Online Non-Monotone DR-Submodular Maximization

Abstract

In this paper, we study fundamental problems of maximizing DR-submodular continuous functions that have real-world applications in the domain of machine learning, economics, operations research and communication systems. It captures a subclass of non-convex optimization that provides both theoretical and practical guarantees. Here, we focus on minimizing regret for online arriving non-monotone DR-submodular functions over down-closed and general convex sets. First, we present an online algorithm that achieves a 1/e-approximation ratio with the regret of O(T^{3/4}) for maximizing DR-submodular functions over any down-closed convex set. Note that, the approximation ratio of 1/e matches the best-known guarantee for the offline version of the problem. Next, we give an online algorithm that achieves an approximation guarantee (depending on the search space) for the problem of maximizing non-monotone continuous DR-submodular functions over a general convex set (not necessarily down-closed). To best of our knowledge, no prior algorithm with approximation guarantee was known for non-monotone DR-submodular maximization in the online setting. Finally we run experiments to verify the performance of our algorithms on problems arising in machine learning domain with the real-world datasets.

Streaming Algorithms for Monotone DR-Submodular Maximization Under a Knapsack Constraint on the Integer Lattice

“…The DR-submodular function is a kind of non-convex function with theoretical guarantees in optimization, which has received much attention in recent years (Bian et al, 2017a;Niazadeh et al, 2020). There are several works considering the online full information or bandit feedback learning of the DR-submodular function (Chen et al, 2018;Zhang et al, 2019;Raut et al, 2020;Thang & Srivastav, 2021;Zhang et al, 2022a;. DR-submodularity is inspired by the submodular set function, and we find it useful for designing submodular bandit algorithms due to its continuity.…”

Online Weakly DR-Submodular Optimization Under Stochastic Cumulative Constraints

“…Guaranteed results for online continuous diminishing-returns (DR) submodular optimization can be found in Refs. [13][14][15], etc.…”