Non-smooth Non-convex Bregman Minimization: Unification and New Algorithms

Abstract

We propose a unifying algorithm for non-smooth non-convex optimization. The algorithm approximates the objective function by a convex model function and finds an approximate (Bregman) proximal point of the convex model. This approximate minimizer of the model function yields a descent direction, along which the next iterate is found. Complemented with an Armijo-like line search strategy, we obtain a flexible algorithm for which we prove (subsequential) convergence to a stationary point under weak assumptions on the growth of the model function error. Special instances of the algorithm with a Euclidean distance function are, for example, Gradient Descent, Forward-Backward Splitting, ProxDescent, without the common requirement of a Lipschitz continuous gradient. In addition, we consider a broad class of Bregman distance functions (generated by Legendre functions), replacing the Euclidean distance. The algorithm has a wide range of applications including many linear and non-linear inverse problems in signal/image processing and machine learning.