Abstract: We develop and analyze an algorithm for nonparametric estimation of divergence functionals and the density ratio of two probability distributions. Our method is based on a non-asymptotic variational characterization of $f$-divergences, which turns the estimation problem into a penalized convex risk minimization problem. We present a derivation of our kernel-based estimation algorithm and an analysis of convergence rates for the estimator. Our simulation results demonstrate the convergence behavior of our method, which compares favorably with existing methods in the literature.