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In recent years, artificial neural networks have developed into a powerfultool for dealing with a multitude of problems for which classical solutionapproaches reach their limits. However, it is still unclear why randomlyinitialized gradient descent optimization algorithms, such as the well-knownbatch gradient descent, are able to achieve zero training loss in manysituations even though the objective function is non-convex and non-smooth. Oneof the most promising approaches to solving this problem in the field ofsupervised learning is the analysis of gradient descent optimization in theso-called overparameterized regime. In this article we provide a furthercontribution to this area of research by considering overparameterizedfully-connected rectified artificial neural networks with biases. Specifically,we show that for a fixed number of training data the mean squared error usingbatch gradient descent optimization applied to such a randomly initializedartificial neural network converges to zero at a linear convergence rate aslong as the width of the artificial neural network is large enough, thelearning rate is small enough, and the training input data are pairwiselinearly independent.

Convergence rates for gradient descent in the training of overparameterized artificial neural networks with biases