# Existence, uniqueness, and convergence rates for gradient flows in the training of artificial neural networks with ReLU activation

arXiv: https://arxiv.org/abs/2108.08106 · HTML: https://mlsanity.com/paper/2108.08106/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 41 references that are in the map.
- First posted on arXiv 18 Aug 2021.

## Builds on (most cited first, 22 of 22)

- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [On Lazy Training in Differentiable Programming](https://mlsanity.com/paper/1812.07956/) (2018)
- [Gradient Descent Provably Optimizes Over-parameterized Neural Networks](https://mlsanity.com/paper/1810.02054/) (2018)
- [Fine-Grained Analysis of Optimization and Generalization for Overparameterized Two-Layer Neural Networks](https://mlsanity.com/paper/1901.08584/) (2019)
- [Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak-Lojasiewicz Condition](https://mlsanity.com/paper/1608.04636/) (2016)
- [On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport](https://mlsanity.com/paper/1805.09545/) (2018)
- [Non-strongly-convex smooth stochastic approximation with convergence rate O(1/n)](https://mlsanity.com/paper/1306.2119/) (2013)
- [Towards a Mathematical Understanding of Neural Network-Based Machine Learning: what we know and what we don't](https://mlsanity.com/paper/2009.10713/) (2020)
- [Fast Convergence of Natural Gradient Descent for Overparameterized Neural Networks](https://mlsanity.com/paper/1905.10961/) (2019)
- [Convergence rates for the stochastic gradient descent method for non-convex objective functions](https://mlsanity.com/paper/1904.01517/) (2019)
- [Sparse Optimization on Measures with Over-parameterized Gradient Descent](https://mlsanity.com/paper/1907.10300/) (2019)
- [A Comparative Analysis of the Optimization and Generalization Property of Two-layer Neural Network and Random Feature Models Under Gradient Descent Dynamics](https://mlsanity.com/paper/1904.04326/) (2019)
- [Stochastic Gradient Descent for Nonconvex Learning without Bounded Gradient Assumptions](https://mlsanity.com/paper/1902.00908/) (2019)
- [Convergence of stochastic gradient descent schemes for Lojasiewicz-landscapes](https://mlsanity.com/paper/2102.09385/) (2021)
- [A Dynamical Central Limit Theorem for Shallow Neural Networks](https://mlsanity.com/paper/2008.09623/) (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)
- [Stopping Criteria for, and Strong Convergence of, Stochastic Gradient Descent on Bottou-Curtis-Nocedal Functions](https://mlsanity.com/paper/2004.00475/) (2020)
- [Landscape analysis for shallow neural networks: complete classification of critical points for affine target functions](https://mlsanity.com/paper/2103.10922/) (2021)
- [Convergence analysis for gradient flows in the training of artificial neural networks with ReLU activation](https://mlsanity.com/paper/2107.04479/) (2021)
- [Unifying abstract inexact convergence theorems and block coordinate variable metric iPiano](https://mlsanity.com/paper/1602.07283/) (2016)
- [A proof of convergence for the gradient descent optimization method with random initializations in the training of neural networks with ReLU activation for piecewise linear target functions](https://mlsanity.com/paper/2108.04620/) (2021)
- [A proof of convergence for stochastic gradient descent in the training of artificial neural networks with ReLU activation for constant target functions](https://mlsanity.com/paper/2104.00277/) (2021)

## Built on by (most cited first, 9 of 9)

- [Gradient flow dynamics of shallow ReLU networks for square loss and orthogonal inputs](https://mlsanity.com/paper/2206.00939/) (2022)
- [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 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 minimizers in shallow residual ReLU neural network optimization landscapes](https://mlsanity.com/paper/2302.14690/) (2023)
- [Topological obstruction to the training of shallow ReLU neural networks](https://mlsanity.com/paper/2410.14837/) (2024)
- [Topology and Geometry of the Learning Space of ReLU Networks: Connectivity and Singularities](https://mlsanity.com/paper/2602.00693/) (2026)
- [On the existence of optimal shallow feedforward networks with ReLU activation](https://mlsanity.com/paper/2303.03950/) (2023)
- [On bounds for norms of reparameterized ReLU artificial neural network parameters: sums of fractional powers of the Lipschitz norm control the network parameter vector](https://mlsanity.com/paper/2206.13646/) (2022)
- [Normalized gradient flow optimization in the training of ReLU artificial neural networks](https://mlsanity.com/paper/2207.06246/) (2022)

## Abstract (arXiv)

The training of artificial neural networks (ANNs) with rectified linear unit (ReLU) activation via gradient descent (GD) type optimization schemes is nowadays a common industrially relevant procedure. Till this day in the scientific literature there is in general no mathematical convergence analysis which explains the numerical success of GD type optimization schemes in the training of ANNs with ReLU activation. GD type optimization schemes can be regarded as temporal discretization methods for the gradient flow (GF) differential equations associated to the considered optimization problem and, in view of this, it seems to be a natural direction of research to first aim to develop a mathematical convergence theory for time-continuous GF differential equations and, thereafter, to aim to extend such a time-continuous convergence theory to implementable time-discrete GD type optimization methods. In this article we establish two basic results for GF differential equations in the training of fully-connected feedforward ANNs with one hidden layer and ReLU activation. In the first main result of this article we establish in the training of such ANNs under the assumption that the probability distribution of the input data of the considered supervised learning problem is absolutely continuous with a bounded density function that every GF differential equation admits for every initial value a solution which is also unique among a suitable class of solutions. In the second main result of this article we prove in the training of such ANNs under the assumption that the target function and the density function of the probability distribution of the input data are piecewise polynomial that every non-divergent GF trajectory converges with an appropriate rate of convergence to a critical point and that the risk of the non-divergent GF trajectory converges with rate 1 to the risk of the critical point.