The stochastic multi-arm bandit problem has been extensively studied under standard assumptions on the arm's distribution (e.g bounded with known support, exponential family, etc). These assumptions are suitable for many real-world problems but sometimes they require knowledge (on tails for instance) that may not be precisely accessible to the practitioner, raising the question of the robustness of bandit algorithms to model misspecification. In this paper we study a generic Dirichlet Sampling (DS) algorithm, based on pairwise comparisons of empirical indices computed with re-sampling of the arms' observations and a data-dependent exploration bonus. We show that different variants of this strategy achieve provably optimal regret guarantees when the distributions are bounded and logarithmic regret for semi-bounded distributions with a mild quantile condition. We also show that a simple tuning achieve robustness with respect to a large class of unbounded distributions, at the cost of slightly worse than logarithmic asymptotic regret. We finally provide numerical experiments showing the merits of DS in a decision-making problem on synthetic agriculture data.

This paper unifies the design and simplifies the analysis of risk-averse Thompson sampling algorithms for the multi-armed bandit problem for a generic class of risk functionals \r{ho} that are continuous. Using the contraction principle in the theory of large deviations, we prove novel concentration bounds for these continuous risk functionals. In contrast to existing works in which the bounds depend on the samples themselves, our bounds only depend on the number of samples. This allows us to sidestep significant analytical challenges and unify existing proofs of the regret bounds of existing Thompson sampling-based algorithms. We show that a wide class of risk functionals as well as "nice" functions of them satisfy the continuity condition. Using our newly developed analytical toolkits, we analyse the algorithms $\rho$-MTS (for multinomial distributions) and $\rho$-NPTS (for bounded distributions) and prove that they admit asymptotically optimal regret bounds of risk-averse algorithms under the mean-variance, CVaR, and other ubiquitous risk measures, as well as a host of newly synthesized risk measures. Numerical simulations show that our bounds are reasonably tight vis-\`a-vis algorithm-independent lower bounds.