Baselines and Score Centering in Policy Gradient Estimation

Reinforcement learning has become a widely used approach to post-training large language models. In a typical implementation, learning is off-policy: The policy generating the responses differs from the policy being trained because of training-inference mismatch or policy staleness in asynchronous training. This discrepancy is commonly addressed by importance sampling, whose ratio is usually regularized in practice through truncation, masking, or clipping. Under regularized importance sampling, the importance-weighted policy gradient is no longer invariant to constant shifts in the reward. This invariance is recovered by using the covariance between the reward and the importance-weighted score as the estimand, which coincides with the policy gradient when the ratio is not regularized. This note develops this covariance perspective for both sequence-level and token-level importance sampling. The covariance perspective reveals the roles of the group-mean baseline and score centering in policy gradient estimation. Both perform centering in covariance estimation, but they differ in the mean being subtracted: The group-mean baseline uses a sample mean, whereas score centering uses a population mean. The two lead to distinct unbiased covariance estimators with different variances. Whether both forms of centering should be used simultaneously depends on whether importance sampling is applied at the sequence level or at the token level. With sequence-level importance sampling, score centering adds no benefit once the group-mean baseline is used; with token-level importance sampling, using the group-mean baseline without score centering generally introduces bias when the importance ratio is regularized.