# Convergence of Deep Neural Networks with General Activation Functions and Pooling

arXiv: https://arxiv.org/abs/2205.06570 · HTML: https://mlsanity.com/paper/2205.06570/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 25 references that are in the map.
- First posted on arXiv 13 May 2022.

## Builds on (most cited first, 14 of 14)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Unsupervised Representation Learning with Deep Convolutional Generative Adversarial Networks](https://mlsanity.com/paper/1511.06434/) (2015)
- [Identity Mappings in Deep Residual Networks](https://mlsanity.com/paper/1603.05027/) (2016)
- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [Universality of Deep Convolutional Neural Networks](https://mlsanity.com/paper/1805.10769/) (2018)
- [Why and When Can Deep -- but Not Shallow -- Networks Avoid the Curse of Dimensionality: a Review](https://mlsanity.com/paper/1611.00740/) (2016)
- [Deep Network Approximation Characterized by Number of Neurons](https://mlsanity.com/paper/1906.05497/) (2019)
- [Optimal Approximation Rate of ReLU Networks in terms of Width and Depth](https://mlsanity.com/paper/2103.00502/) (2021)
- [Lipschitz Certificates for Layered Network Structures Driven by Averaged Activation Operators](https://mlsanity.com/paper/1903.01014/) (2019)
- [Error bounds for deep ReLU networks using the Kolmogorov--Arnold superposition theorem](https://mlsanity.com/paper/1906.11945/) (2019)
- [Exponential Convergence of the Deep Neural Network Approximation for Analytic Functions](https://mlsanity.com/paper/1807.00297/) (2018)
- [Convergence of Deep ReLU Networks](https://mlsanity.com/paper/2107.12530/) (2021)
- [Convergence of Deep Convolutional Neural Networks](https://mlsanity.com/paper/2109.13542/) (2021)
- [A Mathematical Introduction to Generative Adversarial Nets (GAN)](https://mlsanity.com/paper/2009.00169/) (2020)

## Abstract (arXiv)

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.