Reasoning over mathematical objects: on-policy reward modeling and test time aggregation
This work addresses the challenge of automated assessment for mathematical reasoning in STEM fields, offering incremental improvements through enhanced training methods.
The paper tackled the problem of improving language models' ability to derive mathematical objects for STEM applications, and found that their training recipes, including on-policy judge training and test-time aggregation, significantly boosted performance on the Principia benchmarks while also improving results on existing numerical and multiple-choice tasks.
The ability to precisely derive mathematical objects is a core requirement for downstream STEM applications, including mathematics, physics, and chemistry, where reasoning must culminate in formally structured expressions. Yet, current LM evaluations of mathematical and scientific reasoning rely heavily on simplified answer formats such as numerical values or multiple choice options due to the convenience of automated assessment. In this paper we provide three contributions for improving reasoning over mathematical objects: (i) we build and release training data and benchmarks for deriving mathematical objects, the Principia suite; (ii) we provide training recipes with strong LLM-judges and verifiers, where we show that on-policy judge training boosts performance; (iii) we show how on-policy training can also be used to scale test-time compute via aggregation. We find that strong LMs such as Qwen3-235B and o3 struggle on Principia, while our training recipes can bring significant improvements over different LLM backbones, while simultaneously improving results on existing numerical and MCQA tasks, demonstrating cross-format generalization of reasoning abilities.