectors in practice. To gain insight into this empirical observation, we study embeddings derived from regularized linear models, where closed-form solutions facilitate analytical insights. We derive analytically how cosine-similarity can yield arbitrary and therefore meaningless `similarities.' For some linear models the similarities are not even unique, while for others they are implicitly controlled by the regularization. We discuss implications beyond linear models: a combination of different regularizations are employed when learning deep models; these have implicit and unintended effects when taking cosine-similarities of the resulting embeddings, rendering results opaque and possibly arbitrary. Based on these insights, we caution against blindly using cosine-similarity and outline alternatives.

Solving Bengali Math Word Problems (MWPs) remains a major challenge in natural language processing (NLP) due to the language's low-resource status and the multi-step reasoning required. Existing models struggle with complex Bengali MWPs, largely because no human-annotated Bengali dataset has previously addressed this task. This gap has limited progress in Bengali mathematical reasoning. To address this, we created SOMADHAN, a dataset of 8792 complex Bengali MWPs with manually written, step-by-step solutions. We designed this dataset to support reasoning-focused evaluation and model development in a linguistically underrepresented context. Using SOMADHAN, we evaluated a range of large language models (LLMs) - including GPT-4o, GPT-3.5 Turbo, LLaMA series models, Deepseek, and Qwen - through both zero-shot and few-shot prompting with and without Chain of Thought (CoT) reasoning. CoT prompting consistently improved performance over standard prompting, especially in tasks requiring multi-step logic. LLaMA-3.3 70B achieved the highest accuracy of 88% with few-shot CoT prompting. We also applied Low-Rank Adaptation (LoRA) to fine-tune models efficiently, enabling them to adapt to Bengali MWPs with minimal computational cost. Our work fills a critical gap in Bengali NLP by providing a high-quality reasoning dataset and a scalable framework for solving complex MWPs. We aim to advance equitable research in low-resource languages and enhance reasoning capabilities in educational and language technologies.

Recent progress has been made towards learning invariant or equivariant representations with self-supervised learning. While invariant methods are evaluated on large scale datasets, equivariant ones are evaluated in smaller, more controlled, settings. We aim at bridging the gap between the two in order to learn more diverse representations that are suitable for a wide range of tasks. We start by introducing a dataset called 3DIEBench, consisting of renderings from 3D models over 55 classes and more than 2.5 million images where we have full control on the transformations applied to the objects. We further introduce a predictor architecture based on hypernetworks to learn equivariant representations with no possible collapse to invariance. We introduce SIE (Split Invariant-Equivariant) which combines the hypernetwork-based predictor with representations split in two parts, one invariant, the other equivariant, to learn richer representations. We demonstrate significant performance gains over existing methods on equivariance related tasks from both a qualitative and quantitative point of view. We further analyze our introduced predictor and show how it steers the learned latent space. We hope that both our introduced dataset and approach will enable learning richer representations without supervision in more complex scenarios. Code and data are available at https://github.com/facebookresearch/SIE.

Modern neural models trained on textual data rely on pre-trained representations that emerge without direct supervision. As these representations are increasingly being used in real-world applications, the inability to control their content becomes an increasingly important problem. We formulate the problem of identifying and erasing a linear subspace that corresponds to a given concept, in order to prevent linear predictors from recovering the concept. We model this problem as a constrained, linear maximin game, and show that existing solutions are generally not optimal for this task. We derive a closed-form solution for certain objectives, and propose a convex relaxation, \method, that works well for others. When evaluated in the context of binary gender removal, the method recovers a low-dimensional subspace whose removal mitigates bias by intrinsic and extrinsic evaluation. We show that the method is highly expressive, effectively mitigating bias in deep nonlinear classifiers while maintaining tractability and interpretability.

We build upon vec2vec, a procedure designed to align text embedding spaces without parallel data. vec2vec finds a near-perfect alignment, but it is expensive and unstable. We present mini-vec2vec, a simple and efficient alternative that requires substantially lower computational cost and is highly robust. Moreover, the learned mapping is a linear transformation. Our method consists of three main stages: a tentative matching of pseudo-parallel embedding vectors, transformation fitting, and iterative refinement. Our linear alternative exceeds the original instantiation of vec2vec by orders of magnitude in efficiency, while matching or exceeding their results. The method's stability and interpretable algorithmic steps facilitate scaling and unlock new opportunities for adoption in new domains and fields.

Automatic bundle construction is a crucial prerequisite step in various bundle-aware online services. Previous approaches are mostly designed to model the bundling strategy of existing bundles. However, it is hard to acquire large-scale well-curated bundle dataset, especially for those platforms that have not offered bundle services before. Even for platforms with mature bundle services, there are still many items that are included in few or even zero bundles, which give rise to sparsity and cold-start challenges in the bundle construction models. To tackle these issues, we target at leveraging multimodal features, item-level user feedback signals, and the bundle composition information, to achieve a comprehensive formulation of bundle construction. Nevertheless, such formulation poses two new technical challenges: 1) how to learn effective representations by optimally unifying multiple features, and 2) how to address the problems of modality missing, noise, and sparsity problems induced by the incomplete query bundles. In this work, to address these technical challenges, we propose a Contrastive Learning-enhanced Hierarchical Encoder method (CLHE). Specifically, we use self-attention modules to combine the multimodal and multi-item features, and then leverage both item- and bundle-level contrastive learning to enhance the representation learning, thus to counter the modality missing, noise, and sparsity problems. Extensive experiments on four datasets in two application domains demonstrate that our method outperforms a list of SOTA methods. The code and dataset are available at https://github.com/Xiaohao-Liu/CLHE.

Short Text Classification (STC) is crucial for processing and understanding the brief but substantial content prevalent on contemporary digital platforms. The STC encounters difficulties in grasping the semantic and syntactic intricacies, an issue that is apparent in traditional pre-trained language models. Although Graph Convolutional Networks enhance performance by integrating external knowledge bases, these methods are limited by the quality and extent of the knowledge applied. Recently, the emergence of Large Language Models (LLMs) and Chain-of-Thought (CoT) has significantly improved the performance of complex reasoning tasks. However, some studies have highlighted the limitations of their application in fundamental NLP tasks. Consequently, this study first employs CoT to investigate and enhance the capabilities of LLMs in STC tasks. We propose the Syntactic and Semantic Enrichment CoT (SSE-CoT) method, effectively decomposing the STC tasks into four distinct steps: (i) essential concept identification, (ii) common-sense knowledge retrieval, (iii) text rewriting, and (iv) classification. Furthermore, recognizing resource constraints in sectors like finance and healthcare, we then introduce the CoT-Driven Multi-Task Learning (CDMT) framework to extend these capabilities to smaller models. This framework begins by extracting rationales from LLMs and subsequently fine-tunes smaller models to optimize their performance. Extensive experimentation across six short-text benchmarks validated the efficacy of the proposed methods. In particular, SSE-CoT achieved state-of-the-art performance with substantial improvements on all datasets, particularly on the Ohsumed and TagMyNews datasets.

We adapt previous research on category theory and topological unsupervised learning to develop a functorial perspective on manifold learning, also known as nonlinear dimensionality reduction. We first characterize manifold learning algorithms as functors that map pseudometric spaces to optimization objectives and that factor through hierarchical clustering functors. We then use this characterization to prove refinement bounds on manifold learning loss functions and construct a hierarchy of manifold learning algorithms based on their equivariants. We express several popular manifold learning algorithms as functors at different levels of this hierarchy, including Metric Multidimensional Scaling, IsoMap, and UMAP. Next, we use interleaving distance to study the stability of a broad class of manifold learning algorithms. We present bounds on how closely the embeddings these algorithms produce from noisy data approximate the embeddings they would learn from noiseless data. Finally, we use our framework to derive a set of novel manifold learning algorithms, which we experimentally demonstrate are competitive with the state of the art.

Transfer learning with a small amount of target data is an effective and common approach to adapting a pre-trained model to distribution shifts. In some situations, target data labels may be expensive to obtain, so we may only have access to a limited number of target data points. To make the most of a very small target dataset, we propose a lightweight, sample-efficient approach that learns a diverse set of features and adapts to a target distribution by interpolating these features. Our approach, Project and Probe (Pro^2), first learns a linear projection that maps a pre-trained embedding onto orthogonal directions while being predictive of labels in the source dataset. The goal of this step is to learn a variety of predictive features, so that at least some of them remain useful after distribution shift. Pro^2 then learns a linear classifier on top of these projected features using a small target dataset. Theoretically, we find that Pro^2 results in more sample-efficient generalization by inducing a favorable bias-variance tradeoff. Our experiments on four datasets, with multiple distribution shift settings for each, show that Pro^2 improves performance by 5-15% when given limited target data compared to prior methods such as standard linear probing.

Large Language Models (LLMs) excel in reasoning tasks through Chain-of-Thought (CoT) prompting. However, CoT prompting greatly increases computational demands, which has prompted growing interest in distilling CoT capabilities into Small Language Models (SLMs). This study systematically examines the factors influencing CoT distillation, including the choice of granularity, format and teacher model. Through experiments involving four teacher models and seven student models across seven mathematical and commonsense reasoning datasets, we uncover three key findings: (1) Unlike LLMs, SLMs exhibit a non-monotonic relationship with granularity, with stronger models benefiting from finer-grained reasoning and weaker models performing better with simpler CoT supervision; (2) CoT format significantly impacts LLMs but has minimal effect on SLMs, likely due to their reliance on supervised fine-tuning rather than pretraining preferences; (3) Stronger teacher models do NOT always produce better student models, as diversity and complexity in CoT supervision can outweigh accuracy alone. These findings emphasize the need to tailor CoT strategies to specific student model, offering actionable insights for optimizing CoT distillation in SLMs. The code and datasets are available at https://github.com/EIT-NLP/Distilling-CoT-Reasoning.

We introduce Invariant Risk Minimization (IRM), a learning paradigm to estimate invariant correlations across multiple training distributions. To achieve this goal, IRM learns a data representation such that the optimal classifier, on top of that data representation, matches for all training distributions. Through theory and experiments, we show how the invariances learned by IRM relate to the causal structures governing the data and enable out-of-distribution generalization.

We present a meta-learning framework for learning new visual concepts quickly, from just one or a few examples, guided by multiple naturally occurring data streams: simultaneously looking at images, reading sentences that describe the objects in the scene, and interpreting supplemental sentences that relate the novel concept with other concepts. The learned concepts support downstream applications, such as answering questions by reasoning about unseen images. Our model, namely FALCON, represents individual visual concepts, such as colors and shapes, as axis-aligned boxes in a high-dimensional space (the "box embedding space"). Given an input image and its paired sentence, our model first resolves the referential expression in the sentence and associates the novel concept with particular objects in the scene. Next, our model interprets supplemental sentences to relate the novel concept with other known concepts, such as "X has property Y" or "X is a kind of Y". Finally, it infers an optimal box embedding for the novel concept that jointly 1) maximizes the likelihood of the observed instances in the image, and 2) satisfies the relationships between the novel concepts and the known ones. We demonstrate the effectiveness of our model on both synthetic and real-world datasets.

