  We study the minimax estimation of $\alpha$-divergences between discrete
distributions for integer $\alpha\ge 1$, which include the Kullback--Leibler
divergence and the $\chi^2$-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 problem-independent linear program based
on moment matching in the non-smooth regime, and applies a problem-dependent
bias-corrected plug-in estimator in the smooth regime, with a soft decision
boundary between these regimes.
