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 ε gg n-1/3. This result improves upon the previous best accuracy threshold of ε gg n-1/4 achievable by polynomial time computable PML-based universal estimators [ACSS21, ACSS20]. Our estimator reaches a theoretical limit for universal symmetric property estimation as [Han21] 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 ε ll n-1/3.