On-policy rollout methods such as GRPO are central to post-training of large language models, yet they frequently suffer from training instabilities, cross-task capability degradation, and response-length inflation. Although prior work has characterized the subspace geometry of aggregate updates, the stepwise variation of this geometry and its relationship to model performance remain unclear. We introduce Principal-Subspace Overlap, a dimension-corrected measure of individual rollout updates relative to the dominant singular subspaces of pretrained weights. Despite low average overlap, transient spikes often precede performance degradation. To address this, we propose GCPO (Geometrically Constrained Policy Optimization), which applies hard bilateral orthogonal projections to constrain updates to the complementary subspaces, preventing such excursions by construction. Across mathematical reasoning, code generation, and tool-use tasks on Qwen3-8B and GLM4-9B, GCPO consistently outperforms GRPO and recent variants, including DAPO and GSPO, improving over the base models and the strongest baseline by up to 27.69 and 2.37 points, respectively. Furthermore, GCPO preserves general capabilities, eliminates response-length inflation, and stabilizes policy entropy. Our findings provide a new diagnostic lens and a principled design perspective for stable reinforcement learning post-training.
Group Relative Policy Optimization (GRPO) has been a key driver of recent progress in reinforcement learning with verifiable rewards (RLVR) for large language models, but it is typically trained in a low-staleness, near-on-policy regime that incurs substantial system overhead. We ask a simple question: How off-policy can GRPO be? We show that GRPO-style algorithms can tolerate substantially larger rollout staleness than previously assumed, and propose Mu-GRPO, an RL training framework that organizes training into a small number (e.g., four) of large sequential generation-optimization stages. This design induces high rollout staleness while greatly reducing rollout-optimization switching overhead. To stabilize learning under stale data, Mu-GRPO combines relaxed clipping, which preserves useful stale-rollout gradients, with negative-advantage veto, which removes destabilizing post-trigger suffix updates in negative-advantage responses. Across five language models and multiple math reasoning benchmarks, Mu-GRPO matches or exceeds the performance of standard GRPO while achieving around 2x speedup in wall-clock training time, establishing a substantially improved performance-efficiency trade-off for LLM reinforcement learning.
Reinforcement learning from verifiable rewards with GRPO is a standard approach for post-training reasoning LLMs. It remains sample inefficient. Each rollout is used for a single gradient update and then discarded. Naive replay is not well suited in this setting because LLM policies drift quickly per gradient step. Stored rollouts therefore become stale and can destabilize training. We propose a rollout-level replay buffer for GRPO that stores and samples individual rollouts rather than whole groups. The buffer bounds staleness through age eviction. Any rollout older than tau_max training steps is removed. The buffer also preserves on-policy data via fresh-anchored composition. Each batch keeps its fresh on-policy rollouts and then concatenates replay rollouts drawn separately from the buffer. We prioritize replay by per-rollout advantage magnitude and recycle individual rollouts whose advantages are large. Across three Qwen3-Base scales on five math benchmarks, our method outperforms GRPO and naive replay baselines. Gains are positive at every scale and reach +1.66 pp on the five-benchmark average at 4B. Under an AES metric that jointly measures accuracy and token efficiency, our method is the only condition with a positive margin over GRPO at every scale.
Reinforcement Learning with Verifiable Rewards (RLVR) has emerged as a promising approach to improve the reasoning abilities of Large Language Models (LLMs). Among RLVR algorithms, Group Relative Policy Optimization (GRPO) and its variants have demonstrated strong performance and high training efficiency. However, GRPO-style objectives exhibit two issues on high accuracy prompts including mastered prompts (rollout accuracy =1) and majority-correct prompts (rollout accuracy in (0.5,1)). For mastered prompts, group-relative advantages vanish, yielding no training signal and unconstrained policy drift that can cause forgetting. For majority-correct prompts, the induced query weight shrinks as accuracy increases, weakening consolidation from partial correctness to mastery. To alleviate this, we propose Mastery-Consolidated Policy Optimization (MCPO), which introduces (i) a hinge-KL regularizer applied exclusively to mastered prompts to bound harmful policy drift between successive gradient steps, and (ii) a weighting mechanism that prioritizes majority-correct prompts to better allocate optimization effort. Extensive experiments across three mathematical benchmarks demonstrate that MCPO consistently improves pass@1 performance. Counter-intuitively, rather than restricting exploration, MCPO boosts pass@k metrics, indicating that mastery consolidation further catalyzes solution diversity.