Advanced applied mathematics problems are underrepresented in existing Large Language Model (LLM) benchmark datasets. To address this, we introduce $\textbf{HARDMath}$, a dataset inspired by a graduate course on asymptotic methods, featuring challenging applied mathematics problems that require analytical approximation techniques. These problems demand a combination of mathematical reasoning, computational tools, and subjective judgment, making them difficult for LLMs. Our framework auto-generates a large number of problems with solutions validated against numerical ground truths. We evaluate both open- and closed-source LLMs on $\textbf{HARDMath-mini}$, a sub-sampled test set of 366 problems, as well as on 40 word problems formulated in applied science contexts. Even leading closed-source models like GPT-4 achieve only 43.8% overall accuracy with few-shot Chain-of-Thought prompting, and all models demonstrate significantly lower performance compared to results on existing mathematics benchmark datasets. We additionally conduct a detailed error analysis to gain insights into the failure cases of LLMs. These results demonstrate the limitations of current LLM performance on advanced graduate-level applied math problems and underscore the importance of datasets like $\textbf{HARDMath}$ to advance mathematical abilities of LLMs.

Reinforcement Learning from Human Feedback (RLHF) is widely used to align models with human preferences, particularly to enhance the safety of responses generated by LLMs. This method traditionally relies on choosing preferred responses from response pairs. However, due to variations in human opinions and the difficulty of making an overall comparison of two responses, there is a growing shift towards a fine-grained annotation approach, assessing responses based on multiple specific metrics or rules. Selecting and applying these rules efficiently while accommodating the diversity of preference data remains a significant challenge. In this paper, we introduce a dynamic approach that adaptively selects the most critical rules for each pair of responses. We develop a mathematical framework that leverages the maximum discrepancy between each paired responses and theoretically show that this strategy optimizes the mutual information between the rule-based labeling and the hidden ground-truth preferences. We then train an 8B reward model using the adaptively labeled preference dataset and evaluate its performance on RewardBench. As of May 25, 2025, our model achieved the highest safety performance on the leaderboard, outperforming various larger models.