Learning compatible representations enables the interchangeable use of semantic features as models are updated over time. This is particularly relevant in search and retrieval systems where it is crucial to avoid reprocessing of the gallery images with the updated model. While recent research has shown promising empirical evidence, there is still a lack of comprehensive theoretical understanding about learning compatible representations. In this paper, we demonstrate that the stationary representations learned by the d-Simplex fixed classifier optimally approximate compatibility representation according to the two inequality constraints of its formal definition. This not only establishes a solid foundation for future works in this line of research but also presents implications that can be exploited in practical learning scenarios. An exemplary application is the now-standard practice of downloading and fine-tuning new pre-trained models. Specifically, we show the strengths and critical issues of stationary representations in the case in which a model undergoing sequential fine-tuning is asynchronously replaced by downloading a better-performing model pre-trained elsewhere. Such a representation enables seamless delivery of retrieval service (i.e., no reprocessing of gallery images) and offers improved performance without operational disruptions during model replacement. Code available at: https://github.com/miccunifi/iamcl2r.

We introduce EmbeddingGemma, a new lightweight, open text embedding model based on the Gemma 3 language model family. Our innovative training recipe strategically captures knowledge from larger models via encoder-decoder initialization and geometric embedding distillation. We improve model robustness and expressiveness with a spread-out regularizer, and ensure generalizability by merging checkpoints from varied, optimized mixtures. Evaluated on the Massive Text Embedding Benchmark (MTEB) across multilingual, English, and code domains, EmbeddingGemma (300M) achieves state-of-the-art results. Notably, it outperforms prior top models, both proprietary and open, with fewer than 500M parameters, and provides performance comparable to models double its size, offering an exceptional performance-to-cost ratio. Remarkably, this lead persists when quantizing model weights or truncating embedding outputs. This makes EmbeddingGemma particularly well-suited for low-latency and high-throughput use cases such as on-device applications. We provide ablation studies exploring our key design choices. We release EmbeddingGemma to the community to promote further research.

Vector space embedding models like word2vec, GloVe, fastText, and ELMo are extremely popular representations in natural language processing (NLP) applications. We present Magnitude, a fast, lightweight tool for utilizing and processing embeddings. Magnitude is an open source Python package with a compact vector storage file format that allows for efficient manipulation of huge numbers of embeddings. Magnitude performs common operations up to 60 to 6,000 times faster than Gensim. Magnitude introduces several novel features for improved robustness like out-of-vocabulary lookups.

Large Language Models (LLMs) achieve superior performance through Chain-of-Thought (CoT) reasoning, but these token-level reasoning chains are computationally expensive and inefficient. In this paper, we introduce Compressed Latent Reasoning (CoLaR), a novel framework that dynamically compresses reasoning processes in latent space through a two-stage training approach. First, during supervised fine-tuning, CoLaR extends beyond next-token prediction by incorporating an auxiliary next compressed embedding prediction objective. This process merges embeddings of consecutive tokens using a compression factor randomly sampled from a predefined range, and trains a specialized latent head to predict distributions of subsequent compressed embeddings. Second, we enhance CoLaR through reinforcement learning (RL) that leverages the latent head's non-deterministic nature to explore diverse reasoning paths and exploit more compact ones. This approach enables CoLaR to: i) perform reasoning at a dense latent level (i.e., silently), substantially reducing reasoning chain length, and ii) dynamically adjust reasoning speed at inference time by simply prompting the desired compression factor. Extensive experiments across four mathematical reasoning datasets demonstrate that CoLaR achieves 14.1% higher accuracy than latent-based baseline methods at comparable compression ratios, and reduces reasoning chain length by 53.3% with only 4.8% performance degradation compared to explicit CoT method. Moreover, when applied to more challenging mathematical reasoning tasks, our RL-enhanced CoLaR demonstrates performance gains of up to 5.4% while dramatically reducing latent reasoning chain length by 82.8%. The code and models will be released upon acceptance.

The manifold hypothesis posits that high-dimensional data often lies on a lower-dimensional manifold and that utilizing this manifold as the target space yields more efficient representations. While numerous traditional manifold-based techniques exist for dimensionality reduction, their application in self-supervised learning has witnessed slow progress. The recent MSimCLR method combines manifold encoding with SimCLR but requires extremely low target encoding dimensions to outperform SimCLR, limiting its applicability. This paper introduces a novel learning paradigm using an unbalanced atlas (UA), capable of surpassing state-of-the-art self-supervised learning approaches. We investigated and engineered the DeepInfomax with an unbalanced atlas (DIM-UA) method by adapting the Spatiotemporal DeepInfomax (ST-DIM) framework to align with our proposed UA paradigm. The efficacy of DIM-UA is demonstrated through training and evaluation on the Atari Annotated RAM Interface (AtariARI) benchmark, a modified version of the Atari 2600 framework that produces annotated image samples for representation learning. The UA paradigm improves existing algorithms significantly as the number of target encoding dimensions grows. For instance, the mean F1 score averaged over categories of DIM-UA is ~75% compared to ~70% of ST-DIM when using 16384 hidden units.

Chain-of-Thought (CoT) reasoning has become a powerful framework for improving complex problem-solving capabilities in Multimodal Large Language Models (MLLMs). However, the verbose nature of textual reasoning introduces significant inefficiencies. In this work, we propose Heima (as hidden llama), an efficient reasoning framework that leverages reasoning CoTs at hidden latent space. We design the Heima Encoder to condense each intermediate CoT into a compact, higher-level hidden representation using a single thinking token, effectively minimizing verbosity and reducing the overall number of tokens required during the reasoning process. Meanwhile, we design corresponding Heima Decoder with traditional Large Language Models (LLMs) to adaptively interpret the hidden representations into variable-length textual sequence, reconstructing reasoning processes that closely resemble the original CoTs. Experimental results across diverse reasoning MLLM benchmarks demonstrate that Heima model achieves higher generation efficiency while maintaining or even better zero-shot task accuracy. Moreover, the effective reconstruction of multimodal reasoning processes with Heima Decoder validates both the robustness and interpretability of our approach.

Few Shot Class Incremental Learning (FSCIL) with few examples per class for each incremental session is the realistic setting of continual learning since obtaining large number of annotated samples is not feasible and cost effective. We present the framework MASIL as a step towards learning the maximal separable classifier. It addresses the common problem i.e forgetting of old classes and over-fitting to novel classes by learning the classifier weights to be maximally separable between classes forming a simplex Equiangular Tight Frame. We propose the idea of concept factorization explaining the collapsed features for base session classes in terms of concept basis and use these to induce classifier simplex for few shot classes. We further adds fine tuning to reduce any error occurred during factorization and train the classifier jointly on base and novel classes without retaining any base class samples in memory. Experimental results on miniImageNet, CIFAR-100 and CUB-200 demonstrate that MASIL outperforms all the benchmarks.

We provide theoretical and empirical evidence that using tighter evidence lower bounds (ELBOs) can be detrimental to the process of learning an inference network by reducing the signal-to-noise ratio of the gradient estimator. Our results call into question common implicit assumptions that tighter ELBOs are better variational objectives for simultaneous model learning and inference amortization schemes. Based on our insights, we introduce three new algorithms: the partially importance weighted auto-encoder (PIWAE), the multiply importance weighted auto-encoder (MIWAE), and the combination importance weighted auto-encoder (CIWAE), each of which includes the standard importance weighted auto-encoder (IWAE) as a special case. We show that each can deliver improvements over IWAE, even when performance is measured by the IWAE target itself. Furthermore, our results suggest that PIWAE may be able to deliver simultaneous improvements in the training of both the inference and generative networks.

Summarization refinement faces challenges when extending to multi-dimension. In this paper, we introduce ReFeed, a powerful summarization refinement pipeline that enhances multiple dimensions through reflective reasoning on feedback. To achieve this, we release SumFeed-CoT, a large-scale Long-CoT-based dataset optimized for training a lightweight model with reflective reasoning. Our experiments reveal how the number of dimensions, feedback exposure, and reasoning policy influence refinement performance, highlighting reflective reasoning and simultaneously addressing multiple feedback is crucial to mitigate trade-off between dimensions. Furthermore, ReFeed is robust to noisy feedback and feedback order. Lastly, our finding emphasizes that creating data with a proper goal and guideline constitutes a fundamental pillar of effective reasoning. The dataset and model will be released.

Learning decompositions of expensive-to-evaluate black-box functions promises to scale Bayesian optimisation (BO) to high-dimensional problems. However, the success of these techniques depends on finding proper decompositions that accurately represent the black-box. While previous works learn those decompositions based on data, we investigate data-independent decomposition sampling rules in this paper. We find that data-driven learners of decompositions can be easily misled towards local decompositions that do not hold globally across the search space. Then, we formally show that a random tree-based decomposition sampler exhibits favourable theoretical guarantees that effectively trade off maximal information gain and functional mismatch between the actual black-box and its surrogate as provided by the decomposition. Those results motivate the development of the random decomposition upper-confidence bound algorithm (RDUCB) that is straightforward to implement - (almost) plug-and-play - and, surprisingly, yields significant empirical gains compared to the previous state-of-the-art on a comprehensive set of benchmarks. We also confirm the plug-and-play nature of our modelling component by integrating our method with HEBO, showing improved practical gains in the highest dimensional tasks from Bayesmark.

Prototypical part network (ProtoPNet) methods have been designed to achieve interpretable classification by associating predictions with a set of training prototypes, which we refer to as trivial prototypes because they are trained to lie far from the classification boundary in the feature space. Note that it is possible to make an analogy between ProtoPNet and support vector machine (SVM) given that the classification from both methods relies on computing similarity with a set of training points (i.e., trivial prototypes in ProtoPNet, and support vectors in SVM). However, while trivial prototypes are located far from the classification boundary, support vectors are located close to this boundary, and we argue that this discrepancy with the well-established SVM theory can result in ProtoPNet models with inferior classification accuracy. In this paper, we aim to improve the classification of ProtoPNet with a new method to learn support prototypes that lie near the classification boundary in the feature space, as suggested by the SVM theory. In addition, we target the improvement of classification results with a new model, named ST-ProtoPNet, which exploits our support prototypes and the trivial prototypes to provide more effective classification. Experimental results on CUB-200-2011, Stanford Cars, and Stanford Dogs datasets demonstrate that ST-ProtoPNet achieves state-of-the-art classification accuracy and interpretability results. We also show that the proposed support prototypes tend to be better localised in the object of interest rather than in the background region.

