Distributed Online Convex Optimization with Improved Dynamic Regret

TL;DR

This paper introduces a distributed online gradient descent algorithm with a novel regret bound that is independent of the time horizon, improving performance in long-horizon online convex optimization tasks.

Contribution

It proposes a new distributed online gradient descent method with a tighter regret bound based on gradient variation, applicable to dynamic and linear predictive scenarios.

Findings

Regret bound independent of time horizon.

Tighter bounds using gradient variation measure.

Numerical experiments validate theoretical improvements.

Abstract

In this paper, we consider the problem of distributed online convex optimization, where a group of agents collaborate to track the global minimizers of a sum of time-varying objective functions in an online manner. Specifically, we propose a novel distributed online gradient descent algorithm that relies on an online adaptation of the gradient tracking technique used in static optimization. We show that the dynamic regret bound of this algorithm has no explicit dependence on the time horizon and, therefore, can be tighter than existing bounds especially for problems with long horizons. Our bound depends on a new regularity measure that quantifies the total change in the gradients at the optimal points at each time instant. Furthermore, when the optimizer is approximatly subject to linear dynamics, we show that the dynamic regret bound can be further tightened by replacing the regularity measure that captures the path length of the optimizer with the accumulated prediction errors, which can be much lower in this special case. We present numerical experiments to corroborate our theoretical results.