Convex relaxation regression: Black-box optimization of smooth functions by learning their convex envelopes

Finding efficient and provable methods to solve non-convex optimization problems is an outstanding challenge in machine learning and optimization theory. A popular approach used to tackle non-convex problems is to use convex relaxation techniques to find a convex surrogate for the problem. Unfortunately, convex relaxations typically must be found on a problemby-problem basis. Thus, providing a generalpurpose strategy to estimate a convex relaxation would have a wide reaching impact. Here, we introduce Convex Relaxation Regression (CoRR), an approach for learning convex relaxations for a class of smooth functions. The idea behind our approach is to estimate the convex envelope of a function f by evaluating f at a set of T random points and then fitting a convex function to these function evaluations. We prove that with probability greater than 1-δ, the solution of our algorithm converges to the global optimizer of f with error O (log(1/δ/T)^α) for some α > 0. Our approach enables the use of convex optimization tools to solve non-convex optimization problems.