We give a simple optimistic algorithm for which it is easy to derive regret bounds of Õ(√t _mixSAT) after T steps in uniformly ergodic Markov decision processes with S states, A actions, and mixing time parameter t _mix. These bounds are the first regret bounds in the general, non-episodic setting with an optimal dependence on all given parameters. They could only be improved by using an alternative mixing time parameter.