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.

Retromorphic Testing: A New Approach to the Test Oracle Problem

A test oracle serves as a criterion or mechanism to assess the correspondence between software output and the anticipated behavior for a given input set. In automated testing, black-box techniques, known for their non-intrusive nature in test oracle construction, are widely used, including notable methodologies like differential testing and metamorphic testing. Inspired by the mathematical concept of inverse function, we present Retromorphic Testing, a novel black-box testing methodology. It leverages an auxiliary program in conjunction with the program under test, which establishes a dual-program structure consisting of a forward program and a backward program. The input data is first processed by the forward program and then its program output is reversed to its original input format using the backward program. In particular, the auxiliary program can operate as either the forward or backward program, leading to different testing modes. The process concludes by examining the relationship between the initial input and the transformed output within the input domain. For example, to test the implementation of the sine function $\sin(x)$, we can employ its inverse function, $\arcsin(x)$, and validate the equation $x = \sin(\arcsin(x)+2k\pi), \forall k \in \mathbb{Z}$. In addition to the high-level concept of Retromorphic Testing, this paper presents its three testing modes with illustrative use cases across diverse programs, including algorithms, traditional software, and AI applications.

Automated Testing and Improvement of Named Entity Recognition Systems

Named entity recognition (NER) systems have seen rapid progress in recent years due to the development of deep neural networks. These systems are widely used in various natural language processing applications, such as information extraction, question answering, and sentiment analysis. However, the complexity and intractability of deep neural networks can make NER systems unreliable in certain circumstances, resulting in incorrect predictions. For example, NER systems may misidentify female names as chemicals or fail to recognize the names of minority groups, leading to user dissatisfaction. To tackle this problem, we introduce TIN, a novel, widely applicable approach for automatically testing and repairing various NER systems. The key idea for automated testing is that the NER predictions of the same named entities under similar contexts should be identical. The core idea for automated repairing is that similar named entities should have the same NER prediction under the same context. We use TIN to test two SOTA NER models and two commercial NER APIs, i.e., Azure NER and AWS NER. We manually verify 784 of the suspicious issues reported by TIN and find that 702 are erroneous issues, leading to high precision (85.0%-93.4%) across four categories of NER errors: omission, over-labeling, incorrect category, and range error. For automated repairing, TIN achieves a high error reduction rate (26.8%-50.6%) over the four systems under test, which successfully repairs 1,056 out of the 1,877 reported NER errors.