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 $\delta$, then plugging in the PML distribution incurs an error $2\varepsilon$ with probability at most $\delta\cdot \exp(3\sqrt{n})$. In this paper, we strengthen the above result and show that using a careful chaining argument, the error probability can be reduced to $\delta^{1-c}\cdot \exp(c'n^{1/3+c})$ 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 $n=\Omega(k/(\varepsilon^2 \log k))$ 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.

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关于轮廓最大似然的竞争分析和高精度优化

[Acharya 等人的惊人结果。2017] 表明，为了估计离散分布的对称特性，插入使观察到的多组频率的可能性最大化的分布，也称为轮廓最大似然 (PML) 分布，与任何估计器相比，无论对称特性如何，都具有竞争力。具体来说，给定来自离散分布的 $n$ 观测值，如果某个估计量产生错误 $\varepsilon$ 且概率最多为 $\delta$，则插入 PML 分布会产生错误 $2\varepsilon$，概率最多为 $\ delta\cdot \exp(3\sqrt{n})$。在本文中，我们加强了上述结果，并表明使用仔细的链接参数，错误概率可以降低到 $\delta^{1-c}\cdot \exp(c'n^{1/3+c} )$ 用于任意小的常量 $c> 0$ 和一些常量 $c'>0$。特别是，我们证明了 PML 分布是排序分布的最佳估计量：对于任何 $n=\Omega，它与支持大小为 $k$ 的真实分布的排序 $\ell_1$ 距离是 $\varepsilon$-close (k/(\varepsilon^2 \log k))$ 和 $\varepsilon \gg n^{-1/3}$，它们是信息理论上的最优样本复杂度和最大误差范围，其中经典经验分布为次优，分别。为了加强对 PML 的分析，一个关键因素是采用 PML 分布的新“连续性”特性，并构建一系列合适的量化 PML，或“覆盖”。我们还为排序分布构建了一个新的基于近似的估计器，具有接近最佳的浓度特性，无需任何样本拆分，