We propose an efficient multikernel adaptive filtering algorithm with double regularizers, providing a novel pathway towards online model selection and learning. The task is the challenging nonlinear adaptive filtering under no knowledge about a suitable kernel. Under this limited-knowledge assumption on an underlying model of a system of interest, many possible kernels are employed and one of the regularizers, a block ℓ1 norm for kernel groups, contributes to selecting a proper model (relevant kernels) in online and adaptive fashion, preventing a nonlinear filter from overfitting to noisy data. The other regularizer is the block ℓ1 norm for data groups, contributing to updating the dictionary adaptively. As the resulting cost function contains two nonsmooth (but proximable) terms, we approximate the latter regularizer by its Moreau envelope and apply the adaptive proximal forwardbackward splitting method to the approximated cost function. Numerical examples show the efficacy of the proposed algorithm.