Optimal entropy estimation on large alphabets via best polynomial approximation

Consider the problem of estimating the Shannon entropy of a distribution on k elements from n independent samples. We show that the minimax mean-square error is within universal multiplicative constant factors of (k/n log k) + log
²
k/n. This implies the recent result of Valiant-Valiant [1] that the minimal sample size for consistent entropy estimation scales according to Θ(k/log k). The apparatus of best polynomial approximation plays a key role in both the minimax lower bound and the construction of optimal estimators.