A Gradient Perspective on RLVR Stability and Winner Advantage Policy Optimization
Reinforcement learning with verifiable rewards (RLVR) improves language-model reasoning, but GRPO-style optimization can be unstable. We analyze this instability through token-level gradient dynamics and derive a taxonomy linking updates to next-token probabilities and entropy. This analysis shows that stability depends jointly on advantage sign and the current token distribution. Based on this insight, we propose Winner Advantage Policy Optimization (WAPO), a simple clipped policy-gradient objective that updates only on positive-advantage completions. Across mathematical reasoning and multi-hop QA benchmarks, WAPO improves stability while matching or outperforming baselines across multiple model families.