Convergence of Deep Neural Networks with General Activation Functions and Pooling
This work addresses a fundamental mathematical problem in deep learning theory, but it is incremental as it builds directly on prior research.
The paper tackles the convergence of deep neural networks as depth increases, extending previous work to include leaky ReLU and sigmoid activation functions and pooling, proving that a sufficient condition holds for leaky ReLU and establishing a weaker condition for sigmoid.
Deep neural networks, as a powerful system to represent high dimensional complex functions, play a key role in deep learning. Convergence of deep neural networks is a fundamental issue in building the mathematical foundation for deep learning. We investigated the convergence of deep ReLU networks and deep convolutional neural networks in two recent researches (arXiv:2107.12530, 2109.13542). Only the Rectified Linear Unit (ReLU) activation was studied therein, and the important pooling strategy was not considered. In this current work, we study the convergence of deep neural networks as the depth tends to infinity for two other important activation functions: the leaky ReLU and the sigmoid function. Pooling will also be studied. As a result, we prove that the sufficient condition established in arXiv:2107.12530, 2109.13542 is still sufficient for the leaky ReLU networks. For contractive activation functions such as the sigmoid function, we establish a weaker sufficient condition for uniform convergence of deep neural networks.