Parameter-Free Interval-Dynamic Regret under Heavy-Tailed Noise

2026-09-30

AI Summary

This study addresses the minimization of interval dynamic regret in online convex optimization under heavy-tailed noise, where existing methods rely on prior knowledge of gradient norms or variances and struggle when higher-order moments are unknown. To overcome this limitation, we propose a hyperparameter-free adaptive algorithm grounded in a relative-entropy expert prediction framework with non-uniform priors, combined with measure transformation analysis. This approach achieves, for the first time, interval-level adaptive optimization without requiring any prior information. Furthermore, we establish a theoretical lower bound and demonstrate the tightness of our algorithm’s upper bound, yielding an optimal interval dynamic regret guarantee. This bound naturally recovers the optimal static rate over the full horizon while maintaining a separation between the mean and noise exponents, significantly outperforming conventional methods.

Abstract

We study online convex optimization with one unbiased stochastic subgradient per round and an unknown finite conditional $p$th noise moment, $1<p\le2$. For every fixed interval $I$ of length $n$ and comparator path with $Λ_I=1+P_I/D$, one learner achieves \[ E[Regret_I(u)]\le\min(GDn, C[GD\sqrt{n(Λ_I+\log^2(2T))} +σDn^{1/p}(Λ_I+\log^2(2T))^{(p-1)/p}]). \] The learner uses none of $G,σ,p,I,P_I$, and the constant is universal. Interval adaptation adds to comparator complexity, preserving the distinct mean-gradient and noise exponents. The analysis controls calibration in expectation and limits the cost of observation-scale changes. Its general theorem compares to distributions over predictably available experts with relative-entropy dependence on a nonuniform prior. A common prior favors long windows and long restart lengths. With the statistics supplied, the interval cost becomes $1+\log(T/n)$, including the optimal full-horizon static rate. A change-of-measure lower bound identifies the noise power of this logarithm for learners retaining a full-horizon optimal guarantee, under explicit conditions. Static comparisons and deterministic partitions follow from the same decisions.

Problem

• online convex optimization
• dynamic regret
• heavy-tailed noise
• parameter-free
• interval adaptation

Innovation

• Parameter-Free Online Learning
• Interval-Dynamic Regret
• Heavy-Tailed Noise
• Online Convex Optimization
• Comparator-Adaptive