A Relative-Budget Theory for Reinforcement Learning with Verifiable Rewards in Large Language Model Reasoning

Reinforcement learning (RL) is a dominant paradigm for improving the reasoning abilities of large language models, yet its effectiveness varies across tasks and compute budgets. We propose a \emph{relative-budget} theory explaining this variation through a single quantity called relative budget $\xi := H/\mathbb{E}[T]$, where $H$ is the generation horizon (token budget) and $T$ denotes the number of tokens until the first correct solution under a base policy. We show that $\xi$ determines sample efficiency by controlling reward variance and the likelihood of informative trajectories. Our analysis reveals three regimes: in the \emph{deficient} regime ($\xi \to 0$), informative trajectories are rare and the sample complexity explodes; in the \emph{balanced} regime ($\xi=\Theta(1)$), informative trajectories occur with non-negligible probability and RL is maximally sample-efficient; and in the \emph{ample} regime ($\xi \to \infty$), learning remains stable but marginal gains per iteration diminish. We further provide finite-sample guarantees for online RL that characterize learning progress across these regimes. Specifically, in a case study under idealized distributional assumptions, we show that the relative budget grows linearly over iterations. Our empirical results confirm these predictions in realistic settings, identifying a budget $\xi \in [1.5, 2.0]$ that maximizes learning efficiency and coincides with peak reasoning performance.