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.