We study the minimax estimation of α-divergences between discrete distributions for integer α ≥ 1, which include the Kullback-Leibler divergence and the χ2-divergences as special examples. Dropping the usual theoretical tricks to acquire independence, we construct the first minimax rate-optimal estimator which does not require any Poissonization, sample splitting, or explicit construction of approximating polynomials. The estimator uses a hybrid approach which solves a problemindependent linear program based on moment matching in the non-smooth regime, and applies a problem-dependent biascorrected plug-in estimator in the smooth regime, with a soft decision boundary between these regimes.