Minimax Rate-Optimal Estimation of Divergences between Discrete Distributions

We study the minimax estimation of $α$-divergences between discrete distributions for integer $α\ge 1$, which include the Kullback--Leibler divergence and the $χ^2$-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.