论文标题
没有差异的最佳平均估计
Optimal Mean Estimation without a Variance
论文作者
论文摘要
我们研究了不存在数据生成分布方差的设置中的重尾平均估计问题。具体而言，给定一个样本$ \ mathbf {x} = \ {x_i \} _ {i = 1}^n $从分布中的$ \ mathcal {d} $ a $ \ mathbb {r}^d $带有平均值$μ$满足以下\ emph {feem-moment} $ nin的平均$ $的nim} $ nin can { \ begin {equation*} \ forall \ | v \ | = 1：\ Mathbb { $ n，d，δ$的功能。对于$α= 1 $的特定情况，Lugosi和Mendelson的基础工作表现出实现subgaussian置信区间的估计器，随后的工作导致了该估计器的计算有效版本。在这里，我们研究了一般$α$的情况，并在最佳可达到的置信区间上建立以下信息理论下限：\ begin {qore*}ω\ left（\ sqrt {\ frac {\ frac {d} {n} {n}}} {n}}}} + \ lest（ \ left（\ frac {\ log 1 /δ} {n} \ right）^{\fracα{（1 +α）}}} \ right）。此外，\ end {equation*}，我们设计了一个实现此下限的计算估计器。
We study the problem of heavy-tailed mean estimation in settings where the variance of the data-generating distribution does not exist. Concretely, given a sample $\mathbf{X} = \{X_i\}_{i = 1}^n$ from a distribution $\mathcal{D}$ over $\mathbb{R}^d$ with mean $μ$ which satisfies the following \emph{weak-moment} assumption for some ${α\in [0, 1]}$: \begin{equation*} \forall \|v\| = 1: \mathbb{E}_{X \thicksim \mathcal{D}}[\lvert \langle X - μ, v\rangle \rvert^{1 + α}] \leq 1, \end{equation*} and given a target failure probability, $δ$, our goal is to design an estimator which attains the smallest possible confidence interval as a function of $n,d,δ$. For the specific case of $α= 1$, foundational work of Lugosi and Mendelson exhibits an estimator achieving subgaussian confidence intervals, and subsequent work has led to computationally efficient versions of this estimator. Here, we study the case of general $α$, and establish the following information-theoretic lower bound on the optimal attainable confidence interval: \begin{equation*} Ω\left(\sqrt{\frac{d}{n}} + \left(\frac{d}{n}\right)^{\fracα{(1 + α)}} + \left(\frac{\log 1 / δ}{n}\right)^{\fracα{(1 + α)}}\right). \end{equation*} Moreover, we devise a computationally-efficient estimator which achieves this lower bound.
