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

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We investigate the learning rate of multiple kernel learning (MKL) with ℓ₁ and elastic-net regularizations. The elastic-net regularization is a composition of an ℓ₁-regularizer for inducing the sparsity and an ℓ₂-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 ℓ₁ 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 ℓ₁-regularization; otherwise, a faster convergence rate for the ℓ₁-regularization is shown.