Universal Online Convex Optimization with $1$ Projection per Round

TL;DR

This paper introduces a universal online convex optimization algorithm that requires only one projection per round, achieving optimal regret bounds across various convex function classes by leveraging surrogate losses and expert advice.

Contribution

It develops a novel universal OCO algorithm with a single projection per round, unifying regret guarantees for multiple convex function types using surrogate losses and expert aggregation.

Findings

Achieves optimal regret bounds for convex, exponentially concave, and strongly convex functions.

Develops a surrogate loss framework that simplifies projections and analysis.

Extends to exploit smoothness for small-loss regret in multiple convex function classes.

Abstract

To address the uncertainty in function types, recent progress in online convex optimization (OCO) has spurred the development of universal algorithms that simultaneously attain minimax rates for multiple types of convex functions. However, for a $T$-round online problem, state-of-the-art methods typically conduct $O(\log T)$ projections onto the domain in each round, a process potentially time-consuming with complicated feasible sets. In this paper, inspired by the black-box reduction of Cutkosky and Orabona (2018), we employ a surrogate loss defined over simpler domains to develop universal OCO algorithms that only require $1$ projection. Embracing the framework of prediction with expert advice, we maintain a set of experts for each type of functions and aggregate their predictions via a meta-algorithm. The crux of our approach lies in a uniquely designed expert-loss for strongly convex functions, stemming from an innovative decomposition of the regret into the meta-regret and the expert-regret. Our analysis sheds new light on the surrogate loss, facilitating a rigorous examination of the discrepancy between the regret of the original loss and that of the surrogate loss, and carefully controlling meta-regret under the strong convexity condition. In this way, with only $1$ projection per round, we establish optimal regret bounds for general convex, exponentially concave, and strongly convex functions simultaneously. Furthermore, we enhance the expert-loss to exploit the smoothness property, and demonstrate that our algorithm can attain small-loss regret for multiple types of convex and smooth functions.