Chain-of-Thought (CoT) has been proven effective in enhancing the reasoning capabilities of large language models (LLMs). Recent advancements, such as OpenAI's o1 and DeepSeek-R1, suggest that scaling up the length of CoT sequences during inference could further boost LLM reasoning performance. However, due to the autoregressive nature of LLM decoding, longer CoT outputs lead to a linear increase in inference latency, adversely affecting user experience, particularly when the CoT exceeds 10,000 tokens. To address this limitation, we analyze the semantic importance of tokens within CoT outputs and reveal that their contributions to reasoning vary. Building on this insight, we propose TokenSkip, a simple yet effective approach that enables LLMs to selectively skip less important tokens, allowing for controllable CoT compression. Extensive experiments across various models and tasks demonstrate the effectiveness of TokenSkip in reducing CoT token usage while preserving strong reasoning performance. Notably, when applied to Qwen2.5-14B-Instruct, TokenSkip reduces reasoning tokens by 40% (from 313 to 181) on GSM8K, with less than a 0.4% performance drop.

We study the task of agnostically learning halfspaces under the Gaussian distribution. Specifically, given labeled examples (x,y) from an unknown distribution on R^n times { pm 1}, whose marginal distribution on x is the standard Gaussian and the labels y can be arbitrary, the goal is to output a hypothesis with 0-1 loss OPT+epsilon, where OPT is the 0-1 loss of the best-fitting halfspace. We prove a near-optimal computational hardness result for this task, under the widely believed sub-exponential time hardness of the Learning with Errors (LWE) problem. Prior hardness results are either qualitatively suboptimal or apply to restricted families of algorithms. Our techniques extend to yield near-optimal lower bounds for related problems, including ReLU regression.

Vector Symbolic Architectures (VSAs) have emerged as a novel framework for enabling interpretable machine learning algorithms equipped with the ability to reason and explain their decision processes. The basic idea is to represent discrete information through high dimensional random vectors. Complex data structures can be built up with operations over vectors such as the "binding" operation involving element-wise vector multiplication, which associates data together. The reverse task of decomposing the associated elements is a combinatorially hard task, with an exponentially large search space. The main algorithm for performing this search is the resonator network, inspired by Hopfield network-based memory search operations. In this work, we introduce a new variant of the resonator network, based on self-attention based update rules in the iterative search problem. This update rule, based on the Hopfield network with log-sum-exp energy function and norm-bounded states, is shown to substantially improve the performance and rate of convergence. As a result, our algorithm enables a larger capacity for associative memory, enabling applications in many tasks like perception based pattern recognition, scene decomposition, and object reasoning. We substantiate our algorithm with a thorough evaluation and comparisons to baselines.

Machine learning models will often fail when deployed in an environment with a data distribution that is different than the training distribution. When multiple environments are available during training, many methods exist that learn representations which are invariant across the different distributions, with the hope that these representations will be transportable to unseen domains. In this work, we present a nonparametric strategy for learning invariant representations based on the recently-proposed Nadaraya-Watson (NW) head. The NW head makes a prediction by comparing the learned representations of the query to the elements of a support set that consists of labeled data. We demonstrate that by manipulating the support set, one can encode different causal assumptions. In particular, restricting the support set to a single environment encourages the model to learn invariant features that do not depend on the environment. We present a causally-motivated setup for our modeling and training strategy and validate on three challenging real-world domain generalization tasks in computer vision.

Many recent methods for unsupervised or self-supervised representation learning train feature extractors by maximizing an estimate of the mutual information (MI) between different views of the data. This comes with several immediate problems: For example, MI is notoriously hard to estimate, and using it as an objective for representation learning may lead to highly entangled representations due to its invariance under arbitrary invertible transformations. Nevertheless, these methods have been repeatedly shown to excel in practice. In this paper we argue, and provide empirical evidence, that the success of these methods cannot be attributed to the properties of MI alone, and that they strongly depend on the inductive bias in both the choice of feature extractor architectures and the parametrization of the employed MI estimators. Finally, we establish a connection to deep metric learning and argue that this interpretation may be a plausible explanation for the success of the recently introduced methods.

Vector Quantisation (VQ) is experiencing a comeback in machine learning, where it is increasingly used in representation learning. However, optimizing the codevectors in existing VQ-VAE is not entirely trivial. A problem is codebook collapse, where only a small subset of codevectors receive gradients useful for their optimisation, whereas a majority of them simply ``dies off'' and is never updated or used. This limits the effectiveness of VQ for learning larger codebooks in complex computer vision tasks that require high-capacity representations. In this paper, we present a simple alternative method for online codebook learning, Clustering VQ-VAE (CVQ-VAE). Our approach selects encoded features as anchors to update the ``dead'' codevectors, while optimising the codebooks which are alive via the original loss. This strategy brings unused codevectors closer in distribution to the encoded features, increasing the likelihood of being chosen and optimized. We extensively validate the generalization capability of our quantiser on various datasets, tasks (e.g. reconstruction and generation), and architectures (e.g. VQ-VAE, VQGAN, LDM). Our CVQ-VAE can be easily integrated into the existing models with just a few lines of code.

In this paper, we consider the task of retrieving documents with predefined topics from an unlabeled document dataset using an unsupervised approach. The proposed unsupervised approach requires only a small number of keywords describing the respective topics and no labeled document. Existing approaches either heavily relied on a large amount of additionally encoded world knowledge or on term-document frequencies. Contrariwise, we introduce a method that learns jointly embedded document and word vectors solely from the unlabeled document dataset in order to find documents that are semantically similar to the topics described by the keywords. The proposed method requires almost no text preprocessing but is simultaneously effective at retrieving relevant documents with high probability. When successively retrieving documents on different predefined topics from publicly available and commonly used datasets, we achieved an average area under the receiver operating characteristic curve value of 0.95 on one dataset and 0.92 on another. Further, our method can be used for multiclass document classification, without the need to assign labels to the dataset in advance. Compared with an unsupervised classification baseline, we increased F1 scores from 76.6 to 82.7 and from 61.0 to 75.1 on the respective datasets. For easy replication of our approach, we make the developed Lbl2Vec code publicly available as a ready-to-use tool under the 3-Clause BSD license.

We consider the problem of simultaneously clustering and learning a linear representation of data lying close to a union of low-dimensional manifolds, a fundamental task in machine learning and computer vision. When the manifolds are assumed to be linear subspaces, this reduces to the classical problem of subspace clustering, which has been studied extensively over the past two decades. Unfortunately, many real-world datasets such as natural images can not be well approximated by linear subspaces. On the other hand, numerous works have attempted to learn an appropriate transformation of the data, such that data is mapped from a union of general non-linear manifolds to a union of linear subspaces (with points from the same manifold being mapped to the same subspace). However, many existing works have limitations such as assuming knowledge of the membership of samples to clusters, requiring high sampling density, or being shown theoretically to learn trivial representations. In this paper, we propose to optimize the Maximal Coding Rate Reduction metric with respect to both the data representation and a novel doubly stochastic cluster membership, inspired by state-of-the-art subspace clustering results. We give a parameterization of such a representation and membership, allowing efficient mini-batching and one-shot initialization. Experiments on CIFAR-10, -20, -100, and TinyImageNet-200 datasets show that the proposed method is much more accurate and scalable than state-of-the-art deep clustering methods, and further learns a latent linear representation of the data.

Contrastive learning -- a modern approach to extract useful representations from unlabeled data by training models to distinguish similar samples from dissimilar ones -- has driven significant progress in foundation models. In this work, we develop a new theoretical framework for analyzing data augmentation-based contrastive learning, with a focus on SimCLR as a representative example. Our approach is based on the concept of approximate sufficient statistics, which we extend beyond its original definition in oko2025statistical for contrastive language-image pretraining (CLIP) using KL-divergence. We generalize it to equivalent forms and general f-divergences, and show that minimizing SimCLR and other contrastive losses yields encoders that are approximately sufficient. Furthermore, we demonstrate that these near-sufficient encoders can be effectively adapted to downstream regression and classification tasks, with performance depending on their sufficiency and the error induced by data augmentation in contrastive learning. Concrete examples in linear regression and topic classification are provided to illustrate the broad applicability of our results.

Topic modeling is used for discovering latent semantic structure, usually referred to as topics, in a large collection of documents. The most widely used methods are Latent Dirichlet Allocation and Probabilistic Latent Semantic Analysis. Despite their popularity they have several weaknesses. In order to achieve optimal results they often require the number of topics to be known, custom stop-word lists, stemming, and lemmatization. Additionally these methods rely on bag-of-words representation of documents which ignore the ordering and semantics of words. Distributed representations of documents and words have gained popularity due to their ability to capture semantics of words and documents. We present top2vec, which leverages joint document and word semantic embedding to find topic vectors. This model does not require stop-word lists, stemming or lemmatization, and it automatically finds the number of topics. The resulting topic vectors are jointly embedded with the document and word vectors with distance between them representing semantic similarity. Our experiments demonstrate that top2vec finds topics which are significantly more informative and representative of the corpus trained on than probabilistic generative models.

Learning features from data is one of the defining characteristics of deep learning, but our theoretical understanding of the role features play in deep learning is still rudimentary. To address this gap, we introduce a new tool, the interaction tensor, for empirically analyzing the interaction between data and model through features. With the interaction tensor, we make several key observations about how features are distributed in data and how models with different random seeds learn different features. Based on these observations, we propose a conceptual framework for feature learning. Under this framework, the expected accuracy for a single hypothesis and agreement for a pair of hypotheses can both be derived in closed-form. We demonstrate that the proposed framework can explain empirically observed phenomena, including the recently discovered Generalization Disagreement Equality (GDE) that allows for estimating the generalization error with only unlabeled data. Further, our theory also provides explicit construction of natural data distributions that break the GDE. Thus, we believe this work provides valuable new insight into our understanding of feature learning.

The change in data distribution over time, also known as concept drift, poses a significant challenge to the reliability of online learning methods. Existing methods typically require model retraining or drift detection, both of which demand high computational costs and are often unsuitable for real-time applications. To address these limitations, a lightweight, fast and efficient random vector functional-link network termed Lite-RVFL is proposed, capable of adapting to concept drift without drift detection and retraining. Lite-RVFL introduces a novel objective function that assigns weights exponentially increasing to new samples, thereby emphasizing recent data and enabling timely adaptation. Theoretical analysis confirms the feasibility of this objective function for drift adaptation, and an efficient incremental update rule is derived. Experimental results on a real-world safety assessment task validate the efficiency, effectiveness in adapting to drift, and potential to capture temporal patterns of Lite-RVFL. The source code is available at https://github.com/songqiaohu/Lite-RVFL.

Transductive Few-Shot Learning (TFSL) has recently attracted increasing attention since it typically outperforms its inductive peer by leveraging statistics of query samples. However, previous TFSL methods usually encode uniform prior that all the classes within query samples are equally likely, which is biased in imbalanced TFSL and causes severe performance degradation. Given this pivotal issue, in this work, we propose a novel Conditional Transport (CT) based imbalanced TFSL model called {\textbf P}rototypes-oriented {\textbf U}nbiased {\textbf T}ransfer {\textbf M}odel (PUTM) to fully exploit unbiased statistics of imbalanced query samples, which employs forward and backward navigators as transport matrices to balance the prior of query samples per class between uniform and adaptive data-driven distributions. For efficiently transferring statistics learned by CT, we further derive a closed form solution to refine prototypes based on MAP given the learned navigators. The above two steps of discovering and transferring unbiased statistics follow an iterative manner, formulating our EM-based solver. Experimental results on four standard benchmarks including miniImageNet, tieredImageNet, CUB, and CIFAR-FS demonstrate superiority of our model in class-imbalanced generalization.

