Reinforcement Learning with Verifiable Rewards

Advanced reasoning in LLMs on challenging domains like mathematical reasoning can be tackled using verifiable rewards based reinforced fine-tuning (ReFT). In standard ReFT frameworks, a behavior model generates multiple completions with answers per problem, for the answer to be then scored by a reward function. While such RL post-training methods demonstrate significant performance improvements across challenging reasoning domains, the computational cost of generating completions during training with multiple inference steps makes the training cost non-trivial. To address this, we draw inspiration from off-policy RL, and speculative decoding to introduce a novel ReFT framework, dubbed Nested-ReFT, where a subset of layers of the target model acts as the behavior model to generate off-policy completions during training. The behavior model configured with dynamic layer skipping per batch during training decreases the inference cost compared to the standard ReFT frameworks. Our theoretical analysis shows that Nested-ReFT yields unbiased gradient estimates with controlled variance. Our empirical analysis demonstrates improved computational efficiency measured as tokens/sec across multiple math reasoning benchmarks and model sizes. Additionally, we explore three variants of bias mitigation to minimize the off-policyness in the gradient updates that allows for maintaining performance that matches the baseline ReFT performance.

Recent advances highlight Reinforcement Learning with Verifiable Rewards (RLVR) as a promising approach to enhancing LLM capabilities. However, it remains unclear whether the current practice of RLVR truly expands models' reasoning boundary or mainly amplifies high-reward outputs that base models already know. We introduce empirical support dynamics, a framework that quantifies how RLVR preserves, expands, and shrinks access to verified solutions. Applied across models, reasoning tasks, and training checkpoints, it matches verified successes question by question to reveal changes that aggregate accuracy can conceal. Across the evaluated model-level benchmark aggregates, the shrinkage of observed support outweighs its expansion: more questions lose a verified success than gain one at the stated sampling budgets. These losses occur alongside gains in pass@1. Some RLVR models increase average token-level entropy while reducing final-answer entropy: their token choices become more uncertain on average even as their final answers become more concentrated. Together, these findings reveal an ``invisible leash'' in current RLVR practice: it can make existing successes more reliable while losing access to other verified solutions. Empirical support dynamics make this imbalance measurable and motivate exploration strategies that reach underrepresented solutions while retaining prior successes.