ICLR 2025 Papers

Diffusion models, recognized for their effectiveness as generative priors, have become essential tools for addressing a wide range of visual challenges. Recently, there has been a surge of interest in leveraging Denoising processes for Regularization (DR) to solve inverse problems. However, existing methods often face issues such as mode collapse, which results in excessive smoothing and diminished diversity. In this study, we perform a comprehensive analysis to pinpoint the root causes of gradient inaccuracies inherent in DR. Drawing on insights from diffusion model distillation, we propose a novel approach called Consistency Regularization (CR), which provides stabilized gradients without the need for ODE simulations. Building on this, we introduce Hybrid Regularization (HR), a unified framework that combines the strengths of both DR and CR, harnessing their synergistic potential. Our approach proves to be effective across a broad spectrum of inverse problems, encompassing both linear and nonlinear scenarios, as well as various measurement noise statistics. Experimental evaluations on benchmark datasets, including FFHQ and ImageNet, demonstrate that our proposed framework not only achieves highly competitive results compared to state-of-the-art methods but also offers significant reductions in wall-clock time and memory consumption.

In this work, we introduce a novel evaluation paradigm for Large Language Models (LLMs) that compels them to transition from a traditional question-answering role, akin to a student, to a solution-scoring role, akin to a teacher. This paradigm, focusing on "reasoning about reasoning," termed meta-reasoning, shifts the emphasis from result-oriented assessments, which often neglect the reasoning process, to a more comprehensive evaluation that effectively distinguishes between the cognitive capabilities of different models. Our meta-reasoning process mirrors "system-2" slow thinking, requiring careful examination of assumptions, conditions, calculations, and logic to identify mistakes. This paradigm enables one to transform existed saturated, non-differentiating benchmarks that might be leaked in data pretraining stage to evaluation tools that are both challenging and robust against data contamination. To prove our point, we applied our paradigm to GSM8K dataset and developed the MR-GSM8K benchmark. Our extensive analysis includes several state-of-the-art models from both open-source and commercial domains, uncovering fundamental deficiencies in their training and evaluation methodologies. Specifically, we found the OpenAI o1 models which possess characteristics of "system-2" thinking excel the other SOTA models by more than 20 absolute points in our benchmark, supporting our deficiency hypothesis.

In cognition theory, human thinking is governed by two systems: the fast and intuitive System 1 and the slower but more deliberative System 2. Analogously, Large Language Models (LLMs) can operate in two reasoning modes: outputting only the solutions (\emph{fast mode}) or both the reasoning chain and the final solution (\emph{slow mode}). We present \dualformer, a single Transformer model that seamlessly integrates both the fast and slow reasoning modes by training on randomized reasoning traces, where different parts of the traces are strategically dropped during training. At inference time, \dualformer can be easily configured to execute in either fast or slow mode, or automatically decide which mode to engage (\emph{auto mode}). It outperforms baselines in both performance and computational efficiency across all three modes: \textbf{(1)} in slow mode, \dualformer achieves $97.6\%$ optimal rate on unseen $30 \times 30$ maze tasks, surpassing the \searchformer baseline (93.3\%) trained on data with complete reasoning traces, with $45.5\%$ fewer reasoning steps; \textbf{(2)} in fast mode, \dualformer achieves $80\%$ optimal rate, significantly outperforming the Solution-Only model trained on solution-only data, which has an optimal rate of only 30\%; \textbf{(3)} in auto mode, \dualformer achieves $96.6\%$ optimal rate with $59.9\%$ fewer steps than \searchformer. For math reasoning problems, our techniques have also achieved improved performance with LLM fine-tuning, demonstrating its generalization beyond task-specific models. We open source our code at https://github.com/facebookresearch/dualformer.

Post-training Large Language Models (LLMs) with explicit reasoning trajectories can enhance their reasoning abilities. However, acquiring such high-quality trajectory data typically demands meticulous supervision from humans or superior models, which can be either expensive or license-constrained. In this paper, we explore how far an LLM can improve its reasoning by self-synthesizing reasoning paths as training data without any additional supervision. Existing self-synthesizing methods, such as STaR, suffer from poor generalization to out-of-domain (OOD) reasoning tasks. We hypothesize it is due to that their self-synthesized reasoning paths are too task-specific, lacking general task-agnostic reasoning guidance. To address this, we propose **Reasoning Generalist via Self-Improvement (ReGenesis)**, a method to *self-synthesize reasoning paths as post-training data by progressing from abstract to concrete*. More specifically, ReGenesis self-synthesizes reasoning paths by converting general reasoning guidelines into task-specific ones, generating reasoning structures, and subsequently transforming these structures into reasoning paths, without the need for human-designed task-specific examples used in existing methods. We show that ReGenesis achieves superior performance on all in-domain and OOD settings tested compared to existing methods. For six OOD tasks specifically, while previous methods exhibited an average performance decrease of approximately 4.6% after post training, ReGenesis delivers around 6.1% performance improvement. We also conduct an in-depth analysis of our framework and show ReGenesis is effective across various language models and design choices.