To adapt a well-trained large model to downstream tasks, we propose constraining learning within its original latent space by leveraging linear combinations of its basis vectors. This approach ensures stable training without compromising the model's capabilities. Traditionally, constructing orthonormal bases from a matrix requires a transfer matrix, which significantly increases storage and computational overhead for parameters and feature maps. In this paper, we introduce Absorb and Decompose for Q, K, V, and O matrices, enabling their orthogonalization without the need for transfer matrices. Furthermore, the Absorb-Decompose operation eliminates redundant vectors, reducing the encoder attention parameters of Whisper-large-v3 by 46.42% without requiring additional training. For parameter-efficient and stable fine-tuning, we orthonormalized Q, K, V, and O and fine-tuned only the singular values, allowing efficient adaptation while constraining changes to the original latent space. When fine-tuning LLaMA-2-7B on eight commonsense reasoning datasets, our method outperforms LoRA by 5.4% and DoRA by 4.4%.

We observe that the mapping between an image's representation in one model to its representation in another can be learned surprisingly well with just a linear layer, even across diverse models. Building on this observation, we propose text-to-concept, where features from a fixed pretrained model are aligned linearly to the CLIP space, so that text embeddings from CLIP's text encoder become directly comparable to the aligned features. With text-to-concept, we convert fixed off-the-shelf vision encoders to surprisingly strong zero-shot classifiers for free, with accuracy at times even surpassing that of CLIP, despite being much smaller models and trained on a small fraction of the data compared to CLIP. We show other immediate use-cases of text-to-concept, like building concept bottleneck models with no concept supervision, diagnosing distribution shifts in terms of human concepts, and retrieving images satisfying a set of text-based constraints. Lastly, we demonstrate the feasibility of concept-to-text, where vectors in a model's feature space are decoded by first aligning to the CLIP before being fed to a GPT-based generative model. Our work suggests existing deep models, with presumably diverse architectures and training, represent input samples relatively similarly, and a two-way communication across model representation spaces and to humans (through language) is viable.

A capsule is a group of neurons whose activity vector represents the instantiation parameters of a specific type of entity such as an object or an object part. We use the length of the activity vector to represent the probability that the entity exists and its orientation to represent the instantiation parameters. Active capsules at one level make predictions, via transformation matrices, for the instantiation parameters of higher-level capsules. When multiple predictions agree, a higher level capsule becomes active. We show that a discrimininatively trained, multi-layer capsule system achieves state-of-the-art performance on MNIST and is considerably better than a convolutional net at recognizing highly overlapping digits. To achieve these results we use an iterative routing-by-agreement mechanism: A lower-level capsule prefers to send its output to higher level capsules whose activity vectors have a big scalar product with the prediction coming from the lower-level capsule.

Self-supervised representation learning solves auxiliary prediction tasks (known as pretext tasks) without requiring labeled data to learn useful semantic representations. These pretext tasks are created solely using the input features, such as predicting a missing image patch, recovering the color channels of an image from context, or predicting missing words in text; yet predicting this known information helps in learning representations effective for downstream prediction tasks. We posit a mechanism exploiting the statistical connections between certain {\em reconstruction-based} pretext tasks that guarantee to learn a good representation. Formally, we quantify how the approximate independence between the components of the pretext task (conditional on the label and latent variables) allows us to learn representations that can solve the downstream task by just training a linear layer on top of the learned representation. We prove the linear layer yields small approximation error even for complex ground truth function class and will drastically reduce labeled sample complexity. Next, we show a simple modification of our method leads to nonlinear CCA, analogous to the popular SimSiam algorithm, and show similar guarantees for nonlinear CCA.

We consider the problem of conformal prediction under covariate shift. Given labeled data from a source domain and unlabeled data from a covariate shifted target domain, we seek to construct prediction sets with valid marginal coverage in the target domain. Most existing methods require estimating the unknown likelihood ratio function, which can be prohibitive for high-dimensional data such as images. To address this challenge, we introduce the likelihood ratio regularized quantile regression (LR-QR) algorithm, which combines the pinball loss with a novel choice of regularization in order to construct a threshold function without directly estimating the unknown likelihood ratio. We show that the LR-QR method has coverage at the desired level in the target domain, up to a small error term that we can control. Our proofs draw on a novel analysis of coverage via stability bounds from learning theory. Our experiments demonstrate that the LR-QR algorithm outperforms existing methods on high-dimensional prediction tasks, including a regression task for the Communities and Crime dataset, an image classification task from the WILDS repository, and an LLM question-answering task on the MMLU benchmark.

We develop a method to generate predictive regions that cover a multivariate response variable with a user-specified probability. Our work is composed of two components. First, we use a deep generative model to learn a representation of the response that has a unimodal distribution. Existing multiple-output quantile regression approaches are effective in such cases, so we apply them on the learned representation, and then transform the solution to the original space of the response. This process results in a flexible and informative region that can have an arbitrary shape, a property that existing methods lack. Second, we propose an extension of conformal prediction to the multivariate response setting that modifies any method to return sets with a pre-specified coverage level. The desired coverage is theoretically guaranteed in the finite-sample case for any distribution. Experiments conducted on both real and synthetic data show that our method constructs regions that are significantly smaller compared to existing techniques.

We study the problem of classification with a reject option for a fixed predictor, applicable in natural language processing. We introduce a new problem formulation for this scenario, and an algorithm minimizing a new surrogate loss function. We provide a complete theoretical analysis of the surrogate loss function with a strong H-consistency guarantee. For evaluation, we choose the decontextualization task, and provide a manually-labelled dataset of 2mathord,000 examples. Our algorithm significantly outperforms the baselines considered, with a sim!!25% improvement in coverage when halving the error rate, which is only sim!! 3 % away from the theoretical limit.

Learning discriminative shape representations is a crucial issue for large-scale 3D shape retrieval. In this paper, we propose the Collaborative Inner Product Loss (CIP Loss) to obtain ideal shape embedding that discriminative among different categories and clustered within the same class. Utilizing simple inner product operation, CIP loss explicitly enforces the features of the same class to be clustered in a linear subspace, while inter-class subspaces are constrained to be at least orthogonal. Compared to previous metric loss functions, CIP loss could provide more clear geometric interpretation for the embedding than Euclidean margin, and is easy to implement without normalization operation referring to cosine margin. Moreover, our proposed loss term can combine with other commonly used loss functions and can be easily plugged into existing off-the-shelf architectures. Extensive experiments conducted on the two public 3D object retrieval datasets, ModelNet and ShapeNetCore 55, demonstrate the effectiveness of our proposal, and our method has achieved state-of-the-art results on both datasets.

Pre-trained Transformers (\eg BERT) have been commonly used in existing dense retrieval methods for parameter initialization, and recent studies are exploring more effective pre-training tasks for further improving the quality of dense vectors. Although various novel and effective tasks have been proposed, their different input formats and learning objectives make them hard to be integrated for jointly improving the model performance. In this work, we aim to unify a variety of pre-training tasks into the bottlenecked masked autoencoder manner, and integrate them into a multi-task pre-trained model, namely MASTER. Concretely, MASTER utilizes a shared-encoder multi-decoder architecture that can construct a representation bottleneck to compress the abundant semantic information across tasks into dense vectors. Based on it, we integrate three types of representative pre-training tasks: corrupted passages recovering, related passages recovering and PLMs outputs recovering, to characterize the inner-passage information, inter-passage relations and PLMs knowledge. Extensive experiments have shown that our approach outperforms competitive dense retrieval methods. Our code and data are publicly released in https://github.com/microsoft/SimXNS.

Driven by the rapid growth of model parameters, parameter-efficient fine-tuning (PEFT) has become essential for adapting large models to diverse downstream tasks under constrained computational resources. Within this paradigm, orthogonal fine-tuning and its variants preserve semantic representations of pre-trained models, but struggle to achieve both expressiveness and efficiency in terms of parameter counts, memory, and computation. To overcome this limitation, we propose efficient Orthogonal Fine-Tuning with Principal Subspace adaptation (PSOFT), which confines orthogonal transformations to the principal subspace of pre-trained weights. Specifically, PSOFT constructs this subspace via matrix decomposition to enable compatible transformations with higher effective rank, establishes a theoretical condition that strictly maintains the geometry of this subspace for essential semantic preservation, and introduces efficient tunable vectors that gradually relax orthogonality during training to enhance adaptability. Extensive experiments on 35 NLP and CV tasks across four representative models demonstrate that PSOFT offers a practical and scalable solution to simultaneously achieve semantic preservation, expressiveness, and multi-dimensional efficiency in PEFT. The code is publicly available at https://github.com/fei407/PSOFT.

Test-time domain adaptation aims to adapt a source pre-trained model to a target domain without using any source data. Existing works mainly consider the case where the target domain is static. However, real-world machine perception systems are running in non-stationary and continually changing environments where the target domain distribution can change over time. Existing methods, which are mostly based on self-training and entropy regularization, can suffer from these non-stationary environments. Due to the distribution shift over time in the target domain, pseudo-labels become unreliable. The noisy pseudo-labels can further lead to error accumulation and catastrophic forgetting. To tackle these issues, we propose a continual test-time adaptation approach~(CoTTA) which comprises two parts. Firstly, we propose to reduce the error accumulation by using weight-averaged and augmentation-averaged predictions which are often more accurate. On the other hand, to avoid catastrophic forgetting, we propose to stochastically restore a small part of the neurons to the source pre-trained weights during each iteration to help preserve source knowledge in the long-term. The proposed method enables the long-term adaptation for all parameters in the network. CoTTA is easy to implement and can be readily incorporated in off-the-shelf pre-trained models. We demonstrate the effectiveness of our approach on four classification tasks and a segmentation task for continual test-time adaptation, on which we outperform existing methods. Our code is available at https://qin.ee/cotta.

Semantic representations of text, i.e. representations of natural language which capture meaning by geometry, are essential for areas such as information retrieval and document grouping. High-dimensional trained dense vectors have received much attention in recent years as such representations. We investigate the structure of semantic spaces that arise from embeddings made with Sentence-BERT and find that the representations suffer from a well-known problem in high dimensions called hubness. Hubness results in asymmetric neighborhood relations, such that some texts (the hubs) are neighbours of many other texts while most texts (so-called anti-hubs), are neighbours of few or no other texts. We quantify the semantic quality of the embeddings using hubness scores and error rate of a neighbourhood based classifier. We find that when hubness is high, we can reduce error rate and hubness using hubness reduction methods. We identify a combination of two methods as resulting in the best reduction. For example, on one of the tested pretrained models, this combined method can reduce hubness by about 75% and error rate by about 9%. Thus, we argue that mitigating hubness in the embedding space provides better semantic representations of text.

