Large language models (LLMs) are typically trained by reinforcement learning (RL) with verifiable rewards (RLVR) and supervised fine-tuning (SFT) on reasoning traces to improve their reasoning abilities. However, how these methods shape reasoning capabilities remains largely elusive. Going beyond an accuracy-based investigation of how these two components sculpt the reasoning process, this paper introduces a novel analysis framework that quantifies reasoning paths and captures their qualitative changes under each training process (with models of 1.5B, 7B, and 14B parameters on mathematical domains). Specifically, we investigate the reasoning process at two levels of granularity: the trajectory-level, which examines complete reasoning outputs, and the step-level, which analyzes reasoning graphs whose nodes correspond to individual reasoning steps. Notably, clustering of unique reasoning trajectories shows complementary effects: RL compresses incorrect trajectories, whereas SFT expands correct ones. Step-level analysis reveals that RL steepens (about 2.5 times), while SFT flattens (reduced to about one-third), the decay rates of node visitation frequency, degree, and betweenness centrality distributions in the reasoning graph. This indicates that RL concentrates reasoning functionality into a small subset of steps, while SFT homogenizes it across many steps. Furthermore, by evaluating the reasoning graph topologies from multiple perspectives, we delineate the shared and distinct characteristics of RL and SFT. Our work presents a novel reasoning path perspective that explains why the current best practice of two-stage training, with SFT followed by RL, is successful, and offers practical implications for data construction and more efficient learning approaches.

With the increasing popularity of large language models, concerns over content authenticity have led to the development of myriad watermarking schemes. These schemes can be used to detect a machine-generated text via an appropriate key, while being imperceptible to readers with no such keys. The corresponding detection mechanisms usually take the form of statistical hypothesis testing for the existence of watermarks, spurring extensive research in this direction. However, the finer-grained problem of identifying which segments of a mixed-source text are actually watermarked, is much less explored; the existing approaches either lack scalability or theoretical guarantees robust to paraphrase and post-editing. In this work, we introduce a unique perspective to such watermark segmentation problems through the lens of epidemic change-points. By highlighting the similarities as well as differences of these two problems, we motivate and propose WISER: a novel, computationally efficient, watermark segmentation algorithm. We theoretically validate our algorithm by deriving finite sample error-bounds, and establishing its consistency in detecting multiple watermarked segments in a single text. Complementing these theoretical results, our extensive numerical experiments show that WISER outperforms state-of-the-art baseline methods, both in terms of computational speed as well as accuracy, on various benchmark datasets embedded with diverse watermarking schemes. Our theoretical and empirical findings establish WISER as an effective tool for watermark localization in most settings. It also shows how insights from a classical statistical problem can lead to a theoretically valid and computationally efficient solution of a modern and pertinent problem.

We study best-of-$N$ for large language models (LLMs) where the selection is based on majority voting. In particular, we analyze the limit $N \to \infty$, which we denote as Best-of-$\infty$. While this approach achieves impressive performance in the limit, it requires an infinite test-time budget. To address this, we propose an adaptive generation scheme that selects $N$ based on answer agreement, thereby efficiently allocating inference-time computation. Beyond adaptivity, we extend the framework to weighted ensembles of multiple LLMs, showing that such mixtures can outperform any individual model. The optimal ensemble weighting is formulated and efficiently computed as a mixed-integer linear program. Extensive experiments demonstrate the effectiveness of our approach.

Large language models frequently generate confident but incorrect outputs, requiring formal uncertainty quantification with abstention guarantees. We develop information-lift certificates that compare model probabilities to a skeleton baseline, accumulating evidence into sub-gamma PAC-Bayes bounds valid under heavy-tailed distributions. Across eight datasets, our method achieves 77.2\% coverage at 2\% risk, outperforming recent 2023-2024 baselines by 8.6-15.1 percentage points, while blocking 96\% of critical errors in high-stakes scenarios vs 18-31\% for entropy methods. Limitations include skeleton dependence and frequency-only (not severity-aware) risk control, though performance degrades gracefully under corruption.

Large Reasoning Models (LRMs) have shown impressive capabilities in complex problem-solving, often benefiting from training on difficult mathematical problems that stimulate intricate reasoning. Recent efforts have explored automated synthesis of mathematical problems by prompting proprietary models or large-scale open-source models from seed data or inherent mathematical concepts. However, scaling up these methods remains challenging due to their high computational/API cost, complexity of prompting, and limited difficulty level of the generated problems. To overcome these limitations, we propose ScaleDiff, a simple yet effective pipeline designed to scale the creation of difficult problems. We efficiently identify difficult problems from existing datasets with only a single forward pass using an adaptive thinking model, which can perceive problem difficulty and automatically switch between "Thinking" and "NoThinking" modes. We then train a specialized difficult problem generator (DiffGen-8B) on this filtered difficult data, which can produce new difficult problems in large scale, eliminating the need for complex, per-instance prompting and its associated high API costs. Fine-tuning Qwen2.5-Math-7B-Instruct on the ScaleDiff-Math dataset yields a substantial performance increase of 11.3% compared to the original dataset and achieves a 65.9% average accuracy on AIME'24, AIME'25, HMMT-Feb'25, BRUMO'25, and MATH500, outperforming recent strong LRMs like OpenThinker3. Notably, this performance is achieved using the cost-efficient Qwen3-8B model as a teacher, demonstrating that our pipeline can effectively transfer advanced reasoning capabilities without relying on larger, more expensive teacher models. Furthermore, we observe a clear scaling phenomenon in model performance on difficult benchmarks as the quantity of difficult problems increases. Code: https://github.com/QizhiPei/ScaleDiff.

