Efficient Multivariate Robust Mean Estimation Under Mean-Shift Contamination

• URL: http://arxiv.org/abs/2502.14772v1
• Date: Thu, 20 Feb 2025 17:53:13 GMT
• Title: Efficient Multivariate Robust Mean Estimation Under Mean-Shift Contamination
• Authors: Ilias Diakonikolas, Giannis Iakovidis, Daniel M. Kane, Thanasis Pittas,
• Abstract summary: We give the first computationally efficient algorithm for high-dimensional robust mean estimation with mean-shift contamination. Our algorithm has near-optimal sample complexity, runs in sample-polynomial time, and approximates the target mean to any desired accuracy.
• Score: 35.67742880001828
• License:
• Abstract: 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 $\alpha<1/2$, the $i$-th sample $x_i$ is obtained as follows: with probability $1-\alpha$, $x_i$ is drawn from $\mathcal{N}(\mu, I)$, where $\mu \in \mathbb{R}^d$ is the target mean; and with probability $\alpha$, $x_i$ 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.

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