Communication-Efficient Decentralized Online Continuous DR-Submodular Maximization

TL;DR

This paper introduces two communication-efficient decentralized online algorithms for monotone continuous DR-submodular maximization, significantly reducing communication and gradient evaluations while achieving near-optimal regret bounds.

Contribution

The paper presents the first one-shot, projection-free decentralized algorithm and an optimal regret algorithm for continuous DR-submodular maximization in an online setting.

Findings

Mono-DMFW achieves a $(1-1/e)$-regret of $O(T^{4/5})$.

DOBGA attains a $(1-1/e)$-regret of $O( oot{2}rom T)$ with minimal gradient queries.

Experimental results demonstrate the effectiveness of both algorithms.

Abstract

Maximizing a monotone submodular function is a fundamental task in machine learning, economics, and statistics. In this paper, we present two communication-efficient decentralized online algorithms for the monotone continuous DR-submodular maximization problem, both of which reduce the number of per-function gradient evaluations and per-round communication complexity from $T^{3/2}$ to $1$. The first one, One-shot Decentralized Meta-Frank-Wolfe (Mono-DMFW), achieves a $(1-1/e)$-regret bound of $O(T^{4/5})$. As far as we know, this is the first one-shot and projection-free decentralized online algorithm for monotone continuous DR-submodular maximization. Next, inspired by the non-oblivious boosting function \citep{zhang2022boosting}, we propose the Decentralized Online Boosting Gradient Ascent (DOBGA) algorithm, which attains a $(1-1/e)$-regret of $O(\sqrt{T})$. To the best of our knowledge, this is the first result to obtain the optimal $O(\sqrt{T})$ against a $(1-1/e)$-approximation with only one gradient inquiry for each local objective function per step. Finally, various experimental results confirm the effectiveness of the proposed methods.