# Optimal Approximation and Learning Rates for Deep Convolutional Neural Networks

arXiv: https://arxiv.org/abs/2308.03259 · HTML: https://mlsanity.com/paper/2308.03259/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 24 references that are in the map.
- First posted on arXiv 7 Aug 2023.

## Builds on (most cited first, 13 of 13)

- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [Nonparametric regression using deep neural networks with ReLU activation function](https://mlsanity.com/paper/1708.06633/) (2017)
- [Universality of Deep Convolutional Neural Networks](https://mlsanity.com/paper/1805.10769/) (2018)
- [Deep Network Approximation for Smooth Functions](https://mlsanity.com/paper/2001.03040/) (2020)
- [Universal Function Approximation by Deep Neural Nets with Bounded Width and ReLU Activations](https://mlsanity.com/paper/1708.02691/) (2017)
- [The phase diagram of approximation rates for deep neural networks](https://mlsanity.com/paper/1906.09477/) (2019)
- [Theory of Deep Convolutional Neural Networks II: Spherical Analysis](https://mlsanity.com/paper/2007.14285/) (2020)
- [Theory of Deep Convolutional Neural Networks III: Approximating Radial Functions](https://mlsanity.com/paper/2107.00896/) (2021)
- [Universal Consistency of Deep Convolutional Neural Networks](https://mlsanity.com/paper/2106.12498/) (2021)
- [Depth Selection for Deep ReLU Nets in Feature Extraction and Generalization](https://mlsanity.com/paper/2004.00245/) (2020)
- [Realization of spatial sparseness by deep ReLU nets with massive data](https://mlsanity.com/paper/1912.07464/) (2019)
- [Realizing data features by deep nets](https://mlsanity.com/paper/1901.00130/) (2019)
- [Deep Convolutional Neural Networks with Zero-Padding: Feature Extraction and Learning](https://mlsanity.com/paper/2307.16203/) (2023)

## Built on by (most cited first, 1 of 1)

- [Component-based Sketching for Deep ReLU Nets](https://mlsanity.com/paper/2409.14174/) (2024)

## Abstract (arXiv)

This paper focuses on approximation and learning performance analysis for deep convolutional neural networks with zero-padding and max-pooling. We prove that, to approximate r-smooth function, the approximation rates of deep convolutional neural networks with depth L are of order  (L2/log L)-2r/d , which is optimal up to a logarithmic factor. Furthermore, we deduce almost optimal learning rates for implementing empirical risk minimization over deep convolutional neural networks.