# Non-asymptotic Excess Risk Bounds for Classification with Deep Convolutional Neural Networks

arXiv: https://arxiv.org/abs/2105.00292 · HTML: https://mlsanity.com/paper/2105.00292/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 82 references that are in the map.
- First posted on arXiv 1 May 2021.

## Builds on (most cited first, 26 of 26)

- [Google's Neural Machine Translation System: Bridging the Gap between Human and Machine Translation](https://mlsanity.com/paper/1609.08144/) (2016)
- [Deep Learning in Neural Networks: An Overview](https://mlsanity.com/paper/1404.7828/) (2014)
- [Spectrally-normalized margin bounds for neural networks](https://mlsanity.com/paper/1706.08498/) (2017)
- [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)
- [The Expressive Power of Neural Networks: A View from the Width](https://mlsanity.com/paper/1709.02540/) (2017)
- [Stronger generalization bounds for deep nets via a compression approach](https://mlsanity.com/paper/1802.05296/) (2018)
- [On the Expressive Power of Deep Neural Networks](https://mlsanity.com/paper/1606.05336/) (2016)
- [Size-Independent Sample Complexity of Neural Networks](https://mlsanity.com/paper/1712.06541/) (2017)
- [Deep learning: a statistical viewpoint](https://mlsanity.com/paper/2103.09177/) (2021)
- [Universality of Deep Convolutional Neural Networks](https://mlsanity.com/paper/1805.10769/) (2018)
- [Optimal approximation of continuous functions by very deep ReLU networks](https://mlsanity.com/paper/1802.03620/) (2018)
- [Deep Network Approximation Characterized by Number of Neurons](https://mlsanity.com/paper/1906.05497/) (2019)
- [Deep ReLU network approximation of functions on a manifold](https://mlsanity.com/paper/1908.00695/) (2019)
- [Nonparametric Regression on Low-Dimensional Manifolds using Deep ReLU Networks : Function Approximation and Statistical Recovery](https://mlsanity.com/paper/1908.01842/) (2019)
- [Fast convergence rates of deep neural networks for classification](https://mlsanity.com/paper/1812.03599/) (2018)
- [Approximation and Non-parametric Estimation of ResNet-type Convolutional Neural Networks](https://mlsanity.com/paper/1903.10047/) (2019)
- [Nonlinear Approximation via Compositions](https://mlsanity.com/paper/1902.10170/) (2019)
- [On Tighter Generalization Bound for Deep Neural Networks: CNNs, ResNets, and Beyond](https://mlsanity.com/paper/1806.05159/) (2018)
- [Theory of Deep Convolutional Neural Networks II: Spherical Analysis](https://mlsanity.com/paper/2007.14285/) (2020)
- [Generalization Bounds for Convolutional Neural Networks](https://mlsanity.com/paper/1910.01487/) (2019)
- [Fast classification rates without standard margin assumptions](https://mlsanity.com/paper/1910.12756/) (2019)
- [Understanding Generalization and Optimization Performance of Deep CNNs](https://mlsanity.com/paper/1805.10767/) (2018)
- [Adaptive Approximation and Generalization of Deep Neural Network with Intrinsic Dimensionality](https://mlsanity.com/paper/1907.02177/) (2019)
- [Sparse Distance Weighted Discrimination](https://mlsanity.com/paper/1501.06066/) (2015)
- [Convex Risk Minimization and Conditional Probability Estimation](https://mlsanity.com/paper/1506.04513/) (2015)

## Built on by (most cited first, 4 of 4)

- [Convergence rates of deep ReLU networks for multiclass classification](https://mlsanity.com/paper/2108.00969/) (2021)
- [Classification with Deep Neural Networks and Logistic Loss](https://mlsanity.com/paper/2307.16792/) (2023)
- [Non-Asymptotic Guarantees for Robust Statistical Learning under Infinite Variance Assumption](https://mlsanity.com/paper/2201.03182/) (2022)
- [A Theory for Conditional Generative Modeling on Multiple Data Sources](https://mlsanity.com/paper/2502.14583/) (2025)

## Abstract (arXiv)

In this paper, we consider the problem of binary classification with a class of general deep convolutional neural networks, which includes fully-connected neural networks and fully convolutional neural networks as special cases. We establish non-asymptotic excess risk bounds for a class of convex surrogate losses and target functions with different modulus of continuity. An important feature of our results is that we clearly define the prefactors of the risk bounds in terms of the input data dimension and other model parameters and show that they depend polynomially on the dimensionality in some important models. We also show that the classification methods with CNNs can circumvent the curse of dimensionality if the input data is supported on an approximate low-dimensional manifold. To establish these results, we derive an upper bound for the covering number for the class of general convolutional neural networks with a bias term in each convolutional layer, and derive new results on the approximation power of CNNs for any uniformly-continuous target functions. These results provide further insights into the complexity and the approximation power of general convolutional neural networks, which are of independent interest and may have other applications. Finally, we apply our general results to analyze the non-asymptotic excess risk bounds for four widely used methods with different loss functions using CNNs, including the least squares, the logistic, the exponential and the SVM hinge losses.