Normalized gradient flow optimization in the training of ReLU artificial neural networks

TL;DR

This paper introduces a modified gradient flow approach for training ReLU neural networks by restricting dynamics to a special submanifold, leading to better regularity and boundedness properties in shallow networks.

Contribution

It proposes a new submanifold-based gradient flow method for ReLU ANNs and proves boundedness of trajectories in shallow cases, addressing open problems in standard training.

Findings

Gradient flow restricted to a submanifold shows improved regularity.

Boundedness of gradient trajectories is proven for shallow networks with Lipschitz targets.

Standard gradient flow's boundedness remains an open problem.

Abstract

The training of artificial neural networks (ANNs) is nowadays a highly relevant algorithmic procedure with many applications in science and industry. Roughly speaking, ANNs can be regarded as iterated compositions between affine linear functions and certain fixed nonlinear functions, which are usually multidimensional versions of a one-dimensional so-called activation function. The most popular choice of such a one-dimensional activation function is the rectified linear unit (ReLU) activation function which maps a real number to its positive part $ \mathbb{R} \ni x \mapsto \max\{ x, 0 \} \in \mathbb{R} $. In this article we propose and analyze a modified variant of the standard training procedure of such ReLU ANNs in the sense that we propose to restrict the negative gradient flow dynamics to a large submanifold of the ANN parameter space, which is a strict $ C^{ \infty } $-submanifold of the entire ANN parameter space that seems to enjoy better regularity properties than the entire ANN parameter space but which is also sufficiently large and sufficiently high dimensional so that it can represent all ANN realization functions that can be represented through the entire ANN parameter space. In the special situation of shallow ANNs with just one-dimensional ANN layers we also prove for every Lipschitz continuous target function that every gradient flow trajectory on this large submanifold of the ANN parameter space is globally bounded. For the standard gradient flow on the entire ANN parameter space with Lipschitz continuous target functions it remains an open problem of research to prove or disprove the global boundedness of gradient flow trajectories even in the situation of shallow ANNs with just one-dimensional ANN layers.