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.

翻译：在不存在数据生成分布差异的环境下,我们研究重尾线平均估算问题。 具体地说, 根据一个样本 $\ mathbf{X} =\ x_i ⁇ i= 1\ n$\ mathb{D} 美元以上的分配 $\ mathb{ R\ d$ 美元以上, 满足以下 leq 1,\ end- qoment 假设 $\ alpha\ in [0, 1] 美元 :\ begin{ quation\\ tall {v} =1\ tirt} = = mathrb} =\ x_\ x_\\\\ xi} =\\\\\\\\\\\ nn> 美元 美元以上, 美元以上的分配 =\\ talphr\ = talbr\ = dhaild}\ leqqq, irealation\\\ lives a lax lax 工作基础, = drodudeal a prodealmax 工作, 和 a promaxxxxxxx a frodeal_