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# A Statistical Analysis of Polyak-Ruppert Averaged Q-Learning

*Proceedings of The 26th International Conference on Artificial Intelligence and Statistics*, PMLR 206:2207-2261, 2023.

#### Abstract

We study Q-learning with Polyak-Ruppert averaging (a.k.a., averaged Q-learning) in a discounted markov decision process in synchronous and tabular settings. Under a Lipschitz condition, we establish a functional central limit theorem for the averaged iteration $\bar{\mathbf{Q}}_T$ and show that its standardized partial-sum process converges weakly to a rescaled Brownian motion. The FCLT implies a fully online inference method for reinforcement learning. Furthermore, we show that $\bar{\mathbf{Q}}_T$ is the regular asymptotically linear (RAL) estimator for the optimal Q-value function $\mathbf{Q}^*$ that has the most efficient influence function. We present a nonasymptotic analysis for the $\ell_{\infty}$ error, $\mathbb{E}\|\bar{\mathbf{Q}}_T-\mathbf{Q}^*\|_{\infty}$, showing that it matches the instance-dependent lower bound for polynomial step sizes. Similar results are provided for entropy-regularized Q-Learning without the Lipschitz condition.