The Complexity of Estimating Rényi Entropy

It was recently shown that estimating the Shannon entropy H(p) of a discrete k-symbol distribution p requires Θ(k / log k) samples, a number that grows near-linearly in the support size. In many applications H(p) can be replaced by the more general Rényi entropy of order α, Hα(p). We determine the number of samples needed to estimate Hα(p) for all α, showing that α < 1 requires a super-linear, roughly k1/α samples, noninteger α> 1 requires a near-linear k samples, but, perhaps surprisingly, integer α> 1 requires only Θ(k1−1/α) samples. In particular, estimating H2(p), which arises in security, DNA reconstruction, closeness testing, and other applications, requires only Θ( k) samples. The estimators achieving these bounds are simple and run in time linear in the number of samples