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 $\mu$ which satisfies the following \emph{weak-moment} assumption for some ${\alpha \in [0, 1]}$: \begin{equation*} \forall \|v\| = 1: \mathbb{E}_{X \thicksim \mathcal{D}}[\lvert \langle X - \mu, v\rangle \rvert^{1 + \alpha}] \leq 1, \end{equation*} and given a target failure probability, $\delta$, our goal is to design an estimator which attains the smallest possible confidence interval as a function of $n,d,\delta$. For the specific case of $\alpha = 1$, 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 $\alpha$, and establish the following information-theoretic lower bound on the optimal attainable confidence interval: \begin{equation*} \Omega \left(\sqrt{\frac{d}{n}} + \left(\frac{d}{n}\right)^{\frac{\alpha}{(1 + \alpha)}} + \left(\frac{\log 1 / \delta}{n}\right)^{\frac{\alpha}{(1 + \alpha)}}\right). \end{equation*} Moreover, we devise a computationally-efficient estimator which achieves this lower bound.

中文翻译：

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无方差的最优均值估计

我们研究了在不存在数据生成分布方差的情况下的重尾均值估计问题。具体而言，假设样本$ \ mathbf {X} = \ {X_i \} _ {i = 1} ^ n $来自分布$ \ mathcal {D} $超过$ \ mathbb {R} ^ d $的均值$ \ mu $满足以下$ \\ emph {weak-moment}假设，适用于某些$ {\ alpha \ in [0，1]} $：\ begin {equation *} \ forall \ | v \ | = 1：\ mathbb {E} _ {X \ thicksim \ mathcal {D}} [\ lvert \ langle X-\ mu，v \ rangle \ rvert ^ {1 + \ alpha}] \ leq 1，\ end {equation *}并给出目标失效概率$ \ delta $，我们的目标是设计一个估计器，该估计器将作为$ n，d，\ delta $的函数的最小置信区间。对于$ \ alpha = 1 $的特定情况，Lugosi和Mendelson的基础工作展示了达到亚高斯置信区间的估计量，随后的工作导致此估算器的计算效率更高。在这里，我们研究一般$ \ alpha $的情况，并在最佳可获得置信区间上建立以下信息理论下限：\ begin {equation *} \ Omega \ left（\ sqrt {\ frac {d} {n }} + \ left（\ frac {d} {n} \ right）^ {\ frac {\ alpha} {（1 + \ alpha）}} + \ left（\ frac {\ log 1 / \ delta} {n } \ right）^ {\ frac {\ alpha} {（1 + \ alpha）}} \ right）。\ end {equation *}此外，我们设计了一种计算效率高的估算器，可实现此下限。\ begin {equation *} \ Omega \ left（\ sqrt {\ frac {d} {n}} + \ left（\ frac {d} {n} \ right）^ {\ frac {\ alpha} {（1 + \ alpha）}} + \ left（\ frac {\ log 1 / \ delta} {n} \ right）^ {\ frac {\ alpha} {（1 + \ alpha）}} \ right）。\ end {equation *}此外，我们设计了一种计算效率高的估算器，可实现此下限。\ begin {equation *} \ Omega \ left（\ sqrt {\ frac {d} {n}} + \ left（\ frac {d} {n} \ right）^ {\ frac {\ alpha} {（1 + \ alpha）}} + \ left（\ frac {\ log 1 / \ delta} {n} \ right）^ {\ frac {\ alpha} {（1 + \ alpha）}} \ right）。\ end {equation *}此外，我们设计了一种计算效率高的估算器，可实现此下限。