Improved Guarantees for Constrained Online Convex Optimization via Self-Contraction

TL;DR

This paper introduces a projection-based algorithm for Constrained Online Convex Optimization that significantly improves the cumulative constraint violation bounds, achieving exponential improvements for strongly convex losses and better bounds for convex losses.

Contribution

The paper presents a simple algorithm that improves constraint violation guarantees in COCO, leveraging a geometric result on self-contracted curves.

Findings

Achieves $O( ext{log } T)$ regret and constraint violation for strongly convex losses.

Improves constraint violation bounds to $O( ext{log } T)$ for strongly convex cases.

Maintains $O( ext{sqrt } T)$ bounds for convex losses.

Abstract

We consider Constrained Online Convex Optimization (COCO) with adversarially chosen constraints. At each round, the learner chooses an action before observing the loss and constraint function for that round. The goal is to achieve small static regret against the best point satisfying all constraints while also controlling cumulative constraint violation ($\mathsf{CCV}$). For strongly convex losses, state-of-the-art algorithms achieve $O(\log T)$ regret and $O(\sqrt{T \log T})$ $\mathsf{CCV}.$ The corresponding best-known bounds for convex losses is $O(\sqrt{T})$ regret and $O(\sqrt{T} \log T)$ $\mathsf{CCV}$. In this paper, we give a simple projection-based algorithm that simultaneously achieves $O(\log T)$ regret and $O(\log T)$ $\mathsf{CCV}$ for strongly-convex losses, yielding an exponential improvement in the $\mathsf{CCV}$. For the convex losses, our algorithm improves the $\mathsf{CCV}$ to $O(\sqrt{T})$ while maintaining the optimal $O(\sqrt{T})$ regret. The key to our improvement is a recent geometric result for self-contracted curves, which may be of independent interest.