Generalization Bounds for Sparse Random Feature Expansions

TL;DR

This paper introduces sparse random feature expansions leveraging compressive sensing to improve generalization in data-scarce scenarios, outperforming shallow networks in scientific machine learning tasks.

Contribution

It proposes a novel sparse random feature method with theoretical guarantees, enhancing accuracy and efficiency in low-data settings compared to existing random feature approaches.

Findings

Sparse features with random sparse weights improve bounds for low order functions.

The method outperforms shallow networks in scientific machine learning tasks.

Generalization bounds depend on spectral properties and structural conditions.

Abstract

Random feature methods have been successful in various machine learning tasks, are easy to compute, and come with theoretical accuracy bounds. They serve as an alternative approach to standard neural networks since they can represent similar function spaces without a costly training phase. However, for accuracy, random feature methods require more measurements than trainable parameters, limiting their use for data-scarce applications or problems in scientific machine learning. This paper introduces the sparse random feature expansion to obtain parsimonious random feature models. Specifically, we leverage ideas from compressive sensing to generate random feature expansions with theoretical guarantees even in the data-scarce setting. In particular, we provide generalization bounds for functions in a certain class (that is dense in a reproducing kernel Hilbert space) depending on the number of samples and the distribution of features. The generalization bounds improve with additional structural conditions, such as coordinate sparsity, compact clusters of the spectrum, or rapid spectral decay. In particular, by introducing sparse features, i.e. features with random sparse weights, we provide improved bounds for low order functions. We show that the sparse random feature expansions outperforms shallow networks in several scientific machine learning tasks.