Minimax Regret Bounds for Reinforcement Learning
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We consider the problem of provably optimal exploration in reinforcement learning for finite horizon MDPs. We show that an optimistic modification to value iteration achieves a regret bound of Õ( √(HSAT) + H^2S^2A+H√(T)) where H is the time horizon, S the number of states, A the number of actions and T the number of time-steps. This result improves over the best previous known bound Õ(HS √(AT)) achieved by the UCRL2 algorithm of Jaksch et al., 2010. The key significance of our new results is that when T≥ H^3S^3A and SA≥ H, it leads to a regret of Õ(√(HSAT)) that matches the established lower bound of Ω(√(HSAT)) up to a logarithmic factor. Our analysis contains two key insights. We use careful application of concentration inequalities to the optimal value function as a whole, rather than to the transitions probabilities (to improve scaling in S), and we define Bernstein-based "exploration bonuses" that use the empirical variance of the estimated values at the next states (to improve scaling in H).
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