  Random Fourier features is a widely used, simple, and effective technique for
scaling up kernel methods. The existing theoretical analysis of the approach,
however, remains focused on specific learning tasks and typically gives
pessimistic bounds which are at odds with the empirical results. We tackle
these problems and provide the first unified risk analysis of learning with
random Fourier features using the squared error and Lipschitz continuous loss
functions. In our bounds, the trade-off between the computational cost and the
expected risk convergence rate is problem specific and expressed in terms of
the regularization parameter and the \emph{number of effective degrees of
freedom}. We study both the standard random Fourier features method for which
we improve the existing bounds on the number of features required to guarantee
the corresponding minimax risk convergence rate of kernel ridge regression, as
well as a data-dependent modification which samples features proportional to
\emph{ridge leverage scores} and further reduces the required number of
features. As ridge leverage scores are expensive to compute, we devise a simple
approximation scheme which provably reduces the computational cost without loss
of statistical efficiency.
