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In spite of the accomplishments of deep learning based algorithms in numerousapplications and very broad corresponding research interest, at the momentthere is still no rigorous understanding of the reasons why such algorithmsproduce useful results in certain situations. A thorough mathematical analysisof deep learning based algorithms seems to be crucial in order to improve ourunderstanding and to make their implementation more effective and efficient. Inthis article we provide a mathematically rigorous full error analysis of deeplearning based empirical risk minimisation with quadratic loss function in theprobabilistically strong sense, where the underlying deep neural networks aretrained using stochastic gradient descent with random initialisation. Theconvergence speed we obtain is presumably far from optimal and suffers underthe curse of dimensionality. To the best of our knowledge, we establish,however, the first full error analysis in the scientific literature for a deeplearning based algorithm in the probabilistically strong sense and, moreover,the first full error analysis in the scientific literature for a deep learningbased algorithm where stochastic gradient descent with random initialisation isthe employed optimisation method.

Overall error analysis for the training of deep neural networks via stochastic gradient descent with random initialisation