Hanze Dong

Reinforcement learning for large language model reasoning often suffers from signal collapse: under uniform small-group sampling, difficult prompts frequently produce uninformative all-fail or all-success groups, causing useful gradients to disappear. We show that this failure is largely a statistical artifact of undersampling rather than an inherent limitation of the model. This suggests that sampling budget should be allocated adaptively across prompts according to their pass rates. We formalize this idea through a family of non-linear RL objectives, which induce difficulty-aware weighting and recover prior log-objective analyses as a special case. Based on this view, we propose Reinforce-Ada, with an estimation-based variant and a model-free sequential variant that adaptively allocate rollouts to recover informative signals. Across multiple models and math reasoning benchmarks, Reinforce-Ada consistently improves over standard GRPO baselines, recovering much of the benefit of large-group training at substantially lower cost.

Mathematical reasoning is widely regarded as a hallmark of human intelligence, and determining whether large language models (LLMs) can meaningfully perform it remains a central question in artificial intelligence and cognitive science. As LLMs are increasingly integrated into scientific workflows, rigorous evaluation of their mathematical capabilities becomes a practical necessity. Existing mathematical reasoning benchmarks, however, are often limited by synthetic problem settings and growing contamination from widely circulated datasets. We present LiveMathematicianBench, a dynamic multiple-choice benchmark for evaluating research-level mathematical reasoning using recent arXiv papers published after model training cutoffs. By grounding evaluation in newly published theorem statements, the benchmark provides a more realistic testbed for assessing whether models can reason about natural mathematical claims beyond memorized benchmark patterns. LiveMathematicianBench introduces a taxonomy of thirteen problem categories based on the logical form of theorem statements, enabling fine-grained evaluation across reasoning types such as implication, equivalence, existence, and uniqueness. It further introduces a proof-sketch-guided distractor generation pipeline, in which proof sketches are used to construct plausible but invalid answer choices that reflect misleading proof directions. This design makes the benchmark more sensitive to genuine mathematical understanding rather than surface-level answer matching. We additionally introduce a substitution-resistant evaluation mechanism designed to distinguish answer recognition from substantive mathematical reasoning. Evaluation of frontier models shows that the benchmark is far from saturated: the best-performing model, Gemini-3.1-pro-preview, achieves only 43.5\% accuracy in the standard setting. Under substitution-resistant evaluation, accuracy drops sharply across all models: GPT-5.4 achieves the highest score at 30.6\%, while Gemini-3.1-pro-preview falls to 17.6\%, below the 20\% random baseline. A dual-mode protocol comparing performance with and without proof-sketch access reveals that sketches yield consistent gains (e.g., +13.6 pp for Gemini-3.1-pro-preview), suggesting that models can leverage high-level proof strategies to improve reasoning. Overall, LiveMathematicianBench offers a scalable and contamination-resistant testbed for studying research-level mathematical reasoning in large language models.

Inference-time scaling techniques have significantly bolstered the reasoning capabilities of large language models (LLMs) by harnessing additional computational effort at inference without retraining. Similarly, Chain-of-Thought (CoT) prompting and its extension, Long CoT, improve accuracy by generating rich intermediate reasoning trajectories, but these approaches incur substantial token costs that impede their deployment in latency-sensitive settings. In this work, we first show that truncated CoT, which stops reasoning before completion and directly generates the final answer, often matches full CoT sampling while using dramatically fewer tokens. Building on this insight, we introduce Fractured Sampling, a unified inference-time strategy that interpolates between full CoT and solution-only sampling along three orthogonal axes: (1) the number of reasoning trajectories, (2) the number of final solutions per trajectory, and (3) the depth at which reasoning traces are truncated. Through extensive experiments on five diverse reasoning benchmarks and several model scales, we demonstrate that Fractured Sampling consistently achieves superior accuracy-cost trade-offs, yielding steep log-linear scaling gains in Pass@k versus token budget. Our analysis reveals how to allocate computation across these dimensions to maximize performance, paving the way for more efficient and scalable LLM reasoning.

Large reasoning models (LRMs) already possess a latent capacity for long chain-of-thought reasoning. Prior work has shown that outcome-based reinforcement learning (RL) can incidentally elicit advanced reasoning behaviors such as self-correction, backtracking, and verification–phenomena often ref...