Improved Approximate Regret for Decentralized Online Continuous Submodular Maximization via Reductions

TL;DR

This paper advances decentralized online learning for continuous submodular rewards (D-OCSM) by introducing two reductions to decentralized online convex optimization (D-OCO): a boosting-based reduction and a decentralized Meta-Frank-Wolfe (DMFW) framework. These reductions enable plugging in state-of-the-art D-OCO algorithms to achieve improved approximate regret bounds that match centralized benchmarks, including $O(n\rho^{-1/4}\sqrt{T\log n})$ for projection-based and $O(nT^{3/4}+n\rho^{-1/4}\sqrt{T\log n})$ for projection-free settings on general functions, as well as $O(n(\log(nT))^{1/3}\rho^{-1/3}T^{2/3})$ for smooth functions with downward-closed sets. Notably, the work provides the first $(1/e)$-regret guarantees for projection-free D-OCSM over downward-closed sets, and demonstrates that projection-free methods can recover centralized optimal rates. The results significantly reduce consensus-error-related gaps and offer practically scalable, communication-efficient strategies for decentralized learning with submodular objectives.

Abstract

To expand the applicability of decentralized online learning, previous studies have proposed several algorithms for decentralized online continuous submodular maximization (D-OCSM) -- a non-convex/non-concave setting with continuous DR-submodular reward functions. However, there exist large gaps between their approximate regret bounds and the regret bounds achieved in the convex setting. Moreover, if focusing on projection-free algorithms, which can efficiently handle complex decision sets, they cannot even recover the approximate regret bounds achieved in the centralized setting. In this paper, we first demonstrate that for D-OCSM over general convex decision sets, these two issues can be addressed simultaneously. Furthermore, for D-OCSM over downward-closed decision sets, we show that the second issue can be addressed while significantly alleviating the first issue. Our key techniques are two reductions from D-OCSM to decentralized online convex optimization (D-OCO), which can exploit D-OCO algorithms to improve the approximate regret of D-OCSM in these two cases, respectively.

Theorems & Definitions (28)

• Lemma 1
• Theorem 1
• Corollary 1
• Corollary 2
• Theorem 2
• Corollary 3
• Theorem 3
• Corollary 4
• Lemma 2
• Lemma 3