Large sample analysis of the median heuristic
This work addresses a theoretical gap for researchers and practitioners using kernel methods, but it is incremental as it builds on existing heuristic usage without introducing a new method.
The paper tackles the lack of theoretical understanding of the median heuristic for setting RBF kernel bandwidths in kernel methods, focusing on kernel two-sample tests, and provides a convergence analysis showing asymptotic normality of the chosen bandwidth along with empirical comparisons to bandwidths maximizing test power.
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.