This paper considers the problems of maximizing a continuous non-monotone submodular function over the hypercube, both with and without coordinate-wise concavity. This family of optimization problems has several applications in machine learning, economics, and communication systems. The main result is
the first 1/2-approximation algorithm for continuous submodular function maximization; this approximation factor of 1/2 is the best possible for algorithms that only query the objective function at polynomially many points. For the special case of DR-submodular maximization, i.e. when the submodular functions are also coordinate-wise concave along all coordinates, we provide a different 1 2-approximation algorithm that runs in quasi-linear time.
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Fast First-Order Methods for Monotone Strongly DR-Submodular Maximization
Continuous DR-submodular functions are a class of functions that satisfy the Diminishing Returns (DR) property, which implies that they are concave along non-negative directions. Existing works have studied monotone continuous DR-submodular maximization subject to a convex constraint and have proposed efficient algorithms with approximation guarantees. However, in many applications, e. g., computing the stability number of a graph and mean-field inference for probabilistic log-submodular models, the DR-submodular function has the additional property of being strongly concave along non-negative directions that could be utilized for obtaining faster convergence rates. In this paper, we first introduce and characterize the class of strongly DR-submodular functions and show how such a property implies strong concavity along non-negative directions. Then, we study L-smooth monotone strongly DR-submodular functions that have bounded curvature, and we show how to exploit such additional structure to obtain algorithms with improved approximation guarantees and faster convergence rates for the maximization problem. In particular, we propose the SDRFW algorithm that matches the provably optimal approximation ratio after only iterations, where c ∈ [0,1] and μ ≥ 0 are the curvature and the strong DR-submodularity parameter. Furthermore, we study the Projected Gradient Ascent (PGA) method for this problem and provide a refined analysis of the algorithm with an improved approximation ratio (compared to ½ in prior works) and a linear convergence rate. Given that both algorithms require knowledge of the smoothness parameter L, we provide a novel characterization of L for DR-submodular functions showing that in many cases, computing L could be formulated as a convex optimization problem, i. e., a geometric program, that could be solved efficiently. Experimental results illustrate and validate the efficiency and effectiveness of our algorithms.
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- Award ID(s):
- 2023166
- NSF-PAR ID:
- 10443284
- Editor(s):
- Berry, Jonathan; Shmoys, David; Cowen, Lenore; Naumann, Uwe
- Date Published:
- Journal Name:
- Proceedings of SIAM Conference on Applied and Computational Discrete Algorithms
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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