Convex Relaxation Regression: Black-Box Optimization of Smooth Functions by Learning Their Convex Envelopes
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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 problem-by-problem basis. Thus, providing a general-purpose 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 main 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( ((1/δ) /T)^α) for some α> 0. Our approach enables the use of convex optimization tools to solve a class of non-convex optimization problems.
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