arXiv 2004.031662020-04math.STcs.DScs.ITmath.ITmath.PRstat.TH

轮廓最大似然法的竞争分析与高精度优化

On the Competitive Analysis and High Accuracy Optimality of Profile Maximum Likelihood

Yanjun Han, Kirankumar Shiragur

备注：: To appear at SODA 2021

摘要：

[Acharya et al. 2017]的一个引人注目的结果表明，为了估计离散分布的对称特性，插入最大化观察到的多组频率的可能性的分布，也称为轮廓最大似然（PML）分布，与任何估计量相比，无论对称特性如何，都具有竞争力。具体地说，给定来自离散分布的$n$观测值，如果某个估计量导致概率至多为$δ$的错误$\varepsilon$，则插入PML分布导致概率至多为$δ\cdot \exp(3\sqrt{n})$的错误$2\varepsilon$。在本文中，我们加强了上述结果，并表明，使用一个仔细的链接参数，错误概率可以减少到$δ^{1-c}\cdot \exp(c'n^{1/3+c})$任意小常数$c>0$和一些常数$c'>0$。特别是，我们表明，PML分布是一个最佳估计的排序分布：它是$\varepsilon$ -接近排序$\ell_1$距离的真实分布与支持大小$k$的任何$n=Ω(k/(\varepsilon^2 \log k))$和$\varepsilon \gg n^{-1/3}$，这是信息理论上最佳的样本复杂性和最大的错误制度，其中经典的经验分布是次优的，分别。为了加强对PML的分析，一个关键因素是采用新的PML分布的“连续性”属性，并构造一个合适的量化PML链，或“覆盖”。我们还构建了一个新的基于近似的估计排序分布的一个接近最优的浓度属性，没有任何样本分裂，作为副产品，我们获得更好的权衡之间的多项式近似误差和最大幅度的系数的泊松近似$1$ -Lipschitz函数。

英文摘要：

A striking result of [Acharya et al. 2017] showed that to estimate symmetric properties of discrete distributions, plugging in the distribution that maximizes the likelihood of observed multiset of frequencies, also known as the profile maximum likelihood (PML) distribution, is competitive compared with any estimators regardless of the symmetric property. Specifically, given $n$ observations from the discrete distribution, if some estimator incurs an error $\varepsilon$ with probability at most $δ$, then plugging in the PML distribution incurs an error $2\varepsilon$ with probability at most $δ\cdot \exp(3\sqrt{n})$. In this paper, we strengthen the above result and show that using a careful chaining argument, the error probability can be reduced to $δ^{1-c}\cdot \exp(c'n^{1/3+c})$ for arbitrarily small constants $c>0$ and some constant $c'>0$. In particular, we show that the PML distribution is an optimal estimator of the sorted distribution: it is $\varepsilon$-close in sorted $\ell_1$ distance to the true distribution with support size $k$ for any $n=Ω(k/(\varepsilon^2 \log k))$ and $\varepsilon \gg n^{-1/3}$, which are the information-theoretically optimal sample complexity and the largest error regime where the classical empirical distribution is sub-optimal, respectively. In order to strengthen the analysis of the PML, a key ingredient is to employ novel "continuity" properties of the PML distributions and construct a chain of suitable quantized PMLs, or "coverings". We also construct a novel approximation-based estimator for the sorted distribution with a near-optimal concentration property without any sample splitting, where as a byproduct we obtain better trade-offs between the polynomial approximation error and the maximum magnitude of coefficients in the Poisson approximation of $1$-Lipschitz functions.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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