Optimal approximation for unconstrained non-submodular minimization
This addresses a gap in optimization for applications like structured sparse learning or batch Bayesian optimization where functions are not exactly submodular, offering the first approximation guarantees for such problems.
The paper tackles the problem of minimizing non-submodular functions, which lack theoretical guarantees, by extending submodularity-convexity relations to provide approximation guarantees based on how close the function is to submodular, including noisy evaluations, with results being optimal due to a matching lower bound.
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 or 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.