Optimal Decentralized Smoothed Online Convex Optimization

TL;DR

This paper introduces ACORD, a decentralized algorithm for multi-agent smoothed online convex optimization that achieves asymptotic optimality, scales efficiently, and adapts to changing communication graphs, outperforming existing methods.

Contribution

The paper presents the first decentralized algorithm for multi-agent SOCO with proven asymptotic optimality and scalable complexity, adaptable to dynamic communication networks.

Findings

ACORD achieves asymptotic optimality in competitive ratio.

The algorithm's complexity scales logarithmically with the number of agents.

ACORD outperforms the state-of-the-art LPC algorithm in experiments.

Abstract

We study the multi-agent Smoothed Online Convex Optimization (SOCO) problem, where $N$ agents interact through a communication graph. In each round, each agent $i$ receives a strongly convex hitting cost function $f^i_t$ in an online fashion and selects an action $x^i_t \in \mathbb{R}^d$. The objective is to minimize the global cumulative cost, which includes the sum of individual hitting costs $f^i_t(x^i_t)$, a temporal "switching cost" for changing decisions, and a spatial "dissimilarity cost" that penalizes deviations in decisions among neighboring agents. We propose the first truly decentralized algorithm ACORD for multi-agent SOCO that provably exhibits asymptotic optimality. Our approach allows each agent to operate using only local information from its immediate neighbors in the graph. For finite-time performance, we establish that the optimality gap in the competitive ratio decreases with time horizon $T$ and can be conveniently tuned based on the per-round computation available to each agent. Our algorithm benefits from a provably scalable computational complexity that depends only logarithmically on the number of agents and almost linearly on their degree within the graph. Moreover, our results hold even when the communication graph changes arbitrarily and adaptively over time. Finally, ACORD, by virtue of its asymptotic-optimality, is shown to be provably superior to the state-of-the-art LPC algorithm, while exhibiting much lower computational complexity. Extensive numerical experiments across various network topologies further corroborate our theoretical claims.