On the Optimality of Perturbations in Stochastic and Adversarial Multi-armed Bandit Problems

We investigate the optimality of perturbation based algorithms in the stochastic and adversarial multi-armed bandit problems. For the stochastic case, we provide a unified analysis for all sub-Weibull perturbations. The sub-Weibull family includes sub-Gaussian and sub-Exponential distributions. Our bounds are instance optimal for a range of the sub-Weibull parameter. For the adversarial setting, we prove rigorous barriers against two natural solution approaches using tools from discrete choice theory and extreme value theory. Our results suggest that the optimal perturbation, if it exists, will be of Frechet-type.