Gap-Dependent Unsupervised Exploration for Reinforcement Learning
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For the problem of task-agnostic reinforcement learning (RL), an agent first collects samples from an unknown environment without the supervision of reward signals, then is revealed with a reward and is asked to compute a corresponding near-optimal policy. Existing approaches mainly concern the worst-case scenarios, in which no structural information of the reward/transition-dynamics is utilized. Therefore the best sample upper bound is ∝𝒪(1/ϵ^2), where ϵ>0 is the target accuracy of the obtained policy, and can be overly pessimistic. To tackle this issue, we provide an efficient algorithm that utilizes a gap parameter, ρ>0, to reduce the amount of exploration. In particular, for an unknown finite-horizon Markov decision process, the algorithm takes only 𝒪 (1/ϵ· (H^3SA / ρ + H^4 S^2 A) ) episodes of exploration, and is able to obtain an ϵ-optimal policy for a post-revealed reward with sub-optimality gap at least ρ, where S is the number of states, A is the number of actions, and H is the length of the horizon, obtaining a nearly quadratic saving in terms of ϵ. We show that, information-theoretically, this bound is nearly tight for ρ < Θ(1/(HS)) and H>1. We further show that ∝𝒪(1) sample bound is possible for H=1 (i.e., multi-armed bandit) or with a sampling simulator, establishing a stark separation between those settings and the RL setting.
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