In average reward Markov decision processes, state-of-the-art algorithms for regret minimization follow a well-established framework: They are model-based, optimistic and episodic. First, they maintain a confidence region from which optimistic policies are computed using a well-known subroutine called Extended Value Iteration (EVI). Second, these policies are used over time windows called episodes, each ended by the Doubling Trick (DT) rule or a variant thereof. In this work, without modifying EVI, we show that there is a significant advantage in replacing the doubling trick by another simple rule, that we call the Vanishing Multiplicative rule (VM). When managing episodes with VM, the algorithm’s regret is, both in theory and in practice, as good if not better than with DT while the one-shot behavior is greatly improved. More specifically, the management of bad episodes (when sub-optimal policies are being used) is much better under VM than DT by making the regret of exploration logarithmic rather than linear. These results are made possible by a new in-depth understanding of the contrasting behaviors of confidence regions during good and bad episodes.

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.

This paper investigates a new learning problem, the identification of Blackwell optimal policies on deterministic MDPs (DMDPs): A learner has to return a Blackwell optimal policy with fixed confidence using a minimal number of queries. First, we characterize the maximal set of DMDPs for which the identification is possible. Then, we focus on the analysis of algorithms based on product-form confidence regions. We minimize the number of queries by efficiently visiting the state-action pairs with respect to the shape of confidence sets. Furthermore, these confidence sets are themselves optimized to achieve better performances. The performances of our methods compare to the lower bounds up to a factor $n^2$ in the worst case – where $n$ is the number of states, and constant in certain classes of DMDPs.

The first contribution of this paper is the introduction of a new performance measure of a RL algorithm that is more discriminating than the regret, that we call the regret of exploration that measures the asymptotic cost of exploration. The second contribution is a new performance test (PT) to end episodes in RL optimistic algorithms. This test is based on the performance of the current policy with respect to the best policy over the current confidence set. This is in contrast with all existing RL algorithms whose episode lengths are only based on the number of visits to the states. This modification does not harm the regret and brings an additional property. We show that while all current episodic RL algorithms have a linear regret of exploration, our method has a $O(\log{T})$ regret of exploration for non-degenerate deterministic MDPs.

In this paper, we examine the long-run behavior of regularized, no-regret learning in finite N-player games. A well-known result in the field states that the empirical frequencies of play under no-regret learning converge to the game’s set of coarse correlated equilibria; however, our understanding of how the players' _actual strategies_ evolve over time is much more limited – and, in many cases, non-existent. This issue is exacerbated further by a series of recent results showing that _only_ strict Nash equilibria are stable and attracting under regularized learning, thus making the relation between learning and _pointwise_ solution concepts particularly elusive. In lieu of this, we take a more general approach and instead seek to characterize the _setwise_ rationality properties of the players' day-to-day trajectory of play. To do so, we focus on one of the most stringent criteria of setwise strategic stability, namely that any unilateral deviation from the set in question incurs a cost to the deviator – a property known as _closedness under better replies_ (club). In so doing, we obtain a remarkable equivalence between strategic and dynamic stability: _a product of pure strategies is closed under better replies if and only if its span is stable and attracting under regularized learning._ In addition, we estimate the rate of convergence to such sets, and we show that methods based on entropic regularization (like the exponential weights algorithm) converge at a geometric rate, while projection-based methods converge within a finite number of iterations, even with bandit, payoff-based feedback.