Risk bounds when learning infinitely many response functions by ordinary linear regression

TL;DR

This paper establishes risk bounds for learning infinitely many response functions simultaneously using high-dimensional linear regression, providing convergence guarantees even as feature space dimension grows with sample size.

Contribution

It introduces new convergence guarantees for multiple response functions in high-dimensional linear regression with infinite response models and finite VC dimension.

Findings

Provides uniform convergence bounds for excess risk

Allows feature space dimension to grow with sample size

Applicable to building multiple surrogate models efficiently

Abstract

Consider the problem of learning a large number of response functions simultaneously based on the same input variables. The training data consist of a single independent random sample of the input variables drawn from a common distribution together with the associated responses. The input variables are mapped into a high-dimensional linear space, called the feature space, and the response functions are modelled as linear functionals of the mapped features, with coefficients calibrated via ordinary least squares. We provide convergence guarantees on the worst-case excess prediction risk by controlling the convergence rate of the excess risk uniformly in the response function. The dimension of the feature map is allowed to tend to infinity with the sample size. The collection of response functions, although potentially infinite, is supposed to have a finite Vapnik-Chervonenkis dimension. The bound derived can be applied when building multiple surrogate models in a reasonable computing time.