Optimal Mean Estimation without a Variance

Yeshwanth Cherapanamjeri, Nilesh Tripuraneni, Peter Bartlett, Michael Jordan
Proceedings of Thirty Fifth Conference on Learning Theory, PMLR 178:356-357, 2022.

Abstract

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 $\bm{X} = \{X_i\}_{i = 1}^n$ from a distribution $\mc{D}$ over $\mb{R}^d$ with mean $\mu$ which satisfies the following \emph{weak-moment} assumption for some ${\alpha \in [0, 1]}$: \begin{equation*} \forall \norm{v} = 1: \mb{E}_{X \ts \mc{D}}[\abs{\inp{X - \mu}{v}}^{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 \emph{optimal} subgaussian confidence intervals, and subsequent work has led to computationally efficient versions of this estimator. Here, we study the case of general $\alpha$, and provide a precise characterization of the optimal achievable confidence interval by establishing the following information-theoretic lower bound: \begin{equation*} \Omega \lprp{\sqrt{\frac{d}{n}} + \lprp{\frac{d}{n}}^{\frac{\alpha}{(1 + \alpha)}} + \lprp{\frac{\log 1 / \delta}{n}}^{\frac{\alpha}{(1 + \alpha)}}}. \end{equation*} and devising an estimator matching the aforementioned lower bound up to constants. Moreover, our estimator is computationally efficient.

Cite this Paper

BibTeX
@InProceedings{pmlr-v178-cherapanamjeri22a, title = {Optimal Mean Estimation without a Variance}, author = {Cherapanamjeri, Yeshwanth and Tripuraneni, Nilesh and Bartlett, Peter and Jordan, Michael}, booktitle = {Proceedings of Thirty Fifth Conference on Learning Theory}, pages = {356--357}, year = {2022}, editor = {Loh, Po-Ling and Raginsky, Maxim}, volume = {178}, series = {Proceedings of Machine Learning Research}, month = {02--05 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v178/cherapanamjeri22a/cherapanamjeri22a.pdf}, url = {https://proceedings.mlr.press/v178/cherapanamjeri22a.html}, abstract = {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 $\bm{X} = \{X_i\}_{i = 1}^n$ from a distribution $\mc{D}$ over $\mb{R}^d$ with mean $\mu$ which satisfies the following \emph{weak-moment} assumption for some ${\alpha \in [0, 1]}$: \begin{equation*} \forall \norm{v} = 1: \mb{E}_{X \ts \mc{D}}[\abs{\inp{X - \mu}{v}}^{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 \emph{optimal} subgaussian confidence intervals, and subsequent work has led to computationally efficient versions of this estimator. Here, we study the case of general $\alpha$, and provide a precise characterization of the optimal achievable confidence interval by establishing the following information-theoretic lower bound: \begin{equation*} \Omega \lprp{\sqrt{\frac{d}{n}} + \lprp{\frac{d}{n}}^{\frac{\alpha}{(1 + \alpha)}} + \lprp{\frac{\log 1 / \delta}{n}}^{\frac{\alpha}{(1 + \alpha)}}}. \end{equation*} and devising an estimator matching the aforementioned lower bound up to constants. Moreover, our estimator is computationally efficient.} }
Endnote
%0 Conference Paper %T Optimal Mean Estimation without a Variance %A Yeshwanth Cherapanamjeri %A Nilesh Tripuraneni %A Peter Bartlett %A Michael Jordan %B Proceedings of Thirty Fifth Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2022 %E Po-Ling Loh %E Maxim Raginsky %F pmlr-v178-cherapanamjeri22a %I PMLR %P 356--357 %U https://proceedings.mlr.press/v178/cherapanamjeri22a.html %V 178 %X 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 $\bm{X} = \{X_i\}_{i = 1}^n$ from a distribution $\mc{D}$ over $\mb{R}^d$ with mean $\mu$ which satisfies the following \emph{weak-moment} assumption for some ${\alpha \in [0, 1]}$: \begin{equation*} \forall \norm{v} = 1: \mb{E}_{X \ts \mc{D}}[\abs{\inp{X - \mu}{v}}^{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 \emph{optimal} subgaussian confidence intervals, and subsequent work has led to computationally efficient versions of this estimator. Here, we study the case of general $\alpha$, and provide a precise characterization of the optimal achievable confidence interval by establishing the following information-theoretic lower bound: \begin{equation*} \Omega \lprp{\sqrt{\frac{d}{n}} + \lprp{\frac{d}{n}}^{\frac{\alpha}{(1 + \alpha)}} + \lprp{\frac{\log 1 / \delta}{n}}^{\frac{\alpha}{(1 + \alpha)}}}. \end{equation*} and devising an estimator matching the aforementioned lower bound up to constants. Moreover, our estimator is computationally efficient.
APA
Cherapanamjeri, Y., Tripuraneni, N., Bartlett, P. & Jordan, M.. (2022). Optimal Mean Estimation without a Variance. Proceedings of Thirty Fifth Conference on Learning Theory, in Proceedings of Machine Learning Research 178:356-357 Available from https://proceedings.mlr.press/v178/cherapanamjeri22a.html.

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