Settling the Sample Complexity of Online Reinforcement Learning
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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 the context of finite-horizon inhomogeneous Markov decision processes. Specifically, we prove that a modified version of Monotonic Value Propagation (MVP), a model-based algorithm proposed by <cit.>, achieves a regret on the order of (modulo log factors) min{√(SAH^3K), HK }, where S is the number of states, A is the number of actions, H is the planning horizon, and K is the total number of episodes. This regret matches the minimax lower bound for the entire range of sample size K≥ 1, essentially eliminating any burn-in requirement. It also translates to a PAC sample complexity (i.e., the number of episodes needed to yield ε-accuracy) of SAH^3/ε^2 up to log factor, which is minimax-optimal for the full ε-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 the development of a new regret decomposition strategy and a novel analysis paradigm to decouple complicated statistical dependency – a long-standing challenge facing the analysis of online RL in the sample-hungry regime.
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