Fast learning rate of multiple kernel learning: Trade-off between sparsity and smoothness

We investigate the learning rate of multiple kernel learning (MKL) with $\ell1$ and elastic-net regularizations. The elastic-net regularization is a composition of an $\ell1$-regularizer for inducing the sparsity and an $\ell2$-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 $\ell1$ 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 $\ell1$-regularization; otherwise, a faster convergence rate for the $\ell1$-regularization is shown.