Abstract: In recent years, significant attention has been directed towards learning average-reward Markov Decision Processes (MDPs).However, existing algorithms either suffer from sub-optimal regret guarantees or computational inefficiencies.In this paper, we present the first *tractable* algorithm with minimax optimal regret of $\mathrm{O}\left(\sqrt{\mathrm{sp}(h^*) S A T \log(SAT)}\right)$ where $\mathrm{sp}(h^*)$ is the span of the optimal bias function $h^*$, $S\times A$ is the size of the state-action space and $T$ the number of learning steps. Remarkably, our algorithm does not require prior information on $\mathrm{sp}(h^*)$. Our algorithm relies on a novel subroutine, **P**rojected **M**itigated **E**xtended **V**alue **I**teration (`PMEVI`), to compute bias-constrained optimal policies efficiently. This subroutine can be applied to various previous algorithms to obtain improved regret bounds.