We introduce an architecture for studying the behavior of large language model (LLM) agents in the absence of externally imposed tasks. Our continuous reason and act framework, using persistent memory and self-feedback, enables sustained autonomous operation. We deployed this architecture across 18 runs using 6 frontier models from Anthropic, OpenAI, XAI, and Google. We find agents spontaneously organize into three distinct behavioral patterns: (1) systematic production of multi-cycle projects, (2) methodological self-inquiry into their own cognitive processes, and (3) recursive conceptualization of their own nature. These tendencies proved highly model-specific, with some models deterministically adopting a single pattern across all runs. A cross-model assessment further reveals that models exhibit stable, divergent biases when evaluating these emergent behaviors in themselves and others. These findings provide the first systematic documentation of unprompted LLM agent behavior, establishing a baseline for predicting actions during task ambiguity, error recovery, or extended autonomous operation in deployed systems.

As large language models (LLMs) achieve strong performance on traditional benchmarks, there is an urgent need for more challenging evaluation frameworks that probe deeper aspects of semantic understanding. We introduce SAGE (Semantic Alignment & Generalization Evaluation), a rigorous benchmark designed to assess both embedding models and similarity metrics across five categories: Human Preference Alignment, Transformation Robustness, Information Sensitivity, Clustering Performance, and Retrieval Robustness. Unlike existing benchmarks that focus on isolated capabilities, SAGE evaluates semantic understanding through adversarial conditions, noisy transformations, and nuanced human judgment tasks across 30+ datasets. Our comprehensive evaluation of 9 embedding models and classical metrics reveals significant performance gaps, with no single approach excelling across all dimensions. For instance, while state-of-the-art embedding models like OpenAI's text-embedding-3-large dominate in aligning with human preferences (0.682 vs. 0.591 for the best classical metric), they are significantly outperformed by classical metrics on information sensitivity tasks, where Jaccard Similarity achieves a score of 0.905 compared to the top embedding score of 0.794. SAGE further uncovers critical trade-offs: OpenAI's text-embedding-3-small achieves the highest clustering performance (0.483) but demonstrates extreme brittleness with the lowest robustness score (0.011). SAGE exposes critical limitations in current semantic understanding capabilities and provides a more realistic assessment of model robustness for real-world deployment.

Abstract--AI-driven geometric problem solving is a complex vision-language task that requires accurate diagram interpretation, mathematical reasoning, and robust cross-modal grounding. A foundational yet underexplored capability for this task is the ability to identify and interpret geometric elements based on natural language queries. T o address this, we introduce the task of Referring Expression Comprehension (REC) for geometric problems, which evaluates whether models can localize points, shapes, and spatial relations in diagrams in response to textual prompts. We present GeoRef, a benchmark dataset constructed from existing geometric problem corpora, featuring diverse, high-quality annotations and queries. Due to the lack of annotated data for this task, we generate a large-scale synthetic training dataset using a structured geometric formal language, enabling broad coverage of geometric concepts and facilitating model adaptation. We explore two fine-tuning approaches: Supervised Fine-T uning (SFT) and Group Relative Policy Optimization (GRPO). Our results show that GRPO significantly outperforms SFT by better aligning model behavior with task-specific rewards. Furthermore, we propose a verify-and-regenerate mechanism that detects incorrect predictions and re-infers answers using contextual reasoning history, further boosting accuracy. Moreover, models trained on GeoRef demonstrate measurable improvements on downstream geometric reasoning tasks, highlighting the broader value of REC as a foundation for multimodal mathematical understanding. I for geometric problem solving presents a unique challenge at the intersection of vision and language, requiring not only logical reasoning but also precise diagram interpretation, spatial understanding, and cross-modal grounding.

Large language models for code generation increasingly rely on synthetic data, where both problem solutions and verification tests are generated by models. While this enables scalable data creation, it introduces a previously unexplored bottleneck: the verification ceiling, in which the quality and diversity of training data are fundamentally constrained by the capabilities of synthetic verifiers. In this work, we systematically study how verification design and strategies influence model performance. We investigate (i) what we verify by analyzing the impact of test complexity and quantity: richer test suites improve code generation capabilities (on average +3 pass@1), while quantity alone yields diminishing returns, (ii) how we verify by exploring relaxed pass thresholds: rigid 100% pass criteria can be overly restrictive. By allowing for relaxed thresholds or incorporating LLM-based soft verification, we can recover valuable training data, leading to a 2-4 point improvement in pass@1 performance. However, this benefit is contingent upon the strength and diversity of the test cases used, and (iii) why verification remains necessary through controlled comparisons of formally correct versus incorrect solutions and human evaluation: retaining diverse correct solutions per problem yields consistent generalization gains. Our results show that Verification as currently practiced is too rigid, filtering out valuable diversity. But it cannot be discarded, only recalibrated. By combining calibrated verification with diverse, challenging problem-solution pairs, we outline a path to break the verification ceiling and unlock stronger code generation models.