Gradient descent provably escapes saddle points in the training of shallow ReLU networks

TL;DR

This paper extends dynamical systems theory to show that gradient descent effectively avoids saddle points and converges to global minima in shallow ReLU and leaky ReLU networks, even under relaxed regularity conditions.

Contribution

It proves a center-stable manifold theorem applicable to non-regular loss functions and demonstrates gradient descent's ability to bypass saddle points in shallow ReLU networks.

Findings

Gradient descent bypasses most saddle points in shallow ReLU networks.

Convergence to global minima is achieved under certain initialization conditions.

The results hold even when regularity conditions are relaxed.

Abstract

Dynamical systems theory has recently been applied in optimization to prove that gradient descent algorithms bypass so-called strict saddle points of the loss function. However, in many modern machine learning applications, the required regularity conditions are not satisfied. In this paper, we prove a variant of the relevant dynamical systems result, a center-stable manifold theorem, in which we relax some of the regularity requirements. We explore its relevance for various machine learning tasks, with a particular focus on shallow rectified linear unit (ReLU) and leaky ReLU networks with scalar input. Building on a detailed examination of critical points of the square integral loss function for shallow ReLU and leaky ReLU networks relative to an affine target function, we show that gradient descent circumvents most saddle points. Furthermore, we prove convergence to global minima under favourable initialization conditions, quantified by an explicit threshold on the limiting loss.