Learning algorithm analysis for deep neural network with ReLu activation functions

In the article, emphasis is put on the modern artificial neural network structure, which in the literature is known as a deep neural network. Network includes more than one hidden layer and comprises many standard modules with ReLu nonlinear activation function. A learning algorithm includes two standard steps, forward and backward, and its effectiveness depends on the way the learning error is transported back through all the layers to the first layer. Taking into account all the dimensionalities of matrixes and the nonlinear characteristics of ReLu activation function, the problem is very difficult from a theoretic point of view. To implement simple assumptions in the analysis, formal formulas are used to describe relations between the structure of every layer and the internal input vector. In practice tasks, neural networks’ internal layer matrixes with ReLu activations function, include a lot of null value of weight coefficients. This phenomenon has a negatives impact on the effectiveness of the learning algorithm convergences. A theoretical analysis could help to build more effective algorithms.