Efficient Multivariate Robust Mean Estimation Under Mean-Shift Contamination
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917931106959360 |
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| author | Diakonikolas, Ilias Iakovidis, Giannis Kane, Daniel M. Pittas, Thanasis |
| author_facet | Diakonikolas, Ilias Iakovidis, Giannis Kane, Daniel M. Pittas, Thanasis |
| contents | We study the algorithmic problem of robust mean estimation of an identity covariance Gaussian in the presence of mean-shift contamination. In this contamination model, we are given a set of points in $\mathbb{R}^d$ generated i.i.d. via the following process. For a parameter $α<1/2$, the $i$-th sample $x_i$ is obtained as follows: with probability $1-α$, $x_i$ is drawn from $\mathcal{N}(μ, I)$, where $μ\in \mathbb{R}^d$ is the target mean; and with probability $α$, $x_i$ is drawn from $\mathcal{N}(z_i, I)$, where $z_i$ is unknown and potentially arbitrary. Prior work characterized the information-theoretic limits of this task. Specifically, it was shown that, in contrast to Huber contamination, in the presence of mean-shift contamination consistent estimation is possible. On the other hand, all known robust estimators in the mean-shift model have running times exponential in the dimension. Here we give the first computationally efficient algorithm for high-dimensional robust mean estimation with mean-shift contamination that can tolerate a constant fraction of outliers. In particular, our algorithm has near-optimal sample complexity, runs in sample-polynomial time, and approximates the target mean to any desired accuracy. Conceptually, our result contributes to a growing body of work that studies inference with respect to natural noise models lying in between fully adversarial and random settings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_14772 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Efficient Multivariate Robust Mean Estimation Under Mean-Shift Contamination Diakonikolas, Ilias Iakovidis, Giannis Kane, Daniel M. Pittas, Thanasis Data Structures and Algorithms Machine Learning Statistics Theory We study the algorithmic problem of robust mean estimation of an identity covariance Gaussian in the presence of mean-shift contamination. In this contamination model, we are given a set of points in $\mathbb{R}^d$ generated i.i.d. via the following process. For a parameter $α<1/2$, the $i$-th sample $x_i$ is obtained as follows: with probability $1-α$, $x_i$ is drawn from $\mathcal{N}(μ, I)$, where $μ\in \mathbb{R}^d$ is the target mean; and with probability $α$, $x_i$ is drawn from $\mathcal{N}(z_i, I)$, where $z_i$ is unknown and potentially arbitrary. Prior work characterized the information-theoretic limits of this task. Specifically, it was shown that, in contrast to Huber contamination, in the presence of mean-shift contamination consistent estimation is possible. On the other hand, all known robust estimators in the mean-shift model have running times exponential in the dimension. Here we give the first computationally efficient algorithm for high-dimensional robust mean estimation with mean-shift contamination that can tolerate a constant fraction of outliers. In particular, our algorithm has near-optimal sample complexity, runs in sample-polynomial time, and approximates the target mean to any desired accuracy. Conceptually, our result contributes to a growing body of work that studies inference with respect to natural noise models lying in between fully adversarial and random settings. |
| title | Efficient Multivariate Robust Mean Estimation Under Mean-Shift Contamination |
| topic | Data Structures and Algorithms Machine Learning Statistics Theory |
| url | https://arxiv.org/abs/2502.14772 |