Near-optimal Approximate Discrete and Continuous Submodular Function Minimization

TL;DR

This paper introduces faster algorithms for approximately minimizing submodular functions and certain nonconvex functions, significantly reducing oracle evaluation complexity compared to previous methods.

Contribution

It presents a randomized algorithm achieving near-optimal oracle complexity for submodular minimization and extends this approach to a class of nonconvex functions with specific curvature and Lipschitz conditions.

Findings

Achieves $ ilde{O}(n/ ext{epsilon}^2)$ oracle evaluations for submodular minimization.

Improves upon previous algorithms with higher oracle complexities.

Provides efficient algorithms for a broad class of nonconvex functions with specific properties.

Abstract

In this paper we provide improved running times and oracle complexities for approximately minimizing a submodular function. Our main result is a randomized algorithm, which given any submodular function defined on $n$-elements with range $[-1, 1]$, computes an $\epsilon$-additive approximate minimizer in $\tilde{O}(n/\epsilon^2)$ oracle evaluations with high probability. This improves over the $\tilde{O}(n^{5/3}/\epsilon^2)$ oracle evaluation algorithm of Chakrabarty \etal~(STOC 2017) and the $\tilde{O}(n^{3/2}/\epsilon^2)$ oracle evaluation algorithm of Hamoudi \etal. Further, we leverage a generalization of this result to obtain efficient algorithms for minimizing a broad class of nonconvex functions. For any function $f$ with domain $[0, 1]^n$ that satisfies $\frac{\partial^2f}{\partial x_i \partial x_j} \le 0$ for all $i \neq j$ and is $L$-Lipschitz with respect to the $L^\infty$-norm we give an algorithm that computes an $\epsilon$-additive approximate minimizer with $\tilde{O}(n \cdot \mathrm{poly}(L/\epsilon))$ function evaluation with high probability.