## AbstractFor undiscounted reinforcement learning we consider the total regret of a learning algorithm in respect to an optimal policy. We present a reinforcement learning algorithm with total regret O( DS sqrt(AT) ) after T steps for any unknown Markov decision process (MDP) with S states, A actions per state, and diameter D. The diameter of an MDP is at most D if for any pair of states s1, s2 there is a policy which moves from s1 to s2 in at most D steps (on average). Our upper bound holds with high probability and it can be converted into a logarithmic regret bound, if a fixed difference between the average reward of the optimal policy and the second optimal policy is assumed. We also present a corresponding lower bound Omega( sqrt(DSAT) ) on the worst case total regret of any learning algorithm.
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