Large language models have been successful at tasks involving basic forms of in-context reasoning, such as generating coherent language, as well as storing vast amounts of knowledge. At the core of the Transformer architecture behind such models are feed-forward and attention layers, which are often associated to knowledge and reasoning, respectively. In this paper, we study this distinction empirically and theoretically in a controlled synthetic setting where certain next-token predictions involve both distributional and in-context information. We find that feed-forward layers tend to learn simple distributional associations such as bigrams, while attention layers focus on in-context reasoning. Our theoretical analysis identifies the noise in the gradients as a key factor behind this discrepancy. Finally, we illustrate how similar disparities emerge in pre-trained models through ablations on the Pythia model family on simple reasoning tasks.

Few-shot Chain-of-Thought (CoT) significantly enhances the reasoning capabilities of large language models (LLMs), functioning as a whole to guide these models in generating reasoning steps toward final answers. However, we observe that isolated segments, words, or tokens within CoT demonstrations can unexpectedly disrupt the generation process of LLMs. The model may overly concentrate on certain local information present in the demonstration, introducing irrelevant noise into the reasoning process and potentially leading to incorrect answers. In this paper, we investigate the underlying mechanism of CoT through dynamically tracing and manipulating the inner workings of LLMs at each output step, which demonstrates that tokens exhibiting specific attention characteristics are more likely to induce the model to take things out of context; these tokens directly attend to the hidden states tied with prediction, without substantial integration of non-local information. Building upon these insights, we propose a Few-shot Attention Intervention method (FAI) that dynamically analyzes the attention patterns of demonstrations to accurately identify these tokens and subsequently make targeted adjustments to the attention weights to effectively suppress their distracting effect on LLMs. Comprehensive experiments across multiple benchmarks demonstrate consistent improvements over baseline methods, with a remarkable 5.91\% improvement on the AQuA dataset, further highlighting the effectiveness of FAI.

Despite the increasing effectiveness of language models, their reasoning capabilities remain underdeveloped. In particular, causal reasoning through counterfactual question answering is lacking. This work aims to bridge this gap. We first derive novel metrics that balance accuracy in factual and counterfactual questions, capturing a more complete view of the reasoning abilities of language models than traditional factual-only based metrics. Second, we propose several fine-tuning approaches that aim to elicit better reasoning mechanisms, in the sense of the proposed metrics. Finally, we evaluate the performance of the fine-tuned language models in a variety of realistic scenarios. In particular, we investigate to what extent our fine-tuning approaches systemically achieve better generalization with respect to the base models in several problems that require, among others, inductive and deductive reasoning capabilities.

We analyse the complexity of the satisfiability problem, or similarly feasibility problem, (trSAT) for transformer encoders (TE), which naturally occurs in formal verification or interpretation, collectively referred to as formal reasoning. We find that trSAT is undecidable when considering TE as they are commonly studied in the expressiveness community. Furthermore, we identify practical scenarios where trSAT is decidable and establish corresponding complexity bounds. Beyond trivial cases, we find that quantized TE, those restricted by fixed-width arithmetic, lead to the decidability of trSAT due to their limited attention capabilities. However, the problem remains difficult, as we establish scenarios where trSAT is NEXPTIME-hard and others where it is solvable in NEXPTIME for quantized TE. To complement our complexity results, we place our findings and their implications in the broader context of formal reasoning.

