Sharp Convergence Rate and Support Consistency of Multiple Kernel Learning with Sparse and Dense Regularization

Abstract: We theoretically investigate the convergence rate and support consistency (i.e., correctly identifying the subset of non-zero coefficients in the large sample limit) of multiple kernel learning (MKL). We focus on MKL with block-l1 regularization (inducing sparse kernel combination), block-l2 regularization (inducing uniform kernel combination), and elastic-net regularization (including both block-l1 and block-l2 regularization). For the case where the true kernel combination is sparse, we show a sharper convergence rate of the block-l1 and elastic-net MKL methods than the existing rate for block-l1 MKL. We further show that elastic-net MKL requires a milder condition for being consistent than block-l1 MKL. For the case where the optimal kernel combination is not exactly sparse, we prove that elastic-net MKL can achieve a faster convergence rate than the block-l1 and block-l2 MKL methods by carefully controlling the balance between the block-l1and block-l2 regularizers. Thus, our theoretical results overall suggest the use of elastic-net regularization in MKL.