A Statistical Analysis of Polyak-Ruppert Averaged Q-learning
This work provides theoretical guarantees for online inference in reinforcement learning, which is incremental but offers rigorous statistical foundations for practitioners.
The paper tackles the problem of analyzing Q-learning with Polyak-Ruppert averaging in discounted Markov decision processes, establishing a functional central limit theorem and showing that the averaged estimator is regular asymptotically linear with efficient influence, while providing nonasymptotic error bounds that match instance-dependent lower bounds.
We study Q-learning with Polyak-Ruppert averaging 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 $\bar{\boldsymbol{Q}}_T$ and show that its standardized partial-sum process converges weakly to a rescaled Brownian motion. The functional central limit theorem implies a fully online inference method for reinforcement learning. Furthermore, we show that $\bar{\boldsymbol{Q}}_T$ is the regular asymptotically linear (RAL) estimator for the optimal Q-value function $\boldsymbol{Q}^*$ that has the most efficient influence function. We present a nonasymptotic analysis for the $\ell_{\infty}$ error, $\mathbb{E}\|\bar{\boldsymbol{Q}}_T-\boldsymbol{Q}^*\|_{\infty}$, 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.