Curing Miracle Steps in LLM Mathematical Reasoning with Rubric Rewards

Large language models for mathematical reasoning are typically trained with outcome-based rewards, which credit only the final answer. In our experiments, we observe that this paradigm is highly susceptible to reward hacking, leading to a substantial overestimation of a model's reasoning ability. This is evidenced by a high incidence of false positives - solutions that reach the correct final answer through an unsound reasoning process. Through a systematic analysis with human verification, we establish a taxonomy of these failure modes, identifying patterns like Miracle Steps - abrupt jumps to a correct output without a valid preceding derivation. Probing experiments suggest a strong association between these Miracle Steps and memorization, where the model appears to recall the answer directly rather than deriving it. To mitigate this systemic issue, we introduce the Rubric Reward Model (RRM), a process-oriented reward function that evaluates the entire reasoning trajectory against problem-specific rubrics. The generative RRM provides fine-grained, calibrated rewards (0-1) that explicitly penalize logical flaws and encourage rigorous deduction. When integrated into a reinforcement learning pipeline, RRM-based training consistently outperforms outcome-only supervision across four math benchmarks. Notably, it boosts Verified Pass@1024 on AIME2024 from 26.7% to 62.6% and reduces the incidence of Miracle Steps by 71%. Our work demonstrates that rewarding the solution process is crucial for building models that are not only more accurate but also more reliable.

Aligning LLMs for FL-free Program Repair

Large language models (LLMs) have achieved decent results on automated program repair (APR). However, the next token prediction training objective of decoder-only LLMs (e.g., GPT-4) is misaligned with the masked span prediction objective of current infilling-style methods, which impedes LLMs from fully leveraging pre-trained knowledge for program repair. In addition, while some LLMs are capable of locating and repairing bugs end-to-end when using the related artifacts (e.g., test cases) as input, existing methods regard them as separate tasks and ask LLMs to generate patches at fixed locations. This restriction hinders LLMs from exploring potential patches beyond the given locations. In this paper, we investigate a new approach to adapt LLMs to program repair. Our core insight is that LLM's APR capability can be greatly improved by simply aligning the output to their training objective and allowing them to refine the whole program without first performing fault localization. Based on this insight, we designed D4C, a straightforward prompting framework for APR. D4C can repair 180 bugs correctly in Defects4J, with each patch being sampled only 10 times. This surpasses the SOTA APR methods with perfect fault localization by 10% and reduces the patch sampling number by 90%. Our findings reveal that (1) objective alignment is crucial for fully exploiting LLM's pre-trained capability, and (2) replacing the traditional localize-then-repair workflow with direct debugging is more effective for LLM-based APR methods. Thus, we believe this paper introduces a new mindset for harnessing LLMs in APR.