Optimal approximation for unconstrained non-submodular minimization

Part of Proceedings of the International Conference on Machine Learning 1 pre-proceedings (ICML 2020)

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Authors

Marwa El Halabi, Stefanie Jegelka

Abstract

Submodular function minimization is well studied, and existing algorithms solve it exactly or up to arbitrary accuracy. However, in many applications, such as structured sparse learning, and batch Bayesian optimization, the objective function is not exactly submodular, but close. In this case, no theoretical guarantees exist. Indeed, submodular minimization algorithms rely on intricate connections between submodularity and convexity. We show how these relations can be extended to obtain approximation guarantees for minimizing non-submodular functions, characterized by how close the function is to submodular. We also extend this result to noisy function evaluations. Our approximation results are the first for minimizing non-submodular functions, and are optimal, as established by our matching lower bound.