Contaminated Online Convex Optimization

TL;DR

This paper introduces a new framework called contaminated online convex optimization, analyzing regret bounds when objective functions change properties over time, bridging gaps between convex and strongly convex cases.

Contribution

It proposes a novel contaminated regime for online convex optimization and derives tight regret bounds that adapt to changing function properties.

Findings

Lower bound of regret: Ω(log T + √k)

Upper bound with universal algorithms: O(log T + √k log T)

Optimal regret with additional information: O(log T + √k)

Abstract

In online convex optimization, some efficient algorithms have been designed for each of the individual classes of objective functions, e.g., convex, strongly convex, and exp-concave. However, existing regret analyses, including those of universal algorithms, are limited to cases in which the objective functions in all rounds belong to the same class and cannot be applied to cases in which the property of objective functions may change in each time step. This paper introduces a novel approach to address such cases, proposing a new regime we term as \textit{contaminated} online convex optimization. For the contaminated case, we demonstrate that the regret is lower bounded by $\Omega(\log T + \sqrt{k})$. Here, $k$ signifies the level of contamination in the objective functions. We also demonstrate that the regret is bounded by $O(\log T+\sqrt{k\log T})$ when universal algorithms are used. When our proposed algorithms with additional information are employed, the regret is bounded by $O(\log T+\sqrt{k})$, which matches the lower bound. These are intermediate bounds between a convex case and a strongly convex or exp-concave case.