When Does Momentum Help in Reinforcement Learning with Verifiable Rewards?

Reinforcement learning with verifiable rewards (RLVR) has emerged as an effective post-training paradigm for improving the reasoning capabilities of large language models. However, existing group-based RLVR methods often suffer from severe sample inefficiency. This inefficiency stems from reliance on point estimation of rewards from a small number of rollouts, leading to high estimation variance, variance collapse, and ineffective utilization of generated responses. In this work, we reformulate RLVR from a statistical estimation perspective by modeling rewards as samples drawn from a policy-induced distribution and casting advantage computation as the problem of estimating the reward distribution from finite data. Building on this view, we propose Discounted Beta-Bernoulli (DBB) reward estimation, which leverages historical reward statistics for the non-stationary distribution. Although biased, the resulting estimator exhibits reduced and stable variance, theoretically avoids estimated variance collapse, and achieves lower mean squared error than standard point estimation. Extensive experiments across six in-distribution and three out-of-distribution reasoning benchmarks demonstrate that GRPO with DBB consistently outperforms naive GRPO, achieving average Acc@8 improvements of 3.22/2.42 points in-distribution and 12.49/6.92 points out-of-distribution on the 1.7B and 8B models, respectively, without additional computational cost or memory usage.

Reinforcement learning with verifiable rewards (RLVR) relies on costly model-generated responses, motivating larger rollout batches and square-root learning-rate scaling. Reported to approximately preserve progress per sample with one optimizer step per rollout, this rule is less understood in practical multi-step training. We develop a two-clock framework that separates the samples needed to make progress from those consumed before collapse. Under a local quadratic model, the square-root rule preserves first-order drift per sample but not progress: larger batches take fewer, longer updates and lose more to curvature, yielding a learning-rate-dependent critical batch beyond which sample efficiency degrades. A drift-threshold hypothesis predicts a complementary failure clock whose sample count remains approximately invariant along the square-root rule. Controlled GRPO experiments on Qwen2.5-Math-1.5B across a $16\times$ batch range support both predictions. Ratio clipping delays failure, especially for larger batches, but the progress gap persists without it. Widening the clip trades stability for a transient speed gain. The progress gap reappears on another dataset, a general-purpose checkpoint, and a 7B model. Finally, we use the two-clock framework to derive a simple multi-step RLVR schedule: start below the critical batch and switch to a larger batch before failure. Our code is available at: https://github.com/WinstonLiyt/two-clocks-rlvr.

Reinforcement Learning with Verifiable Rewards (RLVR) is widely used for post-training Large Language Models, but practical verifiers can make errors. We study how the rate and structure of reward noise affect RLVR in code generation, with a preliminary scientific-reasoning check. In multi-seed Qwen3 8B experiments on MBPP, mean validation reward over two fixed late evaluations is within 1 percentage point of the clean baseline at the tested resampled group-rollout noise rates through 20%, and within about 2 points at 30%. Confidence intervals allow larger losses; these point estimates do not establish a general tolerance threshold. We also examine full-program pass@k, four controlled noise structures, two model-based verifiers, and policy models from three families spanning 4B-9B parameters. We derive conditional advantage distributions for symmetric and asymmetric group noise, including retained format penalties, and show why clipping limits simple gradient-scaling arguments. The analysis identifies information preserved by whole-group corruption and limits on interpreting our asymmetric sweep as a precision-recall comparison. Overall, the results indicate that imperfect verification can support effective RLVR in the tested settings, while aggregate error rates alone do not characterize the learning signal.