Convergence of Deep Convolutional Neural Networks
This work addresses a foundational mathematical problem in deep learning, providing theoretical insights for researchers, but it is incremental as it builds on prior studies of ReLU networks.
The paper tackles the problem of proving convergence for deep convolutional neural networks as depth increases, establishing sufficient conditions for convergence of infinite products of matrices with increasing sizes, which leads to piecewise convergence for general ReLU networks and pointwise convergence for convolutional networks.
Convergence of deep neural networks as the depth of the networks tends to infinity is fundamental in building the mathematical foundation for deep learning. In a previous study, we investigated this question for deep ReLU networks with a fixed width. This does not cover the important convolutional neural networks where the widths are increasing from layer to layer. For this reason, we first study convergence of general ReLU networks with increasing widths and then apply the results obtained to deep convolutional neural networks. It turns out the convergence reduces to convergence of infinite products of matrices with increasing sizes, which has not been considered in the literature. We establish sufficient conditions for convergence of such infinite products of matrices. Based on the conditions, we present sufficient conditions for piecewise convergence of general deep ReLU networks with increasing widths, and as well as pointwise convergence of deep ReLU convolutional neural networks.