Consistency Analysis of an Empirical Minimum Error Entropy Algorithm

TL;DR

This paper analyzes the consistency properties of an empirical minimum error entropy algorithm in regression, revealing conditions under which it approximates the true regression function and highlighting differences between homoskedastic and heteroskedastic models.

Contribution

It introduces two types of consistency for the MEE algorithm and establishes their relationships, including surprising results about regression consistency with varying bandwidth parameters.

Findings

Error entropy consistency always holds as bandwidth tends to zero.

Regression consistency is implied by error entropy consistency in homoskedastic models.

Regression consistency holds with large bandwidth, regardless of noise heteroskedasticity.

Abstract

In this paper we study the consistency of an empirical minimum error entropy (MEE) algorithm in a regression setting. We introduce two types of consistency. The error entropy consistency, which requires the error entropy of the learned function to approximate the minimum error entropy, is shown to be always true if the bandwidth parameter tends to 0 at an appropriate rate. The regression consistency, which requires the learned function to approximate the regression function, however, is a complicated issue. We prove that the error entropy consistency implies the regression consistency for homoskedastic models where the noise is independent of the input variable. But for heteroskedastic models, a counterexample is used to show that the two types of consistency do not coincide. A surprising result is that the regression consistency is always true, provided that the bandwidth parameter tends to infinity at an appropriate rate. Regression consistency of two classes of special models is shown to hold with fixed bandwidth parameter, which further illustrates the complexity of regression consistency of MEE. Fourier transform plays crucial roles in our analysis.