  The mean shift (MS) algorithm seeks a mode of the kernel density estimate
(KDE). This study presents a convergence guarantee of the mode estimate
sequence generated by the MS algorithm and an evaluation of the convergence
rate, under fairly mild conditions, with the help of the argument concerning
the {\L}ojasiewicz inequality. Our findings extend existing ones covering
analytic kernels and the Epanechnikov kernel. Those are significant in that
they cover the biweight kernel, which is optimal among non-negative kernels in
terms of the asymptotic statistical efficiency for the KDE-based mode
estimation.
