Two-stage Sampled Learning Theory on Distributions

TL;DR

This paper introduces a simple, kernel-based method for distribution regression with theoretical guarantees, addressing the two-stage sampling challenge and establishing the consistency of set kernels in this context.

Contribution

It provides the first consistency proof for a kernel embedding approach to distribution regression under two-stage sampling, including classical and recent kernels.

Findings

Proves consistency of the proposed kernel embedding method.

Derives explicit convergence rates based on sample size and problem complexity.

Establishes the classical set kernel's consistency in regression.

Abstract

We focus on the distribution regression problem: regressing to a real-valued response from a probability distribution. Although there exist a large number of similarity measures between distributions, very little is known about their generalization performance in specific learning tasks. Learning problems formulated on distributions have an inherent two-stage sampled difficulty: in practice only samples from sampled distributions are observable, and one has to build an estimate on similarities computed between sets of points. To the best of our knowledge, the only existing method with consistency guarantees for distribution regression requires kernel density estimation as an intermediate step (which suffers from slow convergence issues in high dimensions), and the domain of the distributions to be compact Euclidean. In this paper, we provide theoretical guarantees for a remarkably simple algorithmic alternative to solve the distribution regression problem: embed the distributions to a reproducing kernel Hilbert space, and learn a ridge regressor from the embeddings to the outputs. Our main contribution is to prove the consistency of this technique in the two-stage sampled setting under mild conditions (on separable, topological domains endowed with kernels). For a given total number of observations, we derive convergence rates as an explicit function of the problem difficulty. As a special case, we answer a 15-year-old open question: we establish the consistency of the classical set kernel [Haussler, 1999; Gartner et. al, 2002] in regression, and cover more recent kernels on distributions, including those due to [Christmann and Steinwart, 2010].