Fast Learning Rate of Multiple Kernel Learning: Trade-Off between Sparsity and Smoothness

We investigate the learning rate of multiple kernel leaning (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 non-zero components of the ground truth is relatively small, and show sharper convergence rates than the learning rates ever shown for both $\ell_1$ and elastic-net regularizations. Our analysis shows there appears a trade-off between the sparsity and the smoothness when it comes to selecting which of $\ell_1$ and elastic-net regularizations to use; if the ground truth is smooth, the elastic-net regularization is preferred, otherwise the $\ell_1$ regularization is preferred.