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Submodular function minimization is well studied, and existing algorithmssolve it exactly or up to arbitrary accuracy. However, in many applications,such as structured sparse learning or batch Bayesian optimization, theobjective function is not exactly submodular, but close. In this case, notheoretical guarantees exist. Indeed, submodular minimization algorithms relyon intricate connections between submodularity and convexity. We show how theserelations can be extended to obtain approximation guarantees for minimizingnon-submodular functions, characterized by how close the function is tosubmodular. We also extend this result to noisy function evaluations. Ourapproximation results are the first for minimizing non-submodular functions,and are optimal, as established by our matching lower bound.

Optimal approximation for unconstrained non-submodular minimization