Large sample analysis of the median heuristic

TL;DR

This paper provides a theoretical and empirical analysis of the median heuristic for setting RBF kernel bandwidths in kernel two-sample tests, demonstrating its asymptotic normality and comparing its performance to power-maximizing methods.

Contribution

It offers the first convergence analysis showing the asymptotic normality of the median heuristic in this context and compares its effectiveness with other bandwidth selection strategies.

Findings

Median heuristic bandwidths are asymptotically normal.

Empirical results compare median heuristic and power-maximizing bandwidths.

Median heuristic performs reliably in simple test settings.

Abstract

In kernel methods, the median heuristic has been widely used as a way of setting the bandwidth of RBF kernels. While its empirical performances make it a safe choice under many circumstances, there is little theoretical understanding of why this is the case. Our aim in this paper is to advance our understanding of the median heuristic by focusing on the setting of kernel two-sample test. We collect new findings that may be of interest for both theoreticians and practitioners. In theory, we provide a convergence analysis that shows the asymptotic normality of the bandwidth chosen by the median heuristic in the setting of kernel two-sample test. Systematic empirical investigations are also conducted in simple settings, comparing the performances based on the bandwidths chosen by the median heuristic and those by the maximization of test power.