Improved learning theory for kernel distribution regression with two-stage sampling

TL;DR

This paper advances the theoretical understanding of kernel distribution regression with two-stage sampling, providing improved error bounds and convergence rates for a broad class of kernels based on Hilbertian embeddings.

Contribution

It introduces a novel near-unbiased condition for Hilbertian embeddings, leading to tighter error bounds and improved convergence rates in kernel distribution regression.

Findings

The near-unbiased condition holds for kernels based on optimal transport and mean embedding.

New error bounds are established for the effect of two-stage sampling.

Convergence rates are strictly improved for the considered kernels.

Abstract

The distribution regression problem encompasses many important statistics and machine learning tasks, and arises in a large range of applications. Among various existing approaches to tackle this problem, kernel methods have become a method of choice. Indeed, kernel distribution regression is both computationally favorable, and supported by a recent learning theory. This theory also tackles the two-stage sampling setting, where only samples from the input distributions are available. In this paper, we improve the learning theory of kernel distribution regression. We address kernels based on Hilbertian embeddings, that encompass most, if not all, of the existing approaches. We introduce the novel near-unbiased condition on the Hilbertian embeddings, that enables us to provide new error bounds on the effect of the two-stage sampling, thanks to a new analysis. We show that this near-unbiased condition holds for three important classes of kernels, based on optimal transport and mean embedding. As a consequence, we strictly improve the existing convergence rates for these kernels. Our setting and results are illustrated by numerical experiments.