Parameter-Free Dynamic Regret under Heavy-Tailed Noise

Abstract

We study online convex optimization with stochastic gradient noise whose conditional $p$-th central moment is bounded by $\sigma^p$, for an unknown $p\in(1,2]$. For losses with Lipschitz bound $G$ on a domain of diameter $D$, we obtain expected universal dynamic regret $\widetilde O(GD\sqrt{T\Lambda}+\sigma DT^{1/p}\Lambda^{(p-1)/p})$, where $\Lambda=1+P_T/D$ and $P_T$ is the path length of a fixed comparator sequence. The algorithm combines restarted AdaGrad experts with an adaptive entropy-regularized master, uses one stochastic gradient per round, and requires no knowledge of $G,\sigma,p$, or $P_T$. Its iterates are invariant under positive rescaling of the gradients. The analysis controls comparator movement within restart blocks before taking expectations, yielding the noise path exponent $(p-1)/p$ rather than the exponent $1/2$ of a direct non-restarted extension. A matching stochastic first-order oracle lower bound, combined with the deterministic dynamic-regret lower bound, identifies the minimax rate up to logarithmic factors as $\min\{GD\sqrt{T\Lambda}+\sigma DT^{1/p}\Lambda^{(p-1)/p},GDT\}$.