Part of Advances in Neural Information Processing Systems 32 (NeurIPS 2019)
Yi Hao, Alon Orlitsky
We study three fundamental statistical-learning problems: distribution estimation, property estimation, and property testing. We establish the profile maximum likelihood (PML) estimator as the first unified sample-optimal approach to a wide range of learning tasks. In particular, for every alphabet size k and desired accuracy ε: \textbf{Distribution estimation} Under ℓ1 distance, PML yields optimal Θ(k/(ε2logk)) sample complexity for sorted-distribution estimation, and a PML-based estimator empirically outperforms the Good-Turing estimator on the actual distribution; \textbf{Additive property estimation} For a broad class of additive properties, the PML plug-in estimator uses just four times the sample size required by the best estimator to achieve roughly twice its error, with exponentially higher confidence; \textbf{α-R\'enyi entropy estimation} For an integer α>1, the PML plug-in estimator has optimal k1−1/α sample complexity; for non-integer α>3/4, the PML plug-in estimator has sample complexity lower than the state of the art; \textbf{Identity testing} In testing whether an unknown distribution is equal to or at least ε far from a given distribution in ℓ1 distance, a PML-based tester achieves the optimal sample complexity up to logarithmic factors of k. With minor modifications, most of these results also hold for a near-linear-time computable variant of PML.