# Optimal Mean Estimation without a Variance

arXiv: https://arxiv.org/abs/2011.12433 · HTML: https://mlsanity.com/paper/2011.12433/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Nov 2020.

## Built on by (most cited first, 13 of 13)

- [From Gradient Clipping to Normalization for Heavy Tailed SGD](https://mlsanity.com/paper/2410.13849/) (2024)
- [Convergence Rates of Stochastic Gradient Descent under Infinite Noise Variance](https://mlsanity.com/paper/2102.10346/) (2021)
- [Outlier-Robust Sparse Mean Estimation for Heavy-Tailed Distributions](https://mlsanity.com/paper/2211.16333/) (2022)
- [Sign Operator for Coping with Heavy-Tailed Noise in Non-Convex Optimization: High Probability Bounds Under (L0, L1)-Smoothness](https://mlsanity.com/paper/2502.07923/) (2025)
- [Nearly-Linear Time and Streaming Algorithms for Outlier-Robust PCA](https://mlsanity.com/paper/2305.02544/) (2023)
- [Near-Optimal Algorithms for Gaussians with Huber Contamination: Mean Estimation and Linear Regression](https://mlsanity.com/paper/2312.01547/) (2023)
- [Order-Optimal Sequential 1-Bit Mean Estimation in General Tail Regimes](https://mlsanity.com/paper/2604.07796/) (2026)
- [Sequential 1-bit Mean Estimation with Near-Optimal Sample Complexity](https://mlsanity.com/paper/2509.21940/) (2025)
- [The Empirical Mean is Minimax Optimal for Local Glivenko-Cantelli](https://mlsanity.com/paper/2410.02835/) (2024)
- [Open Problem: Is Interaction Necessary for Order-Optimal 1-bit Mean Estimation?](https://mlsanity.com/paper/2607.02896/) (2026)
- [Interaction Is Not Necessary for Order-Optimal 1-Bit Mean Estimation](https://mlsanity.com/paper/2608.02538/) (2026)
- [Robust Mean Estimation Without Moments for Symmetric Distributions](https://mlsanity.com/paper/2302.10844/) (2023)
- [Online Heavy-tailed Change-point detection](https://mlsanity.com/paper/2306.09548/) (2023)

## Abstract (arXiv)

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 X = Xii = 1n from a distribution D over ℝd with mean μ which satisfies the following weak-moment assumption for some α ∈ [0, 1]: beginequation* forall |v| = 1: mathbbEX thicksim D[lvert langle X - μ, vrangle rvert1 + α] ≤ 1, endequation* 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: beginequation* Omega left(√fracdn + left(fracdnright)^fracα(1 + α) + left(fraclog 1 / δnright)^fracα(1 + α)right). endequation* Moreover, we devise a computationally-efficient estimator which achieves this lower bound.