Distribution-Free Distribution Regression

TL;DR

This paper develops distribution-free methods for distribution regression, allowing for unknown error distributions and providing theoretical guarantees on prediction risk convergence under certain conditions.

Contribution

It introduces distribution-free distribution regression methods with proven risk convergence rates without assuming specific error or covariate distributions.

Findings

Risk converges to zero at a polynomial rate when the effective dimension is small.

Theoretical framework applies to models with unknown error and covariate distributions.

Provides conditions under which distribution-free distribution regression is effective.

Abstract

`Distribution regression' refers to the situation where a response Y depends on a covariate P where P is a probability distribution. The model is Y=f(P) + mu where f is an unknown regression function and mu is a random error. Typically, we do not observe P directly, but rather, we observe a sample from P. In this paper we develop theory and methods for distribution-free versions of distribution regression. This means that we do not make distributional assumptions about the error term mu and covariate P. We prove that when the effective dimension is small enough (as measured by the doubling dimension), then the excess prediction risk converges to zero with a polynomial rate.