Theoretical Foundations of $\max$@$k$ Reinforcement Learning

Abstract:Reinforcement Learning is a cornerstone technique for modern large reasoning models. Usually, for difficult tasks such as code generation and theorem proving, the agent is evaluated by generating $K$ responses rather than sampling a single response, and performance is then measured using a retry-aware metric such as $\max$@$k$. Despite their practical importance, the theoretical foundations of learning under such criteria remain limited. In this work, we provide a theoretical study of the $\max$@$k$ learning problem in finite-horizon reinforcement learning. We show that optimizing the $\max$@$k$ objectives is fundamentally different from standard expected-return maximization. In particular, we prove that Markovian policies are in general insufficient, identify a compact state augmentation that restores optimality, and explicitly characterize the performance gap that can arise between history-dependent and non-history-dependent policies. Moreover, we show that learning $\max$@$k$-optimal policies is statistically harder than standard reinforcement learning and provide an efficient algorithm that achieves the optimal sample complexity rate.

Machine Learning (cs.LG)

arXiv:2607.17823 [cs.LG]

(or arXiv:2607.17823v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2607.17823 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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