Minimizing approximately submodular functions

Submodular function minimization is a well studied problem; existing algorithms solve it exactly or up to arbitrary accuracy. However, in many applications, the objective function is not exactly submodular. While submodular minimization algorithms rely on intricate connections between submodularity and convexity, we show that these relations can be extended sufficiently to obtain guaranteed approximations for approximately submodular minimization. In particular, we prove how a projected subgradient method can perform well even for a class of non-submodular functions. This class includes important examples, such as objectives for structured sparse learning and variance reduction in Bayesian optimization. We also extend this result to noisy function evaluations. Our results provide the first approximation guarantee for unconstrained minimization of approximately submodular functions, and greatly extend the scope of submodular minimization algorithms.