Invexifying Regularization of Non-Linear Least-Squares Problems

Abstract

We consider regularization of non-convex optimization problems involving a non-linear leastsquares objective. By adding an auxiliary set of variables, we introduce a novel regularization framework whose corresponding objective function is not only provably invex, but it also satisfies the highly desirable Polyak-Lojasiewicz inequality for any choice of the regularization parameter. Although our novel framework is entirely different from the classical 2 -regularization, an interesting connection is established for the special case of under-determined linear least-squares. In particular, we show that gradient descent applied to our novel regularized formulation converges to the same solution as the linear ridge-regression problem. Numerical experiments corroborate our theoretical results and demonstrate the method's performance in practical situations as compared to the typical 2 -regularization.