Solving the Offline and Online Min-Max Problem of Non-smooth Submodular-Concave Functions: A Zeroth-Order Approach

TL;DR

This paper introduces a zeroth-order method for solving non-smooth min-max problems involving submodular-concave functions, providing convergence guarantees and performance analysis in both offline and online settings.

Contribution

It proposes a novel zeroth-order algorithm leveraging Lovász extension and Gaussian smoothing for non-smooth submodular-concave problems, with theoretical convergence and complexity analysis.

Findings

Proves convergence to an $psilon$-saddle point in offline scenarios.

Achieves $O(\u221a{Nar{P}_N})$ online duality gap.

Provides hyperparameter selection guidelines and numerical validation.

Abstract

We consider max-min and min-max problems with objective functions that are possibly non-smooth, submodular with respect to the minimiser and concave with respect to the maximiser. We investigate the performance of a zeroth-order method applied to this problem. The method is based on the subgradient of the Lov\'asz extension of the objective function with respect to the minimiser and based on Gaussian smoothing to estimate the smoothed function gradient with respect to the maximiser. In expectation sense, we prove the convergence of the algorithm to an $\epsilon$-saddle point in the offline case. Moreover, we show that, in the expectation sense, in the online setting, the algorithm achieves $O(\sqrt{N\bar{P}_N})$ online duality gap, where $N$ is the number of iterations and $\bar{P}_N$ is the path length of the sequence of optimal decisions. The complexity analysis and hyperparameter selection are presented for all the cases. The theoretical results are illustrated via numerical examples.