Generalized Continual Category Discovery (GCCD) tackles learning from sequentially arriving, partially labeled datasets while uncovering new categories. Traditional methods depend on feature distillation to prevent forgetting the old knowledge. However, this strategy restricts the model's ability to adapt and effectively distinguish new categories. To address this, we introduce a novel technique integrating a learnable projector with feature distillation, thus enhancing model adaptability without sacrificing past knowledge. The resulting distribution shift of the previously learned categories is mitigated with the auxiliary category adaptation network. We demonstrate that while each component offers modest benefits individually, their combination - dubbed CAMP (Category Adaptation Meets Projected distillation) - significantly improves the balance between learning new information and retaining old. CAMP exhibits superior performance across several GCCD and Class Incremental Learning scenarios. The code is available at https://github.com/grypesc/CAMP.

Learning representations that transfer well to diverse downstream tasks remains a central challenge in representation learning. Existing paradigms -- contrastive learning, self-supervised masking, and denoising auto-encoders -- balance this challenge with different trade-offs. We introduce the {contrastive Mutual Information Machine} (cMIM), a probabilistic framework that extends the Mutual Information Machine (MIM) with a contrastive objective. While MIM maximizes mutual information between inputs and latents and promotes clustering of codes, it falls short on discriminative tasks. cMIM addresses this gap by imposing global discriminative structure while retaining MIM's generative fidelity. Our contributions are threefold. First, we propose cMIM, a contrastive extension of MIM that removes the need for positive data augmentation and is substantially less sensitive to batch size than InfoNCE. Second, we introduce {informative embeddings}, a general technique for extracting enriched features from encoder-decoder models that boosts discriminative performance without additional training and applies broadly beyond MIM. Third, we provide empirical evidence across vision and molecular benchmarks showing that cMIM outperforms MIM and InfoNCE on classification and regression tasks while preserving competitive reconstruction quality. These results position cMIM as a unified framework for representation learning, advancing the goal of models that serve both discriminative and generative applications effectively.

Vector space word representations are learned from distributional information of words in large corpora. Although such statistics are semantically informative, they disregard the valuable information that is contained in semantic lexicons such as WordNet, FrameNet, and the Paraphrase Database. This paper proposes a method for refining vector space representations using relational information from semantic lexicons by encouraging linked words to have similar vector representations, and it makes no assumptions about how the input vectors were constructed. Evaluated on a battery of standard lexical semantic evaluation tasks in several languages, we obtain substantial improvements starting with a variety of word vector models. Our refinement method outperforms prior techniques for incorporating semantic lexicons into the word vector training algorithms.

Large Language Models (LLMs) have revolutionized natural language processing and hold immense potential for advancing Artificial Intelligence. However, the core architecture of most mainstream LLMs -- the Transformer -- has inherent limitations in computational depth, rendering them theoretically incapable of solving many reasoning tasks that demand increasingly deep computations. Chain of Thought (CoT) prompting has emerged as a technique to address these architectural limitations, as evidenced by several theoretical studies. It offers a promising approach to solving complex reasoning tasks that were previously beyond the capabilities of these models. Despite its successes, CoT and its variants (such as Tree of Thought, Graph of Thought, etc.) rely on a "one-prompt-for-all" approach, using a single prompt structure (e.g., "think step by step") for a wide range of tasks -- from counting and sorting to solving mathematical and algorithmic problems. This approach poses significant challenges for models to generate the correct reasoning steps, as the model must navigate through a vast prompt template space to find the appropriate template for each task. In this work, we build upon previous theoretical analyses of CoT to demonstrate how the one-prompt-for-all approach can negatively affect the computability of LLMs. We partition the solution search space into two: the prompt space and the answer space. Our findings show that task-specific supervision is essential for navigating the prompt space accurately and achieving optimal performance. Through experiments with state-of-the-art LLMs, we reveal a gap in reasoning performance when supervision is applied versus when it is not.

One-hot vectors, a common method for representing discrete/categorical data, in machine learning are widely used because of their simplicity and intuitiveness. However, one-hot vectors suffer from a linear increase in dimensionality, posing computational and memory challenges, especially when dealing with datasets containing numerous categories. In this paper, we focus on tabular data generation, and reveal the multinomial diffusion faces the mode collapse phenomenon when the cardinality is high. Moreover, due to the limitations of one-hot vectors, the training phase takes time longer in such a situation. To address these issues, we propose Residual Bit Vectors (ResBit), a technique for densely representing categorical data. ResBit is an extension of analog bits and overcomes limitations of analog bits when applied to tabular data generation. Our experiments demonstrate that ResBit not only accelerates training but also maintains performance when compared with the situations before applying ResBit. Furthermore, our results indicate that many existing methods struggle with high-cardinality data, underscoring the need for lower-dimensional representations, such as ResBit and latent vectors.

In this paper, we contend that a natural objective of representation learning is to compress and transform the distribution of the data, say sets of tokens, towards a low-dimensional Gaussian mixture supported on incoherent subspaces. The goodness of such a representation can be evaluated by a principled measure, called sparse rate reduction, that simultaneously maximizes the intrinsic information gain and extrinsic sparsity of the learned representation. From this perspective, popular deep network architectures, including transformers, can be viewed as realizing iterative schemes to optimize this measure. Particularly, we derive a transformer block from alternating optimization on parts of this objective: the multi-head self-attention operator compresses the representation by implementing an approximate gradient descent step on the coding rate of the features, and the subsequent multi-layer perceptron sparsifies the features. This leads to a family of white-box transformer-like deep network architectures, named CRATE, which are mathematically fully interpretable. We show, by way of a novel connection between denoising and compression, that the inverse to the aforementioned compressive encoding can be realized by the same class of CRATE architectures. Thus, the so-derived white-box architectures are universal to both encoders and decoders. Experiments show that these networks, despite their simplicity, indeed learn to compress and sparsify representations of large-scale real-world image and text datasets, and achieve performance very close to highly engineered transformer-based models: ViT, MAE, DINO, BERT, and GPT2. We believe the proposed computational framework demonstrates great potential in bridging the gap between theory and practice of deep learning, from a unified perspective of data compression. Code is available at: https://ma-lab-berkeley.github.io/CRATE .

A crucial component of many deep learning applications (such as FAQ and RAG) is dense retrieval, in which embedding models are used to convert raw text to numerical vectors and then get the most similar text by MIPS (Maximum Inner Product Search). Some text embedding benchmarks (e.g. MTEB, BEIR, and AIR-Bench) have been established to evaluate embedding models accurately. Thanks to these benchmarks, we can use SOTA models; however, the deployment and application of these models in industry were hampered by their large vector dimensions and numerous parameters. To alleviate this problem, 1) we present a distillation technique that can enable a smaller student model to achieve good performance. 2) Inspired by MRL we present a training approach of reducing the vector dimensions based on its own vectors or its teacher vectors. 3) We do simple yet effective alignment training between images and text to make our model a multimodal encoder. We trained Stella and Jasper models using the technologies above and achieved high scores on the MTEB leaderboard. We release the model and data at Hugging Face Hub (https://huggingface.co/infgrad/jasper_en_vision_language_v1) and the training logs are at https://api.wandb.ai/links/dunnzhang0/z8jqoqpb.

Within the field of causal inference, we consider the problem of estimating heterogeneous treatment effects from data. We propose and validate a novel approach for learning feature representations to aid the estimation of the conditional average treatment effect or CATE. Our method focuses on an intermediate layer in a neural network trained to predict the outcome from the features. In contrast to previous approaches that encourage the distribution of representations to be treatment-invariant, we leverage a genetic algorithm that optimizes over representations useful for predicting the outcome to select those less useful for predicting the treatment. This allows us to retain information within the features useful for predicting outcome even if that information may be related to treatment assignment. We validate our method on synthetic examples and illustrate its use on a real life dataset.

Long chain-of-thought (CoT) prompting helps Large Language Models (LLMs) solve difficult problems, but very long traces often slow or even degrade performance on fast, intuitive "System-1" tasks. We introduce Connector-Aware Compact CoT (CAC-CoT) -- a method that deliberately restricts reasoning to a small, fixed set of connector phrases, steering the model toward concise and well -- structured explanations. Despite its simplicity, our synthetic method with Gemini-2.0-Flash yields a high-quality training quality. CAC-CoT achieves approximately 85% on GSM8K and approximately 40% on GPQA (System-2) while retaining approximately 90% on S1-Bench (System-1). Its reasoning traces average approximately 300 tokens(ART), about one-third the length of baseline traces, delivering higher efficiency without loss of accuracy.

Large Language Models (LLMs) have demonstrated remarkable performance in many applications, including challenging reasoning problems via chain-of-thoughts (CoTs) techniques that generate ``thinking tokens'' before answering the questions. While existing theoretical works demonstrate that CoTs with discrete tokens boost the capability of LLMs, recent work on continuous CoTs lacks a theoretical understanding of why it outperforms discrete counterparts in various reasoning tasks such as directed graph reachability, a fundamental graph reasoning problem that includes many practical domain applications as special cases. In this paper, we prove that a two-layer transformer with D steps of continuous CoTs can solve the directed graph reachability problem, where D is the diameter of the graph, while the best known result of constant-depth transformers with discrete CoTs requires O(n^2) decoding steps where n is the number of vertices (D<n). In our construction, each continuous thought vector is a superposition state that encodes multiple search frontiers simultaneously (i.e., parallel breadth-first search (BFS)), while discrete CoTs must choose a single path sampled from the superposition state, which leads to sequential search that requires many more steps and may be trapped into local solutions. We also performed extensive experiments to verify that our theoretical construction aligns well with the empirical solution obtained via training dynamics. Notably, encoding of multiple search frontiers as a superposition state automatically emerges in training continuous CoTs, without explicit supervision to guide the model to explore multiple paths simultaneously.

