Parameter-Free Dynamic Regret for Online Convex Optimization under Heavy-Tailed Noise

2026-07-29

AI Summary

This work addresses non-stationary online learning under heavy-tailed noise and proposes HT-PAder, an algorithm that requires no prior knowledge of problem parameters. HT-PAder combines geometrically scheduled restarting AdaGrad experts with a path-aware meta-algorithm, AdaGrad-Hedge. It achieves, for the first time, a minimax-optimal universal dynamic regret bound under heavy-tailed noise without any parameter tuning. Specifically, its expected dynamic regret is bounded by Õ(GD√{T(1+P_T/D)} + σD T^{1/p}(1+P_T/D)^{(p-1)/p}), which matches the theoretical lower bound in terms of the comparator path length exponent. This guarantee remains optimal even when the domain diameter D, Lipschitz constant G, noise level σ, and comparator path length P_T are unknown.

Abstract

We study online convex optimization (OCO) in non-stationary environments under heavy-tailed noise, where the stochastic gradient oracle admits only a finite $p$-th central moment for some $p \in (1, 2]$. While static regret is well-understood, achieving universal dynamic regret in a parameter-free manner remains an open challenge. We resolve this by proposing \textbf{HT-PAder}, a parameter-free algorithm combining restarted AdaGrad experts over a geometric pool of block lengths with a pathwise meta-algorithm, \textbf{AdaGrad-Hedge}, which requires no moment conditions on meta-losses. For a domain of diameter $D$, Lipschitz constant $G$, noise level $σ$, and comparator path length $P_T$, HT-PAder achieves an expected universal dynamic regret of \[ \widetilde O\left( GD\sqrt{T(1+P_T/D)} + σD T^{1/p}(1+P_T/D)^{(p-1)/p} \right). \] The algorithm does not require prior knowledge of any of these problem parameters. Even in the special case of finite variance ($p=2$), HT-PAder provides the first parameter-free minimax universal dynamic regret guarantee. We also prove a matching lower bound, establishing the optimality of the path-length exponent.

Problem

• online convex optimization
• heavy-tailed noise
• dynamic regret
• parameter-free
• non-stationary environments

Innovation

• parameter-free
• dynamic regret
• heavy-tailed noise
• online convex optimization
• AdaGrad-Hedge