The Plug-in Approach for Average-Reward and Discounted MDPs: Optimal Sample Complexity Analysis

Abstract: We study the sample complexity of the plug-in approach for learning $\varepsilon$-optimal policies in average-reward Markov decision processes (MDPs) with a generative model. The plug-in approach constructs a model estimate then computes an average-reward optimal policy in the estimated model. Despite representing arguably the simplest algorithm for this problem, the plug-in approach has never been theoretically analyzed. Unlike the more well-studied discounted MDP reduction method, the plug-in approach requires no prior problem information or parameter tuning. Our results fill this gap and address the limitations of prior approaches, as we show that the plug-in approach is optimal in several well-studied settings without using prior knowledge. Specifically it achieves the optimal diameter- and mixing-based sample complexities of $\widetilde{O}\left(SA \frac{D}{\varepsilon^2}\right)$ and $\widetilde{O}\left(SA \frac{\tau_{\mathrm{unif}}}{\varepsilon^2}\right)$, respectively, without knowledge of the diameter $D$ or uniform mixing time $\tau_{\mathrm{unif}}$. We also obtain span-based bounds for the plug-in approach, and complement them with algorithm-specific lower bounds suggesting that they are unimprovable. Our results require novel techniques for analyzing long-horizon problems which may be broadly useful and which also improve results for the discounted plug-in approach, removing effective-horizon-related sample size restrictions and obtaining the first optimal complexity bounds for the full range of sample sizes without reward perturbation.