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
Abstract
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 simulations for the training of ANNs with ReLU activation, it remains  even in the simplest situation of the plain vanilla GD optimization method with random initializations and ANNs with one hidden layer  an open problem to prove (or disprove) the conjecture that the risk of the GD optimization method converges in the training of such ANNs to zero as the width of the ANNs, the number of independent random initializations, and the number of GD steps increase to infinity. In this article we prove this conjecture in the situation where the probability distribution of the input data is equivalent to the continuous uniform distribution on a compact interval, where the probability distributions for the random initializations of the ANN parameters are standard normal distributions, and where the target function under consideration is continuous and piecewise affine linear. Roughly speaking, the key ingredients in our mathematical convergence analysis are (i) to prove that suitable sets of global minima of the risk functions are \emph{twice continuously differentiable submanifolds of the ANN parameter spaces}, (ii) to prove that the Hessians of the risk functions on these sets of global minima satisfy an appropriate \emph{maximal rank condition}, and, thereafter, (iii) to apply the machinery in [Fehrman, B., Gess, B., Jentzen, A., Convergence rates for the stochastic gradient descent method for nonconvex objective functions. J. Mach. Learn. Res. 21(136): 148, 2020] to establish convergence of the GD optimization method with random initializations.
 Publication:

arXiv eprints
 Pub Date:
 August 2021
 arXiv:
 arXiv:2108.04620
 Bibcode:
 2021arXiv210804620J
 Keywords:

 Mathematics  Optimization and Control;
 Computer Science  Machine Learning;
 Mathematics  Numerical Analysis
 EPrint:
 44 pages. arXiv admin note: text overlap with arXiv:2107.04479