Minimax rates of entropy estimation on large alphabets via best polynomial approximation

Consider the problem of estimating the Shannon entropy of a distribution over $k$ elements from $n$ independent samples. We show that the minimax mean-square error is within universal multiplicative constant factors of $$\Big(\frac{k }{n \log k}\Big)² + \frac{\log² k}{n}$$ if $n$ exceeds a constant factor of $\frac{k}{\log k}$; otherwise there exists no consistent estimator. This refines the recent result of Valiant-Valiant \cite{VV11} that the minimal sample size for consistent entropy estimation scales according to $\Theta(\frac{k}{\log k})$. The apparatus of best polynomial approximation plays a key role in both the construction of optimal estimators and, via a duality argument, the minimax lower bound.