Papers › Estimation of KL Divergence: Optimal Minimax Rate
Estimation of KL Divergence: Optimal Minimax Rate
Yuheng Bu, Shaofeng Zou, Yingbin Liang, Venugopal V. Veeravalli
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The problem of estimating the Kullback-Leibler divergence D(PQ) 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 (k/m+(kf(k))/n)²+(log² f(k))/m+(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 (k/mlogk+(kf(k))/nlogk)²+(log² f(k))/m+(f(k))/n, if m and n exceed a constant factor of k/log(k) and kf(k)/logk, 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.
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