On the Efficient Implementation of High Accuracy Optimality of Profile Maximum Likelihood

We provide an efficient unified plug-in approach for estimating symmetric properties of distributions given n independent samples. Our estimator is based on profile-maximum-likelihood (PML) and is sample optimal for estimating various symmetric properties when the estimation error \epsilon \gg n {-1/3} . This result improves upon the previous best accuracy threshold of \epsilon \gg n {-1/4} achievable by polynomial time computable PML-based universal estimators \cite{ACSS20, ACSS20b}. Our estimator reaches a theoretical limit for universal symmetric property estimation as \cite{Han20} shows that a broad class of universal estimators (containing many well known approaches including ours) cannot be sample optimal for every 1 -Lipschitz property when \epsilon \ll n {-1/3} .