# Problem 81: Rényi Entropy Estimation

For any $\alpha \geq 0$, the Rényi entropy of order $\alpha$ of a probability distribution $p$ over a discrete domain $\Omega$ is defined as $$H_\alpha(p) = \frac{1}{1-\alpha}\log\sum_{x\in\Omega} p(x)^\alpha$$ for $\alpha\neq 1$, and $H_1(p) = \lim_{\alpha\to 1} H_\alpha(p)$. In particular, $H_1$ corresponds to the Shannon entropy $H(p)=-\sum_{x\in\Omega} p(x) \log p(x)$. The problem of estimation Rényi entropy of a distribution $p$, given i.i.d. samples from it, was studied in [AcharyaOST-17], where the authors obtain tight bounds for every integer $\alpha\neq 1$, and nearly-tight ones for every non-integer $\alpha \geq 0$. In the later case, however, the sample complexity is only known up to a subpolynomial factor: resolving the exact dependence on $\alpha$ and the alphabet size would be interesting.