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# Settling the sample complexity of online reinforcement learning

*Proceedings of Thirty Seventh Conference on Learning Theory*, PMLR 247:5213-5219, 2024.

#### Abstract

A central issue lying at the heart of online reinforcement learning (RL) is data efficiency. While a number of recent works achieved asymptotically minimal regret in online RL, the optimality of these results is only guaranteed in a “large-sample” regime, imposing enormous burn-in cost in order for their algorithms to operate optimally. How to achieve minimax-optimal regret without incurring any burn-in cost has been an open problem in RL theory. We settle this problem for finite-horizon inhomogeneous Markov decision processes. Specifically, we prove that a modified version of MVP (Monotonic Value Propagation), an optimistic model-based algorithm proposed by Zhang et al., achieves a regret on the order of $$\min\big\{ \sqrt{SAH^3K}, \,HK \big\},$$ where $S$ is the number of states, $A$ is the number of actions, $H$ is the horizon length, and $K$ is the total number of episodes. This regret matches the minimax lower bound for the entire range of sample size K, essentially eliminating any burn-in requirement. It also translates to a PAC sample complexity (i.e., the number of episodes needed to yield $\varepsilon$-accuracy) of $\frac{SAH^3}{\varepsilon^2}$ up to log factor, which is minimax-optimal for the full epsilon-range. Further, we extend our theory to unveil the influences of problem-dependent quantities like the optimal value/cost and certain variances. The key technical innovation lies in a novel analysis paradigm to decouple complicated statistical dependency — a long-standing challenge facing the analysis of online RL in the sample-hungry regime.