Bilevel Optimization and Machine Learning

Abstract

We examine the interplay of optimization and machine learning. Great progress has been made in machine learning by cleverly reducing machine learning problems to convex optimization problems with one or more hyper-parameters. The availability of powerful convex-programming theory and algorithms has enabled a flood of new research in machine learning models and methods. But many of the steps necessary for successful machine learning models fall outside of the convex machine learning paradigm. Thus we now propose framing machine learning problems as Stackelberg games. The resulting bilevel optimization problem allows for efficient systematic search of large numbers of hyper-parameters. We discuss recent progress in solving these bilevel problems and the many interesting optimization challenges that remain. Finally, we investigate the intriguing possibility of novel machine learning models enabled by bilevel programming.

This work was supported in part by the Office of Naval Research under grant no. N00014-06-1-0014. The authors are grateful to Professor Olvi Mangasarian for his suggestions on the penalty approach.