Implicitly normalized forecaster with clipping for linear and non-linear heavy-tailed multi-armed bandits

Implicitly Normalized Forecaster (online mirror descent with Tsallis entropy as prox-function) is known to be an optimal algorithm for adversarial multi-armed problems (MAB). However, most of the complexity results rely on bounded rewards or other restrictive assumptions. Recently closely related best-of-both-worlds algorithm were proposed for both adversarial and stochastic heavy-tailed MAB settings. This algorithm is known to be optimal in both settings, but fails to exploit data fully. In this paper, we propose Implicitly Normalized Forecaster with clipping for MAB problems with heavy-tailed distribution on rewards. We derive convergence results under mild assumptions on rewards distribution and show that the proposed method is optimal for both linear and non-linear heavy-tailed stochastic MAB problems. Also we show that algorithm usually performs better compared to best-of-two-worlds algorithm.