Self-supervised learning (SSL) has emerged as a powerful paradigm for learning representations without labeled data. Most SSL approaches rely on strong, well-established, handcrafted data augmentations to generate diverse views for representation learning. However, designing such augmentations requires domain-specific knowledge and implicitly imposes representational invariances on the model, which can limit generalization. In this work, we propose an unsupervised representation learning method that replaces augmentations by generating views using orthonormal bases and overcomplete frames. We show that embeddings learned from orthonormal and overcomplete spaces reside on distinct manifolds, shaped by the geometric biases introduced by representing samples in different spaces. By jointly leveraging the complementary geometry of these distinct manifolds, our approach achieves superior performance without artificially increasing data diversity through strong augmentations. We demonstrate the effectiveness of our method on nine datasets across five temporal sequence tasks, where signal-specific characteristics make data augmentations particularly challenging. Without relying on augmentation-induced diversity, our method achieves performance gains of up to 15--20\% over existing self-supervised approaches. Source code: https://github.com/eth-siplab/Learning-with-FrameProjections

A common approach for sequence tagging tasks based on contextual word representations is to train a machine learning classifier directly on these embedding vectors. This approach has two shortcomings. First, such methods consider single input sequences in isolation and are unable to put an individual embedding vector in relation to vectors outside the current local context of use. Second, the high performance of these models relies on fine-tuning the embedding model in conjunction with the classifier, which may not always be feasible due to the size or inaccessibility of the underlying feature-generation model. It is thus desirable, given a collection of embedding vectors of a corpus, i.e., a datastore, to find features of each vector that describe its relation to other, similar vectors in the datastore. With this in mind, we introduce complexity measures of the local topology of the latent space of a contextual language model with respect to a given datastore. The effectiveness of our features is demonstrated through their application to dialogue term extraction. Our work continues a line of research that explores the manifold hypothesis for word embeddings, demonstrating that local structure in the space carved out by word embeddings can be exploited to infer semantic properties.

One way to enhance the reasoning capability of Large Language Models (LLMs) is to conduct Supervised Fine-Tuning (SFT) using Chain-of-Thought (CoT) annotations. This approach does not show sufficiently strong generalization ability, however, because the training only relies on the given CoT data. In math problem-solving, for example, there is usually only one annotated reasoning path for each question in the training data. Intuitively, it would be better for the algorithm to learn from multiple annotated reasoning paths given a question. To address this issue, we propose a simple yet effective approach called Reinforced Fine-Tuning (ReFT) to enhance the generalizability of learning LLMs for reasoning, with math problem-solving as an example. ReFT first warmups the model with SFT, and then employs on-line reinforcement learning, specifically the PPO algorithm in this paper, to further fine-tune the model, where an abundance of reasoning paths are automatically sampled given the question and the rewards are naturally derived from the ground-truth answers. Extensive experiments on GSM8K, MathQA, and SVAMP datasets show that ReFT significantly outperforms SFT, and the performance can be potentially further boosted by combining inference-time strategies such as majority voting and re-ranking. Note that ReFT obtains the improvement by learning from the same training questions as SFT, without relying on extra or augmented training questions. This indicates a superior generalization ability for ReFT.

We provide a construction for categorical representation learning and introduce the foundations of "categorifier". The central theme in representation learning is the idea of everything to vector. Every object in a dataset S can be represented as a vector in R^n by an encoding map E: Obj(S)toR^n. More importantly, every morphism can be represented as a matrix E: Hom(S)toR^{n}_{n}. The encoding map E is generally modeled by a deep neural network. The goal of representation learning is to design appropriate tasks on the dataset to train the encoding map (assuming that an encoding is optimal if it universally optimizes the performance on various tasks). However, the latter is still a set-theoretic approach. The goal of the current article is to promote the representation learning to a new level via a category-theoretic approach. As a proof of concept, we provide an example of a text translator equipped with our technology, showing that our categorical learning model outperforms the current deep learning models by 17 times. The content of the current article is part of the recent US patent proposal (patent application number: 63110906).

Although chain-of-thought (CoT) prompting combined with language models has achieved encouraging results on complex reasoning tasks, the naive greedy decoding used in CoT prompting usually causes the repetitiveness and local optimality. To address this shortcoming, ensemble-optimization tries to obtain multiple reasoning paths to get the final answer assembly. However, current ensemble-optimization methods either simply employ rule-based post-processing such as self-consistency, or train an additional model based on several task-related human annotations to select the best one among multiple reasoning paths, yet fail to generalize to realistic settings where the type of input questions is unknown or the answer format of reasoning paths is unknown. To avoid their limitations, we propose self-agreement, a generalizable ensemble-optimization method applying in almost all scenarios where the type of input questions and the answer format of reasoning paths may be known or unknown. Self-agreement firstly samples from language model's decoder to generate a diverse set of reasoning paths, and subsequently prompts the language model one more time to determine the optimal answer by selecting the most agreed answer among the sampled reasoning paths. Self-agreement simultaneously achieves remarkable performance on six public reasoning benchmarks and superior generalization capabilities.

This paper presents SimCSE, a simple contrastive learning framework that greatly advances state-of-the-art sentence embeddings. We first describe an unsupervised approach, which takes an input sentence and predicts itself in a contrastive objective, with only standard dropout used as noise. This simple method works surprisingly well, performing on par with previous supervised counterparts. We find that dropout acts as minimal data augmentation, and removing it leads to a representation collapse. Then, we propose a supervised approach, which incorporates annotated pairs from natural language inference datasets into our contrastive learning framework by using "entailment" pairs as positives and "contradiction" pairs as hard negatives. We evaluate SimCSE on standard semantic textual similarity (STS) tasks, and our unsupervised and supervised models using BERT base achieve an average of 76.3% and 81.6% Spearman's correlation respectively, a 4.2% and 2.2% improvement compared to the previous best results. We also show -- both theoretically and empirically -- that the contrastive learning objective regularizes pre-trained embeddings' anisotropic space to be more uniform, and it better aligns positive pairs when supervised signals are available.

Large Language Models (LLMs) have demonstrated remarkable capabilities but typically require extensive computational resources and memory for inference. Post-training quantization (PTQ) can effectively reduce these demands by storing weights in lower bit-width formats. However, standard uniform quantization often leads to notable performance degradation, particularly in low-bit scenarios. In this work, we introduce a Grouped Lattice Vector Quantization (GLVQ) framework that assigns each group of weights a customized lattice codebook, defined by a learnable generation matrix. To address the non-differentiability of the quantization process, we adopt Babai rounding to approximate nearest-lattice-point search during training, which enables stable optimization of the generation matrices. Once trained, decoding reduces to a simple matrix-vector multiplication, yielding an efficient and practical quantization pipeline. Experiments on multiple benchmarks show that our approach achieves a better trade-off between model size and accuracy compared to existing post-training quantization baselines, highlighting its effectiveness in deploying large models under stringent resource constraints. Our source code is available on GitHub repository: https://github.com/xzhang9308/GLVQ.

Given a matrix Ain R^{ntimes d} and a vector bin R^n, we consider the regression problem with ell_infty guarantees: finding a vector x'in R^d such that |x'-x^*|_infty leq epsilon{d}cdot |Ax^*-b|_2cdot |A^dagger| where x^*=argmin_{xin R^d}|Ax-b|_2. One popular approach for solving such ell_2 regression problem is via sketching: picking a structured random matrix Sin R^{mtimes n} with mll n and SA can be quickly computed, solve the ``sketched'' regression problem argmin_{xin R^d} |SAx-Sb|_2. In this paper, we show that in order to obtain such ell_infty guarantee for ell_2 regression, one has to use sketching matrices that are dense. To the best of our knowledge, this is the first user case in which dense sketching matrices are necessary. On the algorithmic side, we prove that there exists a distribution of dense sketching matrices with m=epsilon^{-2}dlog^3(n/delta) such that solving the sketched regression problem gives the ell_infty guarantee, with probability at least 1-delta. Moreover, the matrix SA can be computed in time O(ndlog n). Our row count is nearly-optimal up to logarithmic factors, and significantly improves the result in [Price, Song and Woodruff, ICALP'17], in which a super-linear in d rows, m=Omega(epsilon^{-2}d^{1+gamma}) for gamma=Theta(frac{loglog n{log d}}) is required. We also develop a novel analytical framework for ell_infty guarantee regression that utilizes the Oblivious Coordinate-wise Embedding (OCE) property introduced in [Song and Yu, ICML'21]. Our analysis is arguably much simpler and more general than [Price, Song and Woodruff, ICALP'17], and it extends to dense sketches for tensor product of vectors.

This paper considers the learning of logical (Boolean) functions with focus on the generalization on the unseen (GOTU) setting, a strong case of out-of-distribution generalization. This is motivated by the fact that the rich combinatorial nature of data in certain reasoning tasks (e.g., arithmetic/logic) makes representative data sampling challenging, and learning successfully under GOTU gives a first vignette of an 'extrapolating' or 'reasoning' learner. We then study how different network architectures trained by (S)GD perform under GOTU and provide both theoretical and experimental evidence that for a class of network models including instances of Transformers, random features models, and diagonal linear networks, a min-degree-interpolator (MDI) is learned on the unseen. We also provide evidence that other instances with larger learning rates or mean-field networks reach leaky MDIs. These findings lead to two implications: (1) we provide an explanation to the length generalization problem (e.g., Anil et al. 2022); (2) we introduce a curriculum learning algorithm called Degree-Curriculum that learns monomials more efficiently by incrementing supports.

This paper proposes a method for the automatic creation of variables (in the case of regression) that complement the information contained in the initial input vector. The method works as a pre-processing step in which the continuous values of the variable to be regressed are discretized into a set of intervals which are then used to define value thresholds. Then classifiers are trained to predict whether the value to be regressed is less than or equal to each of these thresholds. The different outputs of the classifiers are then concatenated in the form of an additional vector of variables that enriches the initial vector of the regression problem. The implemented system can thus be considered as a generic pre-processing tool. We tested the proposed enrichment method with 5 types of regressors and evaluated it in 33 regression datasets. Our experimental results confirm the interest of the approach.

Understanding the locus of semantic representation in large language models (LLMs) is crucial for interpretability and architectural innovation. The dominant paradigm posits that trainable input embeddings serve as foundational "meaning vectors." This paper challenges that view. We construct Transformer models where the embedding layer is entirely frozen, with vectors derived not from data, but from the visual structure of Unicode glyphs. These non-semantic, precomputed visual embeddings are fixed throughout training. Our method is compatible with any tokenizer, including a novel Unicode-centric tokenizer we introduce to ensure universal text coverage. Despite the absence of trainable, semantically initialized embeddings, our models converge, generate coherent text, and, critically, outperform architecturally identical models with trainable embeddings on the MMLU reasoning benchmark. We attribute this to "representational interference" in conventional models, where the embedding layer is burdened with learning both structural and semantic features. Our results indicate that high-level semantics are not inherent to input embeddings but are an emergent property of the Transformer's compositional architecture and data scale. This reframes the role of embeddings from meaning containers to structural primitives. We release all code and models to foster further research.

We present Probabilistic Structure Integration (PSI), a system for learning richly controllable and flexibly promptable world models from data. PSI consists of a three-step cycle. The first step, Probabilistic prediction, involves building a probabilistic graphical model Psi of the data, in the form of a random-access autoregressive sequence model. Psi supports a complete set of learned conditional distributions describing the dependence of any variables in the data on any other set of variables. In step 2, Structure extraction, we show how to extract underlying low-dimensional properties in the data, corresponding to a diverse set of meaningful "intermediate structures", in a zero-shot fashion via causal inference on Psi. Step 3, Integration, completes the cycle by converting these structures into new token types that are then continually mixed back into the training diet as conditioning signals and prediction targets. Each such cycle augments the capabilities of Psi, both allowing it to model the underlying data better, and creating new control handles -- akin to an LLM-like universal prompting language. We train an instance of Psi on 1.4 trillion tokens of internet video data; we use it to perform a variety of useful video prediction and understanding inferences; we extract state-of-the-art optical flow, self-supervised depth and object segmentation; and we use these structures to support a full cycle of predictive improvements.

