# Improving Value-based Process Verifier via Low-Cost Variance Reduction

arXiv: https://arxiv.org/abs/2508.10539 · HTML: https://mlsanity.com/paper/2508.10539/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 18 references that are in the map.
- First posted on arXiv 14 Aug 2025.

## Builds on (most cited first, 13 of 13)

- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [Let's Verify Step by Step](https://mlsanity.com/paper/2305.20050/) (2023)
- [Inner Monologue: Embodied Reasoning through Planning with Language Models](https://mlsanity.com/paper/2207.05608/) (2022)
- [Math-Shepherd: Verify and Reinforce LLMs Step-by-step without Human Annotations](https://mlsanity.com/paper/2312.08935/) (2023)
- [Solving math word problems with process- and outcome-based feedback](https://mlsanity.com/paper/2211.14275/) (2022)
- [Generative Verifiers: Reward Modeling as Next-Token Prediction](https://mlsanity.com/paper/2408.15240/) (2024)
- [Improve Mathematical Reasoning in Language Models by Automated Process Supervision](https://mlsanity.com/paper/2406.06592/) (2024)
- [The Lessons of Developing Process Reward Models in Mathematical Reasoning](https://mlsanity.com/paper/2501.07301/) (2025)
- [ProcessBench: Identifying Process Errors in Mathematical Reasoning](https://mlsanity.com/paper/2412.06559/) (2024)
- [Making Large Language Models Better Reasoners with Step-Aware Verifier](https://mlsanity.com/paper/2206.02336/) (2022)
- [Teaching Algorithmic Reasoning via In-context Learning](https://mlsanity.com/paper/2211.09066/) (2022)
- [Making Large Language Models Better Reasoners with Alignment](https://mlsanity.com/paper/2309.02144/) (2023)

## Abstract (arXiv)

Large language models (LLMs) have achieved remarkable success in a wide range of tasks. However, their reasoning capabilities, particularly in complex domains like mathematics, remain a significant challenge. Value-based process verifiers, which estimate the probability of a partial reasoning chain leading to a correct solution, are a promising approach for improving reasoning. Nevertheless, their effectiveness is often hindered by estimation error in their training annotations, a consequence of the limited number of Monte Carlo (MC) samples feasible due to the high cost of LLM inference. In this paper, we identify that the estimation error primarily arises from high variance rather than bias, and the MC estimator is a Minimum Variance Unbiased Estimator (MVUE). To address the problem, we propose the textscCompound textscMonte textscCarlo textscSampling (ComMCS) method, which constructs an unbiased estimator by linearly combining the MC estimators from the current and subsequent steps. Theoretically, we show that our method leads to a predictable reduction in variance, while maintaining an unbiased estimation without additional LLM inference cost. We also perform empirical experiments on the MATH-500 and GSM8K benchmarks to demonstrate the effectiveness of our method. Notably, ComMCS outperforms regression-based optimization method by 2.8 points, the non-variance-reduced baseline by 2.2 points on MATH-500 on Best-of-32 sampling experiment.