Lazy OCO: Online Convex Optimization on a Switching Budget

TL;DR

This paper introduces efficient algorithms for online convex optimization with limited decision switches, achieving regret bounds that improve with fewer switches, and provides matching lower bounds for some cases.

Contribution

It fills the gap in oblivious adversary settings by providing computationally efficient algorithms with regret bounds depending on the number of switches allowed.

Findings

Regret bound of O(T/S) for general convex losses

Regret bound of ~O(T/S^2) for strongly convex losses

Algorithms with logarithmic switches and regret overhead

Abstract

We study a variant of online convex optimization where the player is permitted to switch decisions at most $S$ times in expectation throughout $T$ rounds. Similar problems have been addressed in prior work for the discrete decision set setting, and more recently in the continuous setting but only with an adaptive adversary. In this work, we aim to fill the gap and present computationally efficient algorithms in the more prevalent oblivious setting, establishing a regret bound of $O(T/S)$ for general convex losses and $\widetilde O(T/S^2)$ for strongly convex losses. In addition, for stochastic i.i.d.~losses, we present a simple algorithm that performs $\log T$ switches with only a multiplicative $\log T$ factor overhead in its regret in both the general and strongly convex settings. Finally, we complement our algorithms with lower bounds that match our upper bounds in some of the cases we consider.