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

Simon Eberle; Arnulf Jentzen; Adrian Riekert; Georg S. Weiss; Simon Eberle; Arnulf Jentzen; Adrian Riekert; Georg S. Weiss

doi:10.3934/era.2023128

Electronic Research Archive

2023, Volume 31, Issue 5: 2519-2554. doi: 10.3934/era.2023128

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

1.
Basque Center of Applied Mathematics, Spain
2.
School of Data Science and Shenzhen Research Institute of Big Data, The Chinese University of Hong Kong, Shenzhen, China
3.
Applied Mathematics: Institute for Analysis and Numerics, University of Münster, Germany
4.
Faculty of Mathematics, AG Analysis of Partial Differential Equations, University of Duisburg-Essen, Germany

Received: 22 August 2022 Revised: 26 January 2023 Accepted: 28 January 2023 Published: 06 March 2023

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. 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. We establish this result by proving that the considered risk function is semialgebraic and, consequently, satisfies the Kurdyka-Łojasiewicz inequality, which allows us to show convergence of every non-divergent GF trajectory.
- deep learning,
- artificial intelligence,
- optimization,
- gradient flow,
- Kurdyka-Łojasiewicz inequalities
Citation: Simon Eberle, Arnulf Jentzen, Adrian Riekert, Georg S. Weiss. Existence, uniqueness, and convergence rates for gradient flows in the training of artificial neural networks with ReLU activation[J]. Electronic Research Archive, 2023, 31(5): 2519-2554. doi: 10.3934/era.2023128

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Abstract

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. 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. We establish this result by proving that the considered risk function is semialgebraic and, consequently, satisfies the Kurdyka-Łojasiewicz inequality, which allows us to show convergence of every non-divergent GF trajectory.

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