Title: Curing "Miracle Steps'' in LLM Math Reasoning with Rubric Rewards
Primary area: alignment, fairness, safety, privacy, and societal considerations
Keywords: faithful chain-of-thought, math reasoning, false positive, rubric

Abstract: 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.

TL;DR: This paper diagnoses how LLMs achieve correct math answers with flawed logic ("false positives") and introduces a "Rubric Reward Model" that rewards the entire problem-solving process to build more trustworthy and accurate reasoners.