## Near-optimal Regret Bounds for Reinforcement Learning

Thomas Jaksch, Ronald Ortner, Peter Auer; 11(51):1563−1600, 2010.

### Abstract

For undiscounted reinforcement learning in Markov decision processes (MDPs) we consider the total regret of a learning algorithm with respect to an optimal policy. In order to describe the transition structure of an MDP we propose a new parameter: An MDP has diameter D if for any pair of states s,s' there is a policy which moves from s to s' in at most D steps (on average). We present a reinforcement learning algorithm with total regret Õ(DS√AT) after T steps for any unknown MDP with S states, A actions per state, and diameter D. A corresponding lower bound of Ω(√DSAT) on the total regret of any learning algorithm is given as well.
These results are complemented by a sample complexity bound on the number of suboptimal steps taken by our algorithm. This bound can be used to achieve a (gap-dependent) regret bound that is logarithmic in T.
Finally, we also consider a setting where the MDP is allowed to change a fixed number of l times. We present a modification of our algorithm that is able to deal with this setting and show a regret bound of Õ(l1/3T2/3DS√A).

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