High-Dimensional Robust Mean Estimation with Untrusted Batches

• URL: http://arxiv.org/abs/2602.20698v1
• Date: Tue, 24 Feb 2026 08:59:37 GMT
• Title: High-Dimensional Robust Mean Estimation with Untrusted Batches
• Authors: Maryam Aliakbarpour, Vladimir Braverman, Yuhan Liu, Junze Yin,
• Abstract summary: We study high-dimensional mean estimation in a collaborative setting where data is contributed by $N$ users in batches of size $n$.<n>We formalize this challenge through a double corruption landscape: an $varepsilon$-fraction of users are entirely adversarial, while the remaining good'' users provide data from distributions that are related to $P$, but deviate by a proximity parameter $$.<n>Our algorithms achieve the minimax-optimal error rate $O(sqrtvarepsilon/n + sqrtd/nN + s
• Score: 38.14592862692954
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• Abstract: We study high-dimensional mean estimation in a collaborative setting where data is contributed by $N$ users in batches of size $n$. In this environment, a learner seeks to recover the mean $μ$ of a true distribution $P$ from a collection of sources that are both statistically heterogeneous and potentially malicious. We formalize this challenge through a double corruption landscape: an $\varepsilon$-fraction of users are entirely adversarial, while the remaining ``good'' users provide data from distributions that are related to $P$, but deviate by a proximity parameter $α$. Unlike existing work on the untrusted batch model, which typically measures this deviation via total variation distance in discrete settings, we address the continuous, high-dimensional regime under two natural variants for deviation: (1) good batches are drawn from distributions with a mean-shift of $\sqrtα$, or (2) an $α$-fraction of samples within each good batch are adversarially corrupted. In particular, the second model presents significant new challenges: in high dimensions, unlike discrete settings, even a small fraction of sample-level corruption can shift empirical means and covariances arbitrarily. We provide two Sum-of-Squares (SoS) based algorithms to navigate this tiered corruption. Our algorithms achieve the minimax-optimal error rate $O(\sqrt{\varepsilon/n} + \sqrt{d/nN} + \sqrtα)$, demonstrating that while heterogeneity $α$ represents an inherent statistical difficulty, the influence of adversarial users is suppressed by a factor of $1/\sqrt{n}$ due to the internal averaging afforded by the batch structure.

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