The Separation Capacity of Random Neural Networks

Abstract

Neural networks with random weights appear in a variety of machine learning applications, most prominently as the initialization of many deep learning algorithms and as a computationally cheap alternative to fully learned neural networks. In the present article, we enhance the theoretical understanding of random neural networks by addressing the following data separation problem: under what conditions can a random neural network make two classes ${\mathcal{X}}^{-},{\mathcal{X}}^{+}\subset {\mathbb{R}}^d$ (with positive distance) linearly separable? We show that a sufficiently large two-layer ReLU-network with standard Gaussian weights and uniformly distributed biases can solve this problem with high probability. Crucially, the number of required neurons is explicitly linked to geometric properties of the underlying sets ${\mathcal{X}}^{-},{\mathcal{X}}^{+}$ and their mutual arrangement. This instance-specific viewpoint allows us to overcome the usual curse of dimensionality (exponential width of the layers) in non-pathological situations where the data carries low-complexity structure. We quantify the relevant structure of the data in terms of a novel notion of mutual complexity (based on a localized version of Gaussian mean width), which leads to sound and informative separation guarantees. We connect our result with related lines of work on approximation, memorization, and generalization.

The current version of the manuscript has been accepted to Journal of Machine Learning Research

References in corpus (8)

• Deep Learning in Neural Networks: An Overview
• Small ReLU networks are powerful memorizers: a tight analysis of memorization capacity
• Random Vector Functional Link Networks for Function Approximation on Manifolds
• A Corrective View of Neural Networks: Representation, Memorization and Learning
• On the Approximation Properties of Random ReLU Features
• The Benefits of Over-parameterization at Initialization in Deep ReLU Networks
• Binarized Johnson-Lindenstrauss embeddings
• On the Approximation Power of Two-Layer Networks of Random ReLUs