  The Implicitly Normalized Forecaster (INF) algorithm is considered to be an
optimal solution for adversarial multi-armed bandit (MAB) problems. However,
most of the existing complexity results for INF rely on restrictive
assumptions, such as bounded rewards. Recently, a related algorithm was
proposed that works for both adversarial and stochastic heavy-tailed MAB
settings. However, this algorithm fails to fully exploit the available data.
  In this paper, we propose a new version of INF called the Implicitly
Normalized Forecaster with clipping (INF-clip) for MAB problems with
heavy-tailed reward distributions. We establish convergence results under mild
assumptions on the rewards distribution and demonstrate that INF-clip is
optimal for linear heavy-tailed stochastic MAB problems and works well for
non-linear ones. Furthermore, we show that INF-clip outperforms the
best-of-both-worlds algorithm in cases where it is difficult to distinguish
between different arms.
