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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)…

Gradient descent (GD) type optimization methods are the standard instrument to train artificial neural networks (ANNs) with rectified linear unit (ReLU) activation. Despite the great success of GD type optimization methods in numerical…

Gradient descent (GD) type optimization schemes are the standard instruments to train fully connected feedforward artificial neural networks (ANNs) with rectified linear unit (ReLU) activation and can be considered as temporal…

In this article we study fully-connected feedforward deep ReLU ANNs with an arbitrarily large number of hidden layers and we prove convergence of the risk of the GD optimization method with random initializations in the training of such…

In many numerical simulations stochastic gradient descent (SGD) type optimization methods perform very effectively in the training of deep neural networks (DNNs) but till this day it remains an open problem of research to provide a…

Deep learning algorithms -- typically consisting of a class of deep artificial neural networks (ANNs) trained by a stochastic gradient descent (SGD) optimization method -- are nowadays an integral part in many areas of science, industry,…

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…

Gradient descent (GD) methods for the training of artificial neural networks (ANNs) belong nowadays to the most heavily employed computational schemes in the digital world. Despite the compelling success of such methods, it remains an open…

The non-convexity of the artificial neural network (ANN) training landscape brings inherent optimization difficulties. While the traditional back-propagation stochastic gradient descent (SGD) algorithm and its variants are effective in…

In this article we study the stochastic gradient descent (SGD) optimization method in the training of fully-connected feedforward artificial neural networks with ReLU activation. The main result of this work proves that the risk of the SGD…

In this article we investigate blow up phenomena for gradient descent optimization methods in the training of artificial neural networks (ANNs). Our theoretical analysis is focused on shallow ANNs with one neuron on the input layer, one…

Stochastic gradient descent (SGD) optimization methods such as the plain vanilla SGD method and the popular Adam optimizer are nowadays the method of choice in the training of artificial neural networks (ANNs). Despite the remarkable…

The field of artificial neural network (ANN) training has garnered significant attention in recent years, with researchers exploring various mathematical techniques for optimizing the training process. In particular, this paper focuses on…

A key challenge in modern deep learning theory is to explain the remarkable success of gradient-based optimization methods when training large-scale, complex deep neural networks. Though linear convergence of such methods has been proved…

Implicit deep learning has received increasing attention recently due to the fact that it generalizes the recursive prediction rules of many commonly used neural network architectures. Its prediction rule is provided implicitly based on the…

Deep learning methods - consisting of a class of deep neural networks (DNNs) trained by a stochastic gradient descent (SGD) optimization method - are nowadays key tools to solve data driven supervised learning problems. Despite the great…

Gradient descent optimization algorithms are the standard ingredients that are used to train artificial neural networks (ANNs). Even though a huge number of numerical simulations indicate that gradient descent optimization methods do indeed…

Implicit deep learning has recently become popular in the machine learning community since these implicit models can achieve competitive performance with state-of-the-art deep networks while using significantly less memory and computational…

We study the problem of training deep neural networks with Rectified Linear Unit (ReLU) activation function using gradient descent and stochastic gradient descent. In particular, we study the binary classification problem and show that for…

Neural networks with REctified Linear Unit (ReLU) activation functions (a.k.a. ReLU networks) have achieved great empirical success in various domains. Nonetheless, existing results for learning ReLU networks either pose assumptions on the…