Prediction with Expert Advice under Heavy-Tailed Losses

We study prediction with expert advice over $T$ rounds with $K$ experts under heavy-tailed losses whose $(1+ε)$-moment is bounded by $θ>0$ for some $ε\in(0,1]$. In this setting, existing algorithms designed for the case $ε=1$ can be suboptimal when $ε<1$, suffering an additional polynomial dependence on $K$. We characterize the minimax pseudo-regret rate as $Θ((θT)^{\frac{1}{1+ε}}(\log K)^{\fracε{1+ε}})$ and establish gap-dependent lower bounds in a self-bounded environment. To attain this minimax rate without prior knowledge of either $ε$ or $θ$, we propose Adaptive Heavy-Tailed FTRL, which uses robust data-dependent scale adaptation together with truncation. We further develop Adaptive Heavy-Tailed OMD, which likewise requires neither $ε$ nor $θ$ and achieves near-optimal regret guarantees in both adversarial and self-bounded environments. Finally, when $ε$ and $θ$ are known, an FTRL algorithm with Shannon-entropy regularization achieves optimal regret guarantees in both environments.