Continuous Non-monotone DR-submodular Maximization with Down-closed Convex Constraint
Shengminjie Chen, Donglei Du, Wenguo Yang, Dachuan Xu, Suixiang Gao
TL;DR
This paper addresses continuous, non-monotone DR-submodular maximization over down-closed convex constraints by showing stationary points can be arbitrarily poor in general, and by extending two notable discrete-domain algorithms to the continuous setting via a Lyapunov framework. It introduces a reduced-domain analysis that yields a $0.309$-approximation bound and develops a Lyapunov-based, discretized $0.385$-approximation algorithm that extends the aided measured continuous greedy to the continuous domain without relying on multilinear or Lovász extensions. The results are backed by numerical experiments across non-monotone quadratic programs, regular coverage, DPPs, and revenue optimization in networks, highlighting both theoretical guarantees and practical effectiveness. The work advances the understanding of how to design and analyze continuous DR-submodular maximization algorithms, providing a path toward better approximations and broader applicability in machine learning and AI tasks.
Abstract
We investigate the continuous non-monotone DR-submodular maximization problem subject to a down-closed convex solvable constraint. Our first contribution is to construct an example to demonstrate that (first-order) stationary points can have arbitrarily bad approximation ratios, and they are usually on the boundary of the feasible domain. These findings are in contrast with the monotone case where any stationary point yields a $1/2$-approximation (Hassani et al. (2017)). Moreover, this example offers insights on how to design improved algorithms by avoiding bad stationary points, such as the restricted continuous local search algorithm (Chekuri et al. (2014)) and the aided measured continuous greedy (Buchbinder and Feldman (2019)). However, the analyses in the last two algorithms only work for the discrete domain because both need to invoke the inequality that the multilinear extension of any submodular set function is bounded from below by its Lovasz extension. Our second contribution, therefore, is to remove this restriction and show that both algorithms can be extended to the continuous domain while retaining the same approximation ratios, and hence offering improved approximation ratios over those in Bian et al. (2017a). for the same problem. At last, we also include numerical experiments to demonstrate our algorithms on problems arising from machine learning and artificial intelligence.