Improved Approximate Regret for Decentralized Online Continuous Submodular Maximization via Reductions

TL;DR

This paper introduces reductions from decentralized online continuous submodular maximization to online convex optimization, improving regret bounds and enabling projection-free algorithms for complex decision sets.

Contribution

It provides novel reductions that enhance regret bounds for D-OCSM and enables projection-free methods for complex decision sets.

Findings

Improved approximate regret bounds for D-OCSM.

Reduction techniques from D-OCSM to D-OCO.

Effective algorithms for downward-closed decision sets.

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.