Non-stationary Online Learning for Curved Losses: Improved Dynamic Regret via Mixability

TL;DR

This paper advances non-stationary online learning by leveraging mixability to achieve significantly improved dynamic regret bounds for curved losses like squared and logistic, surpassing prior convex-focused results.

Contribution

It introduces a novel analysis framework that exploits mixability to improve dynamic regret bounds for curved losses in non-stationary environments.

Findings

Achieves $ ilde{O}(d T^{1/3} P_T^{2/3})$ dynamic regret for mixable losses.

Improves upon previous regret bounds that depended more heavily on dimensionality.

Provides a simple analytical approach avoiding complex KKT-based analysis.

Abstract

Non-stationary online learning has drawn much attention in recent years. Despite considerable progress, dynamic regret minimization has primarily focused on convex functions, leaving the functions with stronger curvature (e.g., squared or logistic loss) underexplored. In this work, we address this gap by showing that the regret can be substantially improved by leveraging the concept of mixability, a property that generalizes exp-concavity to effectively capture loss curvature. Let $d$ denote the dimensionality and $P_T$ the path length of comparators that reflects the environmental non-stationarity. We demonstrate that an exponential-weight method with fixed-share updates achieves an $\mathcal{O}(d T^{1/3} P_T^{2/3} \log T)$ dynamic regret for mixable losses, improving upon the best-known $\mathcal{O}(d^{10/3} T^{1/3} P_T^{2/3} \log T)$ result (Baby and Wang, 2021) in $d$. More importantly, this improvement arises from a simple yet powerful analytical framework that exploits the mixability, which avoids the Karush-Kuhn-Tucker-based analysis required by existing work.