On the Efficient Implementation of High Accuracy Optimality of Profile Maximum Likelihood
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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≫ n^-1/3. This result improves upon the previous best accuracy threshold of ϵ≫ 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≪ n^-1/3.
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