Learning Deep Models: Critical Points and Local Openness

Maher Nouiehed, Meisam Razaviyayn

Abstract: With the increasing interest in deeper understanding of the loss surface of many non-convex deep models, this paper presents a unifying framework to study the local/global optima equivalence of the optimization problems arising from training of such non-convex models. Using the "local openness" property of the underlying training models, we provide simple sufficient conditions under which any local optimum of the resulting optimization problem is globally optimal. We first completely characterize the local openness of matrix multiplication mapping in its range. Then we use our characterization to: 1) show that every local optimum of two layer linear networks is globally optimal. Unlike many existing results in the literature, our result requires no assumption on the target data matrix Y, and input data matrix X. 2) develop almost complete characterization of the local/global optima equivalence of multi-layer linear neural networks. We provide various counterexamples to show the necessity of each of our assumptions. 3) show global/local optima equivalence of non-linear deep models having certain pyramidal structure. Unlike some existing works, our result requires no assumption on the differentiability of the activation functions and can go beyond "full-rank" cases.

Keywords: Training Deep Models, Non-convex Optimization, Local and Global Equivalence, Local Openness