Consistency Analysis of an Empirical Minimum Error Entropy Algorithm
This work addresses theoretical guarantees for MEE algorithms in machine learning regression, providing insights into consistency conditions that are incremental to existing statistical learning theory.
The paper analyzes the consistency of an empirical minimum error entropy (MEE) algorithm in regression, showing that error entropy consistency always holds with bandwidth tending to 0, while regression consistency holds for homoskedastic models but not always for heteroskedastic ones, with a surprising result that it always holds if bandwidth tends to infinity.
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.