Estimating Renyi Entropy of Discrete Distributions

It was recently shown that estimating the Shannon entropy $H(p)$ of a discrete $k$-symbol distribution $p$ requires $\Theta(k/\log k)$ samples, a number that grows near-linearly in the support size. In many applications $H(p)$ can be replaced by the more general Renyi entropy of order $\alpha$, $H_\alpha(p)$. We determine the number of samples needed to estimate $H_\alpha(p)$ for all $\alpha$, showing that $\alpha < 1$ requires a super-linear, roughly $k^{1/\alpha}$ samples, noninteger $\alpha>1$ requires a near-linear $k$ samples, but, perhaps surprisingly, integer $\alpha>1$ requires only $\Theta(k^{1-1/\alpha})$ samples. Furthermore, developing on a recently established connection between polynomial approximation and estimation of additive functions of the form $\sum_x f (p_x)$, we reduce the sample complexity for noninteger values of $\alpha$ by a factor of $\log k$ compared to the empirical estimator. The estimators achieving these bounds are simple and run in time linear in the number of samples. Our lower bound provides an explicit construction of distributions with different Renyi entropies that are hard to distinguish.