自适应特性估计的高精度极限
On the High Accuracy Limitation of Adaptive Property Estimation
摘要：
近年来，自适应（或统一）方法在估计离散分布的对称性质方面取得了成功，即首先获得独立于目标性质的分布估计量，然后将估计量插入目标性质作为最终估计量。已经提出了几种这样的方法，并证明是自适应最优的，即，它们在低精度内实现了对于大类属性的最优样本复杂度，特别是对于大估计误差$\varepsilon\gg n^{-1/3}$，其中$n$是样本量。在本文中，我们的特点是高精度的限制，或适应的惩罚，所有这样的方法。具体来说，我们表明，在一个温和的假设下，分布估计是接近真实的排序分布的期望，任何自适应方法不能实现最佳的样本复杂度为每个$1$ -Lipschitz属性内的准确性$\varepsilon \ll n^{-1/3}$。特别是，该结果反驳了[Acharya et al. 2017]中的猜想，即轮廓最大似然（PML）插件方法在$\varepsilon$的所有范围的属性估计中均为最佳，并证实了[Han and Shiragur，2021]中的猜想，即他们对PML的竞争分析是严格的。
英文摘要：
Recent years have witnessed the success of adaptive (or unified) approaches in estimating symmetric properties of discrete distributions, where one first obtains a distribution estimator independent of the target property, and then plugs the estimator into the target property as the final estimator. Several such approaches have been proposed and proved to be adaptively optimal, i.e. they achieve the optimal sample complexity for a large class of properties within a low accuracy, especially for a large estimation error $\varepsilon\gg n^{-1/3}$ where $n$ is the sample size. In this paper, we characterize the high accuracy limitation, or the penalty for adaptation, for all such approaches. Specifically, we show that under a mild assumption that the distribution estimator is close to the true sorted distribution in expectation, any adaptive approach cannot achieve the optimal sample complexity for every $1$-Lipschitz property within accuracy $\varepsilon \ll n^{-1/3}$. In particular, this result disproves a conjecture in [Acharya et al. 2017] that the profile maximum likelihood (PML) plug-in approach is optimal in property estimation for all ranges of $\varepsilon$, and confirms a conjecture in [Han and Shiragur, 2021] that their competitive analysis of the PML is tight.
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