Estimation of KL Divergence: Optimal Minimax Rate

Abstract

The problem of estimating the Kullback-Leibler divergence $D(P\|Q)$ between two unknown distributions $P$ and $Q$ is studied, under the assumption that the alphabet size $k$ of the distributions can scale to infinity. The estimation is based on $m$ independent samples drawn from $P$ and $n$ independent samples drawn from $Q$. It is first shown that there does not exist any consistent estimator that guarantees asymptotically small worst-case quadratic risk over the set of all pairs of distributions. A restricted set that contains pairs of distributions, with density ratio bounded by a function $f(k)$ is further considered. {An augmented plug-in estimator is proposed, and its worst-case quadratic risk is shown to be within a constant factor of $(\frac{k}{m}+\frac{kf(k)}{n})^2+\frac{\log ^2 f(k)}{m}+\frac{f(k)}{n}$, if $m$ and $n$ exceed a constant factor of $k$ and $kf(k)$, respectively.} Moreover, the minimax quadratic risk is characterized to be within a constant factor of $(\frac{k}{m\log k}+\frac{kf(k)}{n\log k})^2+\frac{\log ^2 f(k)}{m}+\frac{f(k)}{n}$, if $m$ and $n$ exceed a constant factor of $k/\log(k)$ and $kf(k)/\log k$, respectively. The lower bound on the minimax quadratic risk is characterized by employing a generalized Le Cam's method. A minimax optimal estimator is then constructed by employing both the polynomial approximation and the plug-in approaches.