The rapid advancements in Vision-Language Models (VLMs) have shown great potential in tackling mathematical reasoning tasks that involve visual context. Unlike humans who can reliably apply solution steps to similar problems with minor modifications, we found that state-of-the-art VLMs like GPT-4o can consistently fail in these scenarios, revealing limitations in their mathematical reasoning capabilities. In this paper, we investigate the **mathematical reasoning robustness** in VLMs and evaluate how well these models perform under different variants of the same question, such as changes in visual numerical values or function graphs. While several vision-based math benchmarks have been developed to assess VLMs' problem-solving capabilities, these benchmarks contain only static sets of problems and cannot easily evaluate mathematical reasoning robustness. To fill this gap, we introduce **DynaMath**, a dynamic visual math benchmark designed for in-depth assessment of VLMs. **DynaMath** includes 501 high-quality, multi-topic *seed* questions, *each represented as a Python program*. Those programs are carefully designed and annotated to enable the automatic generation of a much larger set of *concrete* questions, including many different types of visual and textual variations. **DynaMath** allows us to evaluate the generalization ability of VLMs, by assessing their performance under varying input conditions of a seed question. We evaluated 14 state-of-the-art VLMs with 5,010 generated concrete questions (10 per seed question). Our results show that the worst-case model accuracy, defined as the percentage of correctly answered seed questions in all 10 variants, is significantly lower than the average-case accuracy. In addition, many models show high consistency in answering these questions -- the incorrectness of a certain variant of a seed question is not only due to inherent randomness. Our analysis emphasizes the need to study the robustness of VLMs' reasoning abilities, and **DynaMath** provides valuable insights to guide the development of more reliable models for mathematical reasoning.

While the scaling laws of large language models (LLMs) training have been extensively studied, optimal inference configurations of LLMs remain underexplored. We study inference scaling laws (aka test-time scaling laws) and compute-optimal inference, focusing on the trade-offs between model sizes and generating additional tokens with different inference strategies. As a first step towards understanding and designing compute-optimal inference methods, we studied cost-performance trade-offs for inference strategies such as greedy search, majority voting, best-of-$n$, weighted voting, and two different tree search algorithms, using different model sizes and compute budgets. Our findings suggest that scaling inference compute with inference strategies can be more computationally efficient than scaling model parameters. Additionally, smaller models combined with advanced inference algorithms offer Pareto-optimal trade-offs in cost and performance. For example, the Llemma-7B model, when paired with our novel tree search algorithm, consistently outperforms the Llemma-34B model across all tested inference strategies on the MATH benchmark. We hope these insights contribute to a deeper understanding of inference scaling laws (test-time scaling laws) for LLMs.

Large Language Models (LLMs) have demonstrated strong reasoning capabilities in solving complex problems. However, current approaches primarily enhance reasoning through the elaboration of thoughts while neglecting the diversity of reasoning types. LLMs typically employ deductive reasoning, proceeding step-by-step from given conditions, which limits their exploration during problem-solving. Our analysis reveals that certain problems are exclusively solvable through specific reasoning strategies like inductive, abductive, or analogical reasoning. However, incorporating diverse reasoning approaches presents two key challenges: identifying the appropriate reasoning type for each problem and exploiting this approach during problem-solving. Therefore, we propose the TypedThinker that predicts suitable reasoning types based on the problem and their previous effectiveness and provides relevant demonstrations to guide LLMs in applying these strategies. Experimental results show significant improvements across multiple benchmarks, with performance gains of 3.4\% for Mistral 7B, 6.5\% for LLaMA3 8B, and 7\% for Qwen 2 7B on logical and mathematical reasoning tasks. TypedThinker enhances LLM reasoning without requiring knowledge distillation from larger models. It can be integrated into more advanced systems like GPT-4o or specialized models like MetaMath to diversify their reasoning approaches and improve their problem-solving capabilities.

Large Language Models (LLMs) have been successful in mathematical reasoning tasks such as formal theorem proving when integrated with interactive proof assistants like Lean. Existing approaches involve training or fine-tuning an LLM on a specific dataset to perform well on particular domains, such as undergraduate-level mathematics. These methods struggle with generalizability to advanced mathematics. A fundamental limitation is that these approaches operate on static domains, failing to capture how mathematicians often work across multiple domains and projects simultaneously or cyclically. We present LeanAgent, a novel lifelong learning framework for formal theorem proving that continuously generalizes to and improves on ever-expanding mathematical knowledge without forgetting previously learned knowledge. LeanAgent introduces several key innovations, including a curriculum learning strategy that optimizes the learning trajectory in terms of mathematical difficulty, a dynamic database for efficient management of evolving mathematical knowledge, and progressive training to balance stability and plasticity. LeanAgent successfully generates formal proofs for 155 theorems across 23 diverse Lean repositories where formal proofs were previously missing, many from advanced mathematics. It performs significantly better than the static LLM baseline, proving challenging theorems in domains like abstract algebra and algebraic topology while showcasing a clear progression of learning from basic concepts to advanced topics. In addition, we analyze LeanAgent's superior performance on key lifelong learning metrics. LeanAgent achieves exceptional scores in stability and backward transfer, where learning new tasks improves performance on previously learned tasks. This emphasizes LeanAgent's continuous generalizability and improvement, explaining its superior theorem-proving performance.

