VC dimension of partially quantized neural networks in the overparametrized regime

TL;DR

This paper introduces hyperplane arrangement neural networks (HANNs), a class of partially quantized networks with VC dimension smaller than their number of weights, explaining their good generalization in overparametrized regimes.

Contribution

The paper demonstrates that HANNs can have significantly smaller VC dimension than their number of weights while maintaining high expressivity and competitive performance.

Findings

HANNs have VC dimension smaller than the number of weights.

Empirical risk minimization over HANNs achieves minimax classification rates.

HANNs match state-of-the-art performance on UCI datasets.

Abstract

Vapnik-Chervonenkis (VC) theory has so far been unable to explain the small generalization error of overparametrized neural networks. Indeed, existing applications of VC theory to large networks obtain upper bounds on VC dimension that are proportional to the number of weights, and for a large class of networks, these upper bound are known to be tight. In this work, we focus on a class of partially quantized networks that we refer to as hyperplane arrangement neural networks (HANNs). Using a sample compression analysis, we show that HANNs can have VC dimension significantly smaller than the number of weights, while being highly expressive. In particular, empirical risk minimization over HANNs in the overparametrized regime achieves the minimax rate for classification with Lipschitz posterior class probability. We further demonstrate the expressivity of HANNs empirically. On a panel of 121 UCI datasets, overparametrized HANNs match the performance of state-of-the-art full-precision models.