用非参数统计方法提升大模型推理中的梯度估计精度。

Recent advances in large language models (LLMs) have increasingly relied on reinforcement learning (RL) to improve their reasoning capabilities. Three types of approaches have been widely adopted: The first relies on a deep neural network to estimate the value function of the learning policy in order to reduce the variance of the policy gradient. However, estimating and maintaining such a value network incurs substantial computational and memory overhead. The second avoids training a value network by approximating the value function using sample averages. However, it samples a large number of reasoning traces per prompt for accurate value function approximation, making it computationally expensive. The third samples only a single reasoning trajectory per prompt, which reduces computational cost but suffers from poor sample efficiency. This paper focuses on a practical, resource-constrained setting in which only a small number of reasoning traces can be sampled per prompt, while low-variance gradient estimation remains essential for high-quality policy learning. To address this challenge, we bring classical nonparametric statistical methods, which are both computationally and statistically efficient, to LLM reasoning. We employ kernel smoothing as a concrete example for value function estimation and the subsequent policy optimization. Numerical and theoretical results demonstrate that our proposal achieves accurate value and gradient estimation, leading to improved policy optimization.