LLM Mathematical Reasoning

While Large language models (LLMs) have proved able to address some complex reasoning tasks, we also know that they are highly sensitive to input variation, which can lead to different solution paths and final answers. Answer consistency across input variations can thus be taken as a sign of stronger confidence. Leveraging this aspect, we introduce a framework where models are systematically pushed to diversify solution paths towards a final answer, thereby testing them for answer consistency across multiple input variations. Focusing on math problems, we induce variations in (i) order of shots in prompt, (ii) problem phrasing, and (iii) languages used. Experiments on a wide range of open-source LLMs of various sizes show that reasoning consistency differs by variation dimension, and that by aggregating consistency across dimensions, our framework enhances mathematical reasoning performance on monolingual datasets GSM8K and MATH500, and the multilingual dataset MGSM. We also discuss the different efficacy of the types of variations induced and how they could be further leveraged in future work.

Large language models have demonstrated remarkable progress in mathematical reasoning, leveraging chain-of-thought and reinforcement learning. However, many open questions remain regarding the interplay between reasoning token usage and accuracy gains. In particular, when comparing models across generations, it is unclear whether improved performance results from longer reasoning chains or more efficient reasoning. We systematically analyze reasoning chain length across o1-mini and o3-mini variants on the Omni-MATH benchmark, finding that o3-mini (m) achieves superior accuracy without requiring longer reasoning chains than o1-mini. Moreover, we show that accuracy generally declines as reasoning chains grow across all models and compute settings, even when controlling for difficulty of the questions. This accuracy drop is significantly smaller in more proficient models, suggesting that new generations of reasoning models use test-time compute more effectively. Finally, we highlight that while o3-mini (h) achieves a marginal accuracy gain over o3-mini (m), it does so by allocating substantially more reasoning tokens across all problems, even the ones that o3-mini (m) can already solve. These findings provide new insights into the relationship between model capability and reasoning length, with implications for efficiency, scaling, and evaluation methodologies.

Recent studies have raised significant concerns regarding the reliability of current mathematics benchmarks, highlighting issues such as simplistic design and potential data contamination. Consequently, developing a reliable benchmark that effectively evaluates large language models' (LLMs) genuine capabilities in mathematical reasoning remains a critical challenge. To address these concerns, we propose RV-Bench, a novel evaluation methodology for Benchmarking LLMs with Random Variables in mathematical reasoning. Specifically, we build question-generating functions to produce random variable questions (RVQs), whose background content mirrors original benchmark problems, but with randomized variable combinations, rendering them "unseen" to LLMs. Models must completely understand the inherent question pattern to correctly answer RVQs with diverse variable combinations. Thus, an LLM's genuine reasoning capability is reflected through its accuracy and robustness on RV-Bench. We conducted extensive experiments on over 30 representative LLMs across more than 1,000 RVQs. Our findings propose that LLMs exhibit a proficiency imbalance between encountered and ``unseen'' data distributions. Furthermore, RV-Bench reveals that proficiency generalization across similar mathematical reasoning tasks is limited, but we verified it can still be effectively elicited through test-time scaling.