Every Local Minimum Value is the Global Minimum Value of Induced Model in Non-convex Machine Learning

TL;DR

This paper proves that in nonconvex machine learning models, every local minimum is globally optimal with respect to the perturbable gradient basis, supporting the theoretical foundation of deep neural networks and related models.

Contribution

It establishes that all differentiable local minima are globally optimal for the perturbable gradient basis model, unifying and extending existing theoretical results.

Findings

Local minima are globally optimal for the perturbable gradient basis.

Results apply directly to practical deep neural networks without modifications.

Improves and unifies previous theoretical results on neural network optimization.

Abstract

For nonconvex optimization in machine learning, this article proves that every local minimum achieves the globally optimal value of the perturbable gradient basis model at any differentiable point. As a result, nonconvex machine learning is theoretically as supported as convex machine learning with a handcrafted basis in terms of the loss at differentiable local minima, except in the case when a preference is given to the handcrafted basis over the perturbable gradient basis. The proofs of these results are derived under mild assumptions. Accordingly, the proven results are directly applicable to many machine learning models, including practical deep neural networks, without any modification of practical methods. Furthermore, as special cases of our general results, this article improves or complements several state-of-the-art theoretical results on deep neural networks, deep residual networks, and overparameterized deep neural networks with a unified proof technique and novel geometric insights. A special case of our results also contributes to the theoretical foundation of representation learning.