Generalisation error in learning with random features and the hidden manifold model

TL;DR

This paper provides a comprehensive theoretical analysis of generalisation errors in high-dimensional learning models, including random features, neural networks, and hidden manifold models, using the replica method to derive explicit formulas.

Contribution

It introduces a novel theoretical framework using the replica method to analytically compute asymptotic generalisation errors across various high-dimensional learning scenarios.

Findings

Double descent behaviour with a peak at the interpolation threshold.

Orthogonal projections outperform Gaussian in random features learning.

Correlations in hidden manifold data significantly affect learning performance.

Abstract

We study generalised linear regression and classification for a synthetically generated dataset encompassing different problems of interest, such as learning with random features, neural networks in the lazy training regime, and the hidden manifold model. We consider the high-dimensional regime and using the replica method from statistical physics, we provide a closed-form expression for the asymptotic generalisation performance in these problems, valid in both the under- and over-parametrised regimes and for a broad choice of generalised linear model loss functions. In particular, we show how to obtain analytically the so-called double descent behaviour for logistic regression with a peak at the interpolation threshold, we illustrate the superiority of orthogonal against random Gaussian projections in learning with random features, and discuss the role played by correlations in the data generated by the hidden manifold model. Beyond the interest in these particular problems, the theoretical formalism introduced in this manuscript provides a path to further extensions to more complex tasks.