Sample Complexity Bounds for Robust Mean Estimation with Mean-Shift Contamination

We study the basic task of mean estimation in the presence of mean-shift contamination. In the mean-shift contamination model, an adversary is allowed to replace a small constant fraction of the clean samples by samples drawn from arbitrarily shifted versions of the base distribution. Prior work characterized the sample complexity of this task for the special cases of the Gaussian and Laplace distributions. Specifically, it was shown that consistent estimation is possible in these cases, a property that is provably impossible in Huber's contamination model. An open question posed in earlier work was to determine the sample complexity of mean estimation in the mean-shift contamination model for general base distributions. In this work, we study and essentially resolve this open question. Specifically, we show that, under mild spectral conditions on the characteristic function of the (potentially multivariate) base distribution, there exists a sample-efficient algorithm that estimates the target mean to any desired accuracy. We complement our upper bound with a qualitatively matching sample complexity lower bound. Our techniques make critical use of Fourier analysis, and in particular introduce the notion of a Fourier witness as an essential ingredient of our upper and lower bounds.

This paper studies mean estimation when the means of the distributions generating a small fraction of the data points have been adversarially corrupted. Unlike in more general corruption models, where corrupted data points may be completely arbitrary, this setting allows for arbitrarily accurate estimation for well-behaved distributions. Previous work has studied this problem for a few special cases, such as Gaussian and Laplace distributions. We provide a much broader characterization: we identify when accurate estimation is possible for general data distributions and determine the number of samples required to achieve it.