Convergence analysis for gradient flows in the training of artificial neural networks with ReLU activation

Abstract

Gradient descent (GD) type optimization schemes are the standard methods to train artificial neural networks (ANNs) with rectified linear unit (ReLU) activation. Such schemes can be considered as discretizations of gradient flows (GFs) associated to the training of ANNs with ReLU activation and most of the key difficulties in the mathematical convergence analysis of GD type optimization schemes in the training of ANNs with ReLU activation seem to be already present in the dynamics of the corresponding GF differential equations. It is the key subject of this work to analyze such GF differential equations in the training of ANNs with ReLU activation and three layers (one input layer, one hidden layer, and one output layer). In particular, in this article we prove in the case where the target function is possibly multi-dimensional and continuous and in the case where the probability distribution of the input data is absolutely continuous with respect to the Lebesgue measure that the risk of every bounded GF trajectory converges to the risk of a critical point. In addition, in this article we show in the case of a 1-dimensional affine linear target function and in the case where the probability distribution of the input data coincides with the standard uniform distribution that the risk of every bounded GF trajectory converges to zero if the initial risk is sufficiently small. Finally, in the special situation where there is only one neuron on the hidden layer (1-dimensional hidden layer) we strengthen the above named result for affine linear target functions by proving that that the risk of every (not necessarily bounded) GF trajectory converges to zero if the initial risk is sufficiently small.

37 pages

References in corpus (4)

• Gradient Dynamics of Shallow Univariate ReLU Networks
• A proof of convergence for gradient descent in the training of artificial neural networks for constant target functions
• A proof of convergence for stochastic gradient descent in the training of artificial neural networks with ReLU activation for constant target functions
• Landscape analysis for shallow neural networks: complete classification of critical points for affine target functions

Cited by in corpus (5)

• On the existence of global minima and convergence analyses for gradient descent methods in the training of deep neural networks
• Existence, uniqueness, and convergence rates for gradient flows in the training of artificial neural networks with ReLU activation
• Predicting the fatigue life of asphalt concrete using neural networks
• 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
• Learning Gradient Flow: Using Equation Discovery to Accelerate Engineering Optimization