The utility of synthetic data to enhance pretraining data quality and hence to improve downstream task accuracy has been widely explored in recent large language models (LLMs). Yet, these approaches fall inadequate in complex, multi-hop and mathematical reasoning tasks as the synthetic data typically fails to add complementary knowledge to the existing raw corpus. In this work, we propose a novel large-scale and diverse Math Informed syNthetic Dialogue (MIND) generation method that improves the mathematical reasoning ability of LLMs. Specifically, using MIND, we generate synthetic conversations based on OpenWebMath (OWM), resulting in a new math corpus, MIND-OWM. Our experiments with different conversational settings reveal that incorporating knowledge gaps between dialog participants is essential for generating high-quality math data. We further identify an effective way to format and integrate synthetic and raw data during pretraining to maximize the gain in mathematical reasoning, emphasizing the need to restructure raw data rather than use it as-is. Compared to pretraining just on raw data, a model pretrained on MIND-OWM shows significant boost in mathematical reasoning (GSM8K: +13.42%, MATH: +2.30%), including superior performance in specialized knowledge (MMLU: +4.55%, MMLU-STEM: +4.28%) and general purpose reasoning tasks (GENERAL REASONING: +2.51%).

Recent advances in language models have demonstrated their capability to solve mathematical reasoning problems, achieving near-perfect accuracy on grade-school level math benchmarks like GSM8K. In this paper, we formally study how language models solve these problems. We design a series of controlled experiments to address several fundamental questions: (1) Can language models truly develop reasoning skills, or do they simply memorize templates? (2) What is the model's hidden (mental) reasoning process? (3) Do models solve math questions using skills similar to or different from humans? (4) Do models trained on GSM8K-like datasets develop reasoning skills beyond those necessary for solving GSM8K problems? (5) What mental process causes models to make reasoning mistakes? (6) How large or deep must a model be to effectively solve GSM8K-level math questions? Our study uncovers many hidden mechanisms by which language models solve mathematical questions, providing insights that extend beyond current understandings of LLMs.

Self-improvement through post-training methods such as iterative preference learning has been acclaimed for enhancing the problem-solving capabilities (e.g., mathematical reasoning) of Large Language Models (LLMs) without human intervention. However, as our exploration deepens, it is crucial to critically assess whether these enhancements indeed signify comprehensive progress or if they could lead to unintended regressions. Through rigorous experimentation and analysis across diverse problem-solving tasks, we uncover nuances in the self-improvement trajectories of LLMs. Our study introduces the concept of \emph{self-improvement reversal}, where models showing improved overall accuracy metrics might paradoxically exhibit declines in broader, essential capabilities. We propose a comprehensive evaluative framework to scrutinize the underlying mechanisms and outcomes of post-training self-improvement, aiming to discern between superficial metric improvements and genuine enhancements in model functionality. The findings emphasize the complexity of technological advancements in LLMs, underscoring the need for a nuanced understanding of the \textit{progress or regress} dichotomy in their development.

In the absence of extensive human-annotated data for complex reasoning tasks, self-improvement -- where models are trained on their own outputs -- has emerged as a primary method for enhancing performance. Recently, the approach to self-improvement has shifted toward a more dynamic, online fashion through iterative training processes. However, the critical factors underlying the mechanism of these self-improving methods remain poorly understood, such as under what conditions self-improvement is effective, and what are the bottlenecks in the current iterations. In this work, we identify and propose methods to monitor two pivotal factors in this iterative process: (1) the model's ability to explore and generate high-quality responses among multiple candidates (exploration); and (2) the reliability of external rewards in selecting the best responses from the generated outputs (exploitation). These factors are inherently moving targets throughout the self-improvement cycles, yet their dynamics are rarely discussed in prior research -- It remains unclear what impedes continual model enhancement after only a few iterations. Using mathematical reasoning as a case study, we begin with a quantitative analysis to track the dynamics of exploration and exploitation, discovering that a model's exploratory capabilities rapidly deteriorate over iterations, and the effectiveness of exploiting external rewards diminishes as well due to shifts in distribution from the original policy. Motivated by these findings, we introduce B-STaR, a Self-Taught Reasoning framework that autonomously adjusts configurations across iterations to Balance exploration and exploitation, thereby optimizing the self-teaching effectiveness based on the current policy model and available rewards. Our experiments in mathematical reasoning demonstrate that B-STaR not only enhances the model's exploratory capabilities throughout training but also achieves a more effective balance between exploration and exploitation, leading to superior performance. Crucially, this work deconstructs the opaque nature of self-training algorithms, elucidating the interpretable dynamics throughout the process and highlighting current limitations for future research to address.