Changing how pre-trained models behave -- e.g., improving their performance on a downstream task or mitigating biases learned during pre-training -- is a common practice when developing machine learning systems. In this work, we propose a new paradigm for steering the behavior of neural networks, centered around task vectors. A task vector specifies a direction in the weight space of a pre-trained model, such that movement in that direction improves performance on the task. We build task vectors by subtracting the weights of a pre-trained model from the weights of the same model after fine-tuning on a task. We show that these task vectors can be modified and combined together through arithmetic operations such as negation and addition, and the behavior of the resulting model is steered accordingly. Negating a task vector decreases performance on the target task, with little change in model behavior on control tasks. Moreover, adding task vectors together can improve performance on multiple tasks at once. Finally, when tasks are linked by an analogy relationship of the form ``A is to B as C is to D", combining task vectors from three of the tasks can improve performance on the fourth, even when no data from the fourth task is used for training. Overall, our experiments with several models, modalities and tasks show that task arithmetic is a simple, efficient and effective way of editing models.

Modeling latent variables with priors and hyperpriors is an essential problem in variational image compression. Formally, trade-off between rate and distortion is handled well if priors and hyperpriors precisely describe latent variables. Current practices only adopt univariate priors and process each variable individually. However, we find inter-correlations and intra-correlations exist when observing latent variables in a vectorized perspective. These findings reveal visual redundancies to improve rate-distortion performance and parallel processing ability to speed up compression. This encourages us to propose a novel vectorized prior. Specifically, a multivariate Gaussian mixture is proposed with means and covariances to be estimated. Then, a novel probabilistic vector quantization is utilized to effectively approximate means, and remaining covariances are further induced to a unified mixture and solved by cascaded estimation without context models involved. Furthermore, codebooks involved in quantization are extended to multi-codebooks for complexity reduction, which formulates an efficient compression procedure. Extensive experiments on benchmark datasets against state-of-the-art indicate our model has better rate-distortion performance and an impressive 3.18times compression speed up, giving us the ability to perform real-time, high-quality variational image compression in practice. Our source code is publicly available at https://github.com/xiaosu-zhu/McQuic.

While paragraph embedding models are remarkably effective for downstream classification tasks, what they learn and encode into a single vector remains opaque. In this paper, we investigate a state-of-the-art paragraph embedding method proposed by Zhang et al. (2017) and discover that it cannot reliably tell whether a given sentence occurs in the input paragraph or not. We formulate a sentence content task to probe for this basic linguistic property and find that even a much simpler bag-of-words method has no trouble solving it. This result motivates us to replace the reconstruction-based objective of Zhang et al. (2017) with our sentence content probe objective in a semi-supervised setting. Despite its simplicity, our objective improves over paragraph reconstruction in terms of (1) downstream classification accuracies on benchmark datasets, (2) faster training, and (3) better generalization ability.

Mikolov et al. (2013a) observed that continuous bag-of-words (CBOW) word embeddings tend to underperform Skip-gram (SG) embeddings, and this finding has been reported in subsequent works. We find that these observations are driven not by fundamental differences in their training objectives, but more likely on faulty negative sampling CBOW implementations in popular libraries such as the official implementation, word2vec.c, and Gensim. We show that after correcting a bug in the CBOW gradient update, one can learn CBOW word embeddings that are fully competitive with SG on various intrinsic and extrinsic tasks, while being many times faster to train.

Normalizing Flows explicitly maximize a full-dimensional likelihood on the training data. However, real data is typically only supported on a lower-dimensional manifold leading the model to expend significant compute on modeling noise. Injective Flows fix this by jointly learning a manifold and the distribution on it. So far, they have been limited by restrictive architectures and/or high computational cost. We lift both constraints by a new efficient estimator for the maximum likelihood loss, compatible with free-form bottleneck architectures. We further show that naively learning both the data manifold and the distribution on it can lead to divergent solutions, and use this insight to motivate a stable maximum likelihood training objective. We perform extensive experiments on toy, tabular and image data, demonstrating the competitive performance of the resulting model.

Semantic prompt caches reduce the latency and cost of large language model (LLM) inference by reusing cached LLM-generated responses for semantically similar prompts. Vector similarity metrics assign a numerical score to quantify the similarity between an embedded prompt and its nearest neighbor in the cache. Existing systems rely on a static threshold to classify whether the similarity score is sufficiently high to result in a cache hit. We show that this one-size-fits-all threshold is insufficient across different prompts. We propose VectorQ, a framework to learn embedding-specific threshold regions that adapt to the complexity and uncertainty of an embedding. Through evaluations on a combination of four diverse datasets, we show that VectorQ consistently outperforms state-of-the-art systems across all static thresholds, achieving up to 12x increases in cache hit rate and error rate reductions up to 92%.

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.

Gaussian processes (GPs) are popular nonparametric statistical models for learning unknown functions and quantifying the spatiotemporal uncertainty in data. Recent works have extended GPs to model scalar and vector quantities distributed over non-Euclidean domains, including smooth manifolds appearing in numerous fields such as computer vision, dynamical systems, and neuroscience. However, these approaches assume that the manifold underlying the data is known, limiting their practical utility. We introduce RVGP, a generalisation of GPs for learning vector signals over latent Riemannian manifolds. Our method uses positional encoding with eigenfunctions of the connection Laplacian, associated with the tangent bundle, readily derived from common graph-based approximation of data. We demonstrate that RVGP possesses global regularity over the manifold, which allows it to super-resolve and inpaint vector fields while preserving singularities. Furthermore, we use RVGP to reconstruct high-density neural dynamics derived from low-density EEG recordings in healthy individuals and Alzheimer's patients. We show that vector field singularities are important disease markers and that their reconstruction leads to a comparable classification accuracy of disease states to high-density recordings. Thus, our method overcomes a significant practical limitation in experimental and clinical applications.

Transformer-based language models (LMs) are powerful and widely-applicable tools, but their usefulness is constrained by a finite context window and the expensive computational cost of processing long text documents. We propose to adapt pre-trained LMs into AutoCompressors. These models are capable of compressing long contexts into compact summary vectors, which are then accessible to the model as soft prompts. Summary vectors are trained with an unsupervised objective, whereby long documents are processed in segments and summary vectors from all previous segments are used in language modeling. We fine-tune OPT models on sequences of up to 30,720 tokens and show that AutoCompressors can utilize long contexts to improve perplexity. We evaluate AutoCompressors on in-context learning by compressing task demonstrations. We find that summary vectors are good substitutes for plain-text demonstrations, increasing accuracy while reducing inference cost. Finally, we explore the benefits of pre-computing summary vectors for large corpora by applying summary vectors to retrieval-augmented language modeling. Overall, AutoCompressors emerge as a simple and inexpensive solution for extending the context window of LMs while speeding up inference over long contexts.

We present a feasibility-seeking approach to neural network training. This mathematical optimization framework is distinct from conventional gradient-based loss minimization and uses projection operators and iterative projection algorithms. We reformulate training as a large-scale feasibility problem: finding network parameters and states that satisfy local constraints derived from its elementary operations. Training then involves projecting onto these constraints, a local operation that can be parallelized across the network. We introduce PJAX, a JAX-based software framework that enables this paradigm. PJAX composes projection operators for elementary operations, automatically deriving the solution operators for the feasibility problems (akin to autodiff for derivatives). It inherently supports GPU/TPU acceleration, provides a familiar NumPy-like API, and is extensible. We train diverse architectures (MLPs, CNNs, RNNs) on standard benchmarks using PJAX, demonstrating its functionality and generality. Our results show that this approach is as a compelling alternative to gradient-based training, with clear advantages in parallelism and the ability to handle non-differentiable operations.

Weak supervision (WS) frameworks are a popular way to bypass hand-labeling large datasets for training data-hungry models. These approaches synthesize multiple noisy but cheaply-acquired estimates of labels into a set of high-quality pseudolabels for downstream training. However, the synthesis technique is specific to a particular kind of label, such as binary labels or sequences, and each new label type requires manually designing a new synthesis algorithm. Instead, we propose a universal technique that enables weak supervision over any label type while still offering desirable properties, including practical flexibility, computational efficiency, and theoretical guarantees. We apply this technique to important problems previously not tackled by WS frameworks including learning to rank, regression, and learning in hyperbolic space. Theoretically, our synthesis approach produces a consistent estimators for learning some challenging but important generalizations of the exponential family model. Experimentally, we validate our framework and show improvement over baselines in diverse settings including real-world learning-to-rank and regression problems along with learning on hyperbolic manifolds.

Large Language Models (LLMs) achieve state-of-the-art performance across various NLP tasks but face deployment challenges due to high computational costs and memory constraints. Knowledge distillation (KD) is a promising solution, transferring knowledge from large teacher models to smaller student models. However, existing KD methods often assume shared vocabularies and tokenizers, limiting their flexibility. While approaches like Universal Logit Distillation (ULD) and Dual-Space Knowledge Distillation (DSKD) address vocabulary mismatches, they overlook the critical reasoning-aware distillation aspect. To bridge this gap, we propose CoT2Align a universal KD framework that integrates Chain-of-Thought (CoT) augmentation and introduces Cross-CoT Alignment to enhance reasoning transfer. Additionally, we extend Optimal Transport beyond token-wise alignment to a sequence-level and layer-wise alignment approach that adapts to varying sequence lengths while preserving contextual integrity. Comprehensive experiments demonstrate that CoT2Align outperforms existing KD methods across different vocabulary settings, improving reasoning capabilities and robustness in domain-specific tasks.

Continuous latent variables (LVs) are a key ingredient of many generative models, as they allow modelling expressive mixtures with an uncountable number of components. In contrast, probabilistic circuits (PCs) are hierarchical discrete mixtures represented as computational graphs composed of input, sum and product units. Unlike continuous LV models, PCs provide tractable inference but are limited to discrete LVs with categorical (i.e. unordered) states. We bridge these model classes by introducing probabilistic integral circuits (PICs), a new language of computational graphs that extends PCs with integral units representing continuous LVs. In the first place, PICs are symbolic computational graphs and are fully tractable in simple cases where analytical integration is possible. In practice, we parameterise PICs with light-weight neural nets delivering an intractable hierarchical continuous mixture that can be approximated arbitrarily well with large PCs using numerical quadrature. On several distribution estimation benchmarks, we show that such PIC-approximating PCs systematically outperform PCs commonly learned via expectation-maximization or SGD.

