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

arXiv: https://arxiv.org/abs/2207.06246 · HTML: https://mlsanity.com/paper/2207.06246/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 21 references that are in the map.
- First posted on arXiv 13 Jul 2022.

## Builds on (most cited first, 9 of 9)

- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Implicit Bias of Gradient Descent for Wide Two-layer Neural Networks Trained with the Logistic Loss](https://mlsanity.com/paper/2002.04486/) (2020)
- [On the Convergence of Gradient Descent Training for Two-layer ReLU-networks in the Mean Field Regime](https://mlsanity.com/paper/2005.13530/) (2020)
- [A proof of convergence for gradient descent in the training of artificial neural networks for constant target functions](https://mlsanity.com/paper/2102.09924/) (2021)
- [On the existence of global minima and convergence analyses for gradient descent methods in the training of deep neural networks](https://mlsanity.com/paper/2112.09684/) (2021)
- [Convergence analysis for gradient flows in the training of artificial neural networks with ReLU activation](https://mlsanity.com/paper/2107.04479/) (2021)
- [Existence, uniqueness, and convergence rates for gradient flows in the training of artificial neural networks with ReLU activation](https://mlsanity.com/paper/2108.08106/) (2021)
- [Convergence proof for stochastic gradient descent in the training of deep neural networks with ReLU activation for constant target functions](https://mlsanity.com/paper/2112.07369/) (2021)
- [On the existence of infinitely many realization functions of non-global local minima in the training of artificial neural networks with ReLU activation](https://mlsanity.com/paper/2202.11481/) (2022)

## Abstract (arXiv)

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  ℝ ni x mapsto max x, 0  ∈ 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 ∞  -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.