Do large language models (LLMs) solve reasoning tasks by learning robust generalizable algorithms, or do they memorize training data? To investigate this question, we use arithmetic reasoning as a representative task. Using causal analysis, we identify a subset of the model (a circuit) that explains most of the model's behavior for basic arithmetic logic and examine its functionality. By zooming in on the level of individual circuit neurons, we discover a sparse set of important neurons that implement simple heuristics. Each heuristic identifies a numerical input pattern and outputs corresponding answers. We hypothesize that the combination of these heuristic neurons is the mechanism used to produce correct arithmetic answers. To test this, we categorize each neuron into several heuristic types---such as neurons that activate when an operand falls within a certain range---and find that the unordered combination of these heuristic types is the mechanism that explains most of the model's accuracy on arithmetic prompts. Finally, we demonstrate that this mechanism appears as the main source of arithmetic accuracy early in training. Overall, our experimental results across several LLMs show that LLMs perform arithmetic using neither robust algorithms nor memorization; rather, they rely on a ``bag of heuristics''.

Multi-modal Large Language Models (MLLMs) have recently showcased superior proficiency in general visual scenarios. However, we identify their mathematical capabilities remain under-explored with three areas to be improved: visual encoding of math diagrams, diagram-language alignment, and chain-of-thought (CoT) reasoning. This draws forth an urgent demand for an effective training paradigm and a large-scale, comprehensive dataset with detailed CoT rationales, which is challenging to collect and costly to annotate manually. To tackle this issue, we propose MAVIS, a MAthematical VISual instruction tuning pipeline for MLLMs, featuring an automatic data engine to efficiently create mathematical visual datasets. We design the data generation process to be entirely independent of human intervention or GPT API usage, while ensuring the diagram-caption correspondence, question-answer correctness, and CoT reasoning quality. With this approach, we curate two datasets, MAVIS-Caption (558K diagram-caption pairs) and MAVIS-Instruct (834K visual math problems with CoT rationales), and propose four progressive stages for training MLLMs from scratch. First, we utilize MAVIS-Caption to fine-tune a math-specific vision encoder (CLIP-Math) through contrastive learning, tailored for improved diagram visual encoding. Second, we also leverage MAVIS-Caption to align the CLIP-Math with a large language model (LLM) by a projection layer, enhancing vision-language alignment in mathematical domains. Third, we adopt MAVIS-Instruct to perform the instruction tuning for robust problem-solving skills, and term the resulting model as MAVIS-7B. Fourth, we apply Direct Preference Optimization (DPO) to enhance the CoT capabilities of our model, further refining its step-wise reasoning performance. On various mathematical benchmarks, our MAVIS-7B achieves leading results among open-source MLLMs, e.g., surpassing other 7B models by +9.3% and the second-best LLaVA-NeXT (110B) by +6.9%, demonstrating the effectiveness of our method.

In-context learning (ICL), the remarkable ability to solve a task from only input exemplars, is often assumed to be a unique hallmark of Transformer models. By examining commonly employed synthetic ICL tasks, we demonstrate that multi-layer perceptrons (MLPs) can also learn in-context. Moreover, MLPs, and the closely related MLP-Mixer models, learn in-context comparably with Transformers under the same compute budget in this setting. We further show that MLPs outperform Transformers on a series of classical tasks from psychology designed to test relational reasoning, which are closely related to in-context classification. These results underscore a need for studying in-context learning beyond attention-based architectures, while also challenging prior arguments against MLPs' ability to solve relational tasks. Altogether, our results highlight the unexpected competence of MLPs in a synthetic setting, and support the growing interest in all-MLP alternatives to Transformer architectures. It remains unclear how MLPs perform against Transformers at scale on real-world tasks, and where a performance gap may originate. We encourage further exploration of these architectures in more complex settings to better understand the potential comparative advantage of attention-based schemes.