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

TL;DR

This paper derives non-asymptotic excess risk bounds for deep convolutional neural networks in binary classification, highlighting their ability to handle high-dimensional data and approximate low-dimensional structures effectively.

Contribution

It provides new non-asymptotic risk bounds for CNNs, including their capacity to bypass the curse of dimensionality on low-dimensional manifolds.

Findings

Risk bounds depend polynomially on data dimension

CNNs can circumvent the curse of dimensionality on low-dimensional manifolds

Derived covering number bounds and approximation results for CNNs

Abstract

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.