Convex Relaxation Regression: Black-Box Optimization of Smooth Functions by Learning Their Convex Envelopes

Abstract

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-, the solution of our algorithm converges to the global optimizer of f with error O ( ((1/) T )^ ) for some > 0. Our approach enables the use of convex optimization tools to solve a class of non-convex optimization problems.

Tasks

Reproductions