Geometric-Mean Policy Optimization

Group Relative Policy Optimization (GRPO) has significantly enhanced the reasoning capability of large language models by optimizing the arithmetic mean of token-level rewards. Unfortunately, GRPO is observed to suffer from unstable policy updates when facing tokens with outlier importance-weighted rewards, which manifest as extreme importance sampling ratios during training. In this study, we propose Geometric-Mean Policy Optimization (GMPO), with the aim to improve the stability of GRPO through suppressing token reward outliers. GMPO is plug-and-play-simply replacing GRPO's arithmetic mean with the geometric mean of token-level rewards, as the latter is inherently less sensitive to outliers. GMPO is theoretically plausible-analysis reveals that both GMPO and GRPO are weighted forms of the policy gradient while the former enjoys more stable weights, which consequently benefits policy optimization and performance. Experiments on multiple mathematical reasoning benchmarks show that GMPO-7B improves the average Pass@1 of GRPO by up to 4.1%, outperforming many state-of-the-art approaches. Code is available at https://github.com/callsys/GMPO . Figure 1 : Comparison between GRPO and our GMPO. GRPO optimizes the arithmetic mean of token-level rewards while GMPO the geometric mean (left). When training with GRPO, the important sample ratio ρ t (θ) = π θ (ot|q,o<t) π θ old (ot|q,o<t) frequently reaches extreme values, leading to unstable policy updates. In contrast, GMPO enjoys more stable important sample ratio with fewer outliers (right).