Variance-Reduced Policy Gradient is Nearly Minimax-Optimal

Several variance-reduced versions of REINFORCE based on importance sampling achieve an improved $O(ε^{-3})$ sample complexity to find an $ε$-stationary point, under an unrealistic assumption on the variance of the importance weights. In this paper, we propose the \algo (Defensive Policy Gradient) algorithm, based on defensive importance sampling, which achieves the same rate without any assumption on the variance of ordinary importance weights. We also establish lower bounds in a generalized black-box policy-optimization model that hides states and actions and permits parameter-dependent rewards. In this model, the optimal rates are $Θ(ε^{-4})$ with bounded-variance one-policy feedback and $Θ(ε^{-3})$ with mean-square-smooth coupled two-policy feedback. Under standard policy-regularity conditions, REINFORCE and \algo realize the corresponding oracle conditions and attain the $O(ε^{-4})$ and $O(ε^{-3})$ upper bounds, respectively. Although the lower bounds do not apply directly to the classical MDP interaction model in which these algorithms operate, this correspondence provides oracle-level evidence that the faster rate of \algo is optimal and genuinely separated from that of vanilla policy gradient.

In this paper, we establish last-iterate convergence rates for off-policy actor--critic methods in reinforcement learning. In particular, under a single-loop, single-timescale implementation and a broad class of policy updates, including approximate policy iteration and natural policy gradient methods, we prove the first $\tilde{\mathcal{O}}(ε^{-2})$ sample complexity guarantee for finding an $ε$-optimal policy under minimal assumptions, namely, the existence of a policy that induces an irreducible Markov chain. This stands in stark contrast to the existing literature, where an $\tilde{\mathcal{O}}(ε^{-2})$ sample complexity is achieved only through nested-loop updates and/or under strong, algorithm-dependent assumptions on the policies, such as uniform mixing and uniform exploration. Technically, to address the challenges posed by the coupled update equations arising from the single-loop implementation, as well as the potentially unbounded iterates induced by off-policy learning, our analysis is based on a coupled Lyapunov drift framework. Specifically, we establish a geometric convergence rate for the actor and an $\tilde{\mathcal{O}}(1/T)$ convergence rate for the critic, and combine the two Lyapunov drift inequalities through a cross-domination property. We believe this analytical framework is of independent interest and may be applicable to other coupled iterative algorithms with unbounded

Reinforcement learning with verifiable rewards spends much of its rollout budget on states that yield no learning signal. We recast this as a question about the sampling distribution itself: under a fixed rollout budget, which distribution over states minimizes the variance of the policy-gradient estimator? The answer admits a closed-form Neyman allocation, $μ_{\mathrm{var}}^π(s) \propto d^π(s)\,σ_π(s)$, which connects naturally to GRPO's group-relative advantage and induces an emergent reverse curriculum as the policy improves, with no externally imposed schedule. Building on this principle, we propose VSR (Variance-driven State Resampling), a plug-in modification to GRPO using a prefix replay buffer and a zero-cost variance proxy. On mathematical reasoning benchmarks and long-horizon agentic tasks (Sokoban, BabyAI), VSR outperforms GRPO and GiGPO on most benchmarks with fewer generated tokens per update.