Research
Near Optimal Exploration-Exploitation in Non-Communicating Markov Decision Processes
December 3, 2018
Abstract
While designing the state space of an MDP, it is common to include states that are transient or not reachable by any policy (e.g., in mountain car, the product space of speed and position contains configurations that are not physically reachable). This results in weakly-communicating or multi-chain MDPs. In this paper, we introduce TUCRL, the first algorithm able to perform efficient exploration-exploitation in any finite Markov Decision Process (MDP) without requiring any form of prior knowledge. In particular, for any MDP with SC communicating states, A actions and ΓC ≤ SC possible communicating next states, we derive a Õ(DC √ ΓCSCAT) regret bound, where DC is the diameter (i.e., the length of the longest shortest path between any two states) of the communicating part of the MDP. This is in contrast with existing optimistic algorithms (e.g., UCRL, Optimistic PSRL) that suffer linear regret in weakly-communicating MDPs, as well as posterior sampling or regularised algorithms (e.g., REGAL), which require prior knowledge on the bias span of the optimal policy to achieve sub-linear regret. We also prove that in weakly-communicating MDPs, no algorithm can ever achieve a logarithmic growth of the regret without first suffering a linear regret for a number of steps that is exponential in the parameters of the MDP. Finally, we report numerical simulations supporting our theoretical findings and showing how TUCRL overcomes the limitations of the state-of-the-art.
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