A Unified Approach for Maximizing Continuous DR-submodular Functions

TL;DR

This paper introduces a versatile Frank-Wolfe based algorithm for maximizing continuous DR-submodular functions across various oracle access models, achieving improved or comparable results and pioneering regret bounds with bandit feedback.

Contribution

It unifies and extends optimization methods for DR-submodular functions, covering multiple oracle types and settings with new theoretical guarantees.

Findings

Improves results in nine out of sixteen cases

Avoids expensive projections in two scenarios

First regret bounds with bandit feedback for stochastic DR-submodular functions

Abstract

This paper presents a unified approach for maximizing continuous DR-submodular functions that encompasses a range of settings and oracle access types. Our approach includes a Frank-Wolfe type offline algorithm for both monotone and non-monotone functions, with different restrictions on the general convex set. We consider settings where the oracle provides access to either the gradient of the function or only the function value, and where the oracle access is either deterministic or stochastic. We determine the number of required oracle accesses in all cases. Our approach gives new/improved results for nine out of the sixteen considered cases, avoids computationally expensive projections in two cases, with the proposed framework matching performance of state-of-the-art approaches in the remaining five cases. Notably, our approach for the stochastic function value-based oracle enables the first regret bounds with bandit feedback for stochastic DR-submodular functions.