  We investigate the learning rate of multiple kernel learning (MKL) with
$\ell_1$ and elastic-net regularizations. The elastic-net regularization is a
composition of an $\ell_1$-regularizer for inducing the sparsity and an
$\ell_2$-regularizer for controlling the smoothness. We focus on a sparse
setting where the total number of kernels is large, but the number of nonzero
components of the ground truth is relatively small, and show sharper
convergence rates than the learning rates have ever shown for both $\ell_1$ and
elastic-net regularizations. Our analysis reveals some relations between the
choice of a regularization function and the performance. If the ground truth is
smooth, we show a faster convergence rate for the elastic-net regularization
with less conditions than $\ell_1$-regularization; otherwise, a faster
convergence rate for the $\ell_1$-regularization is shown.
