Networks with Finite VC Dimension: Pro and Contra

TL;DR

This paper examines the trade-offs of using neural networks with finite VC dimension, highlighting their approximation capabilities, learning consistency, and the impact of network depth on performance in large data settings.

Contribution

It provides a theoretical analysis of the effects of finite VC dimension on neural network approximation and learning, including new insights into their deterministic behavior and practical limitations.

Findings

Finite VC dimension aids uniform convergence but may limit approximation.

Approximation and empirical errors behave almost deterministically in high dimensions.

Network depth with ReLU units influences accuracy and learning consistency.

Abstract

Approximation and learning of classifiers of large data sets by neural networks in terms of high-dimensional geometry and statistical learning theory are investigated. The influence of the VC dimension of sets of input-output functions of networks on approximation capabilities is compared with its influence on consistency in learning from samples of data. It is shown that, whereas finite VC dimension is desirable for uniform convergence of empirical errors, it may not be desirable for approximation of functions drawn from a probability distribution modeling the likelihood that they occur in a given type of application. Based on the concentration-of-measure properties of high dimensional geometry, it is proven that both errors in approximation and empirical errors behave almost deterministically for networks implementing sets of input-output functions with finite VC dimensions in processing large data sets. Practical limitations of the universal approximation property, the trade-offs between the accuracy of approximation and consistency in learning from data, and the influence of depth of networks with ReLU units on their accuracy and consistency are discussed.