  We investigate the learning rate of multiple kernel leaning (MKL) with
elastic-net regularization, which consists 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 non-zero components of the ground truth is relatively
small, and prove that elastic-net MKL achieves the minimax learning rate on the
$\ell_2$-mixed-norm ball. Our bound is sharper than the convergence rates ever
shown, and has a property that the smoother the truth is, the faster the
convergence rate is.
