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 $$Big(frac{k }{n log k}Big)^2 + frac{log^2 k}{n}$$ 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.