Learning representations that are useful for unknown downstream tasks is a fundamental challenge in representation learning. Prominent approaches in this domain include contrastive learning, self-supervised masking, and denoising auto-encoders. In this paper, we introduce a novel method, termed contrastive Mutual Information Machine (cMIM), which aims to enhance the utility of learned representations for downstream tasks. cMIM integrates a new contrastive learning loss with the Mutual Information Machine (MIM) learning framework, a probabilistic auto-encoder that maximizes the mutual information between inputs and latent representations while clustering the latent codes. Despite MIM's potential, initial experiments indicated that the representations learned by MIM were less effective for discriminative downstream tasks compared to state-of-the-art (SOTA) models. The proposed cMIM method directly addresses this limitation. The main contributions of this work are twofold: (1) We propose a novel contrastive extension to MIM for learning discriminative representations which eliminates the need for data augmentation and is robust to variations in the number of negative examples (i.e., batch size). (2) We introduce a generic method for extracting informative embeddings from encoder-decoder models, which significantly improves performance in discriminative downstream tasks without requiring additional training. This method is applicable to any pre-trained encoder-decoder model. By presenting cMIM, we aim to offer a unified generative model that is effective for both generative and discriminative tasks. Our results demonstrate that the learned representations are valuable for downstream tasks while maintaining the generative capabilities of MIM.

To fully understand the behavior of a large language model (LLM) requires our understanding of its input space. If this input space differs from our assumption, our understanding of and conclusions about the LLM is likely flawed, regardless of its architecture. Here, we elucidate the structure of the token embeddings, the input domain for LLMs, both empirically and theoretically. We present a generalized and statistically testable model where the neighborhood of each token splits into well-defined signal and noise dimensions. This model is based on a generalization of a manifold called a fiber bundle, so we denote our hypothesis test as the ``fiber bundle null.'' Failing to reject the null is uninformative, but rejecting it at a specific token indicates that token has a statistically significant local structure, and so is of interest to us. By running our test over several open-source LLMs, each with unique token embeddings, we find that the null is frequently rejected, and so the token subspace is provably not a fiber bundle and hence also not a manifold. As a consequence of our findings, when an LLM is presented with two semantically equivalent prompts, and if one prompt contains a token implicated by our test, that prompt will likely exhibit more output variability proportional to the local signal dimension of the token.

Despite their widespread success, the application of deep neural networks to functional data remains scarce today. The infinite dimensionality of functional data means standard learning algorithms can be applied only after appropriate dimension reduction, typically achieved via basis expansions. Currently, these bases are chosen a priori without the information for the task at hand and thus may not be effective for the designated task. We instead propose to adaptively learn these bases in an end-to-end fashion. We introduce neural networks that employ a new Basis Layer whose hidden units are each basis functions themselves implemented as a micro neural network. Our architecture learns to apply parsimonious dimension reduction to functional inputs that focuses only on information relevant to the target rather than irrelevant variation in the input function. Across numerous classification/regression tasks with functional data, our method empirically outperforms other types of neural networks, and we prove that our approach is statistically consistent with low generalization error. Code is available at: https://github.com/jwyyy/AdaFNN.

The recent tremendous success of unsupervised word embeddings in a multitude of applications raises the obvious question if similar methods could be derived to improve embeddings (i.e. semantic representations) of word sequences as well. We present a simple but efficient unsupervised objective to train distributed representations of sentences. Our method outperforms the state-of-the-art unsupervised models on most benchmark tasks, highlighting the robustness of the produced general-purpose sentence embeddings.

We introduce ReALLM, a novel approach for compression and memory-efficient adaptation of pre-trained language models that encompasses most of the post-training quantization and fine-tuning methods for a budget of <4 bits. Pre-trained matrices are decomposed into a high-precision low-rank component and a vector-quantized latent representation (using an autoencoder). During the fine-tuning step, only the low-rank components are updated. Our results show that pre-trained matrices exhibit different patterns. ReALLM adapts the shape of the encoder (small/large embedding, high/low bit VQ, etc.) to each matrix. ReALLM proposes to represent each matrix with a small embedding on b bits and a neural decoder model D_phi with its weights on b_phi bits. The decompression of a matrix requires only one embedding and a single forward pass with the decoder. Our weight-only quantization algorithm yields the best results on language generation tasks (C4 and WikiText-2) for a budget of 3 bits without any training. With a budget of 2 bits, ReALLM achieves state-of-the art performance after fine-tuning on a small calibration dataset.

Modular and composable transfer learning is an emerging direction in the field of Parameter Efficient Fine-Tuning, as it enables neural networks to better organize various aspects of knowledge, leading to improved cross-task generalization. In this paper, we introduce a novel approach Customized Polytropon C-Poly that combines task-common skills and task-specific skills, while the skill parameters being highly parameterized using low-rank techniques. Each task is associated with a customizable number of exclusive specialized skills and also benefits from skills shared with peer tasks. A skill assignment matrix is jointly learned. To evaluate our approach, we conducted extensive experiments on the Super-NaturalInstructions and the SuperGLUE benchmarks. Our findings demonstrate that C-Poly outperforms fully-shared, task-specific, and skill-indistinguishable baselines, significantly enhancing the sample efficiency in multi-task learning scenarios.

We introduce a new type of deep contextualized word representation that models both (1) complex characteristics of word use (e.g., syntax and semantics), and (2) how these uses vary across linguistic contexts (i.e., to model polysemy). Our word vectors are learned functions of the internal states of a deep bidirectional language model (biLM), which is pre-trained on a large text corpus. We show that these representations can be easily added to existing models and significantly improve the state of the art across six challenging NLP problems, including question answering, textual entailment and sentiment analysis. We also present an analysis showing that exposing the deep internals of the pre-trained network is crucial, allowing downstream models to mix different types of semi-supervision signals.

When two different parties use the same learning rule on their own data, how can we test whether the distributions of the two outcomes are similar? In this paper, we study the similarity of outcomes of learning rules through the lens of the Total Variation (TV) distance of distributions. We say that a learning rule is TV indistinguishable if the expected TV distance between the posterior distributions of its outputs, executed on two training data sets drawn independently from the same distribution, is small. We first investigate the learnability of hypothesis classes using TV indistinguishable learners. Our main results are information-theoretic equivalences between TV indistinguishability and existing algorithmic stability notions such as replicability and approximate differential privacy. Then, we provide statistical amplification and boosting algorithms for TV indistinguishable learners.

This paper presents TS2Vec, a universal framework for learning representations of time series in an arbitrary semantic level. Unlike existing methods, TS2Vec performs contrastive learning in a hierarchical way over augmented context views, which enables a robust contextual representation for each timestamp. Furthermore, to obtain the representation of an arbitrary sub-sequence in the time series, we can apply a simple aggregation over the representations of corresponding timestamps. We conduct extensive experiments on time series classification tasks to evaluate the quality of time series representations. As a result, TS2Vec achieves significant improvement over existing SOTAs of unsupervised time series representation on 125 UCR datasets and 29 UEA datasets. The learned timestamp-level representations also achieve superior results in time series forecasting and anomaly detection tasks. A linear regression trained on top of the learned representations outperforms previous SOTAs of time series forecasting. Furthermore, we present a simple way to apply the learned representations for unsupervised anomaly detection, which establishes SOTA results in the literature. The source code is publicly available at https://github.com/yuezhihan/ts2vec.

Semantic word embeddings represent the meaning of a word via a vector, and are created by diverse methods. Many use nonlinear operations on co-occurrence statistics, and have hand-tuned hyperparameters and reweighting methods. This paper proposes a new generative model, a dynamic version of the log-linear topic model of~mnih2007three. The methodological novelty is to use the prior to compute closed form expressions for word statistics. This provides a theoretical justification for nonlinear models like PMI, word2vec, and GloVe, as well as some hyperparameter choices. It also helps explain why low-dimensional semantic embeddings contain linear algebraic structure that allows solution of word analogies, as shown by~mikolov2013efficient and many subsequent papers. Experimental support is provided for the generative model assumptions, the most important of which is that latent word vectors are fairly uniformly dispersed in space.

Embedding models have been crucial in enabling various downstream tasks such as semantic similarity, information retrieval, and clustering. Recently, there has been a surge of interest in developing universal text embedding models that can generalize across tasks (e.g., MTEB). However, progress in learning universal multimodal embedding models has been relatively slow despite their importance. In this work, we aim to explore the potential for building universal embeddings capable of handling a wide range of downstream tasks. Our contributions are twofold: (1) MMEB (Massive Multimodal Embedding Benchmark), which covers 4 meta-tasks (i.e. classification, visual question answering, multimodal retrieval, and visual grounding) and 36 datasets, including 20 training and 16 evaluation datasets, and (2) VLM2Vec (Vision-Language Model -> Vector), a contrastive training framework that converts any state-of-the-art vision-language model into an embedding model via training on MMEB. Unlike previous models such as CLIP and BLIP, VLM2Vec can process any combination of images and text to generate a fixed-dimensional vector based on task instructions. We build a series of VLM2Vec models on Phi-3.5-V and evaluate them on MMEB's evaluation split. Our results show that \model achieves an absolute average improvement of 10% to 20% over existing multimodal embedding models on both in-distribution and out-of-distribution datasets in MMEB.

This paper presents E5, a family of state-of-the-art text embeddings that transfer well to a wide range of tasks. The model is trained in a contrastive manner with weak supervision signals from our curated large-scale text pair dataset (called CCPairs). E5 can be readily used as a general-purpose embedding model for any tasks requiring a single-vector representation of texts such as retrieval, clustering, and classification, achieving strong performance in both zero-shot and fine-tuned settings. We conduct extensive evaluations on 56 datasets from the BEIR and MTEB benchmarks. For zero-shot settings, E5 is the first model that outperforms the strong BM25 baseline on the BEIR retrieval benchmark without using any labeled data. When fine-tuned, E5 obtains the best results on the MTEB benchmark, beating existing embedding models with 40x more parameters.

In this work, we focus on a lightweight convolutional architecture that creates fixed-size vector embeddings of sentences. Such representations are useful for building NLP systems, including conversational agents. Our work derives from a recently proposed recursive convolutional architecture for auto-encoding text paragraphs at byte level. We propose alternations that significantly reduce training time, the number of parameters, and improve auto-encoding accuracy. Finally, we evaluate the representations created by our model on tasks from SentEval benchmark suite, and show that it can serve as a better, yet fairly low-resource alternative to popular bag-of-words embeddings.

We address the problem of self-supervised learning on discrete event sequences generated by real-world users. Self-supervised learning incorporates complex information from the raw data in low-dimensional fixed-length vector representations that could be easily applied in various downstream machine learning tasks. In this paper, we propose a new method "CoLES", which adapts contrastive learning, previously used for audio and computer vision domains, to the discrete event sequences domain in a self-supervised setting. We deployed CoLES embeddings based on sequences of transactions at the large European financial services company. Usage of CoLES embeddings significantly improves the performance of the pre-existing models on downstream tasks and produces significant financial gains, measured in hundreds of millions of dollars yearly. We also evaluated CoLES on several public event sequences datasets and showed that CoLES representations consistently outperform other methods on different downstream tasks.