Shrinking the Variance: Shrinkage Baselines for Reinforcement Learning with Verifiable Rewards

Abstract

Reinforcement Learning with Verifiable Rewards (RLVR) has emerged as a powerful paradigm for post-training large reasoning models (LRMs) using policy-gradient methods such as GRPO. To stabilize training, these methods typically center trajectory rewards by subtracting the empirical mean reward for each prompt. Statistically, this centering acts as a control variate (baseline), reducing the variance of the policy-gradient estimator. In practice, the mean reward is estimated using per-prompt empirical averages computed from the generations for each prompt in a batch. Motivated by Stein's paradox, we propose shrinkage estimators that combine per-prompt and across-prompt means to improve per-prompt mean estimation accuracy, especially in the low-generation regime typical of RLVR. Theoretically, we construct a shrinkage-based baseline that provably yields lower-variance policy-gradient estimators across algorithms. Our baseline is a drop-in replacement for standard per-prompt mean baselines and requires no additional hyperparameters or computation. Empirically, shrinkage baselines consistently outperform empirical-mean baselines, producing lower-variance gradient updates and improved training stability.