Non-monotone DR-submodular Maximization over General Convex Sets

Abstract

<jats:p>Many real-world problems can often be cast as the optimization of DR-submodular functions defined over a convex domain. These functions play an important role with applications in many areas of applied mathematics, such as machine learning, computer vision, operation research, communication systems or economics. In addition, they capture a subclass of non-convex optimization that provides both practical and theoretical guarantees. In this paper, we show that for maximizing non-monotone DR-submodular functions over a general convex set (such as up-closed convex sets, conic convex set, etc) the Frank-Wolfe algorithm achieves an approximation guarantee which depends on the convex set. To the best of our knowledge, this is the first approximation guarantee. Finally we benchmark our algorithm on problems arising in machine learning domain with the real-world datasets.</jats:p>

An improved approximation algorithm for maximizing a DR-submodular function over a convex set

“…In particular, Durr et al (2021) show that, under the previous assumptions (i)-(iv), there exists a Frank-Wolfe type of algorithm with…”

“…While it is highly desirable to design efficient approximation algorithms under this general setting where neither the objective function is monotonic nor the feasible set is down-closed, unfortunately, Vondrák (2013) shows that such a problem admits no constant approximation under the value oracle model. Our main contribution is to present a 1 4 = 0.25-approximation Frank-Wolfe type of algorithm with a sub-exponential time-complexity under the value oracle model, improving a previous approximation ratio of 1 3 √ 3 ≈ 0.1924 with the same order of time-complexity by Durr et al (2021).…”

“…In this work, we consider the problem without any of the two assumptions. This problem was first studied in (Durr et al, 2021) who design an α-approximation algorithm with convergence rate β:…”

Lyapunov function approach for approximation algorithm design and analysis: with applications in submodular maximization

“…This problem was studied in Durr et al (2021) and Du et al (2022), with the latter improving the approximation ratio in the former. Algorithm 3 recovers the algorithm in Du et al (2022).…”

Resolving the Approximability of Offline and Online Non-monotone DR-Submodular Maximization over General Convex Sets

“…Motivated by the above-mentioned situation, a few recent works started to consider DRsubmodular maximization subject to a general (not necessarily down-closed) convex set constraint K. In general, no constant approximation ratio can be guaranteed for this problem in subexponential time due to an hardness result by Vondrák [27]. However, Durr et al [13] showed that this inapproximability result can be bypassed when the convex set constraint K includes points whose ℓ ∞ -norm is less than the maximal value of 1. Specifically, Durr et al [13] showed a sub-exponential time offline algorithm guaranteeing 1 3 √ 3 (1 − m)-approximation for this problem, where m is the minimal ℓ ∞ -norm of any vector in K. Later, Th áng & Srivastav [25] showed how to obtain a similar result in an online (regret minimization) setting, and an improved sub-exponential offline algorithm obtaining 1 4 (1 − m)-approximation was suggested by Du et al [12].…”