Mittagsseminar (in cooperation with J. Lengler, A. Steger, and D. Steurer)

Date and Time: Tuesday, May 02, 2023, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Gleb Novikov

Robust Mean Estimation Without a Mean: Dimension-Independent Error in Polynomial Time for Symmetric Distributions

We study the problem of robustly estimating the mean/location parameter of distributions without moment bounds. We focus on large classes of distributions extensively studied in statistics: symmetric product distributions and elliptical distributions. Concretely, suppose an adversary can arbitrarily corrupt an $\varepsilon$-fraction of the observed d-dimensional samples. For every natural number k, we design an estimator that, using \tilde{O}(d^k) samples, achieves error O(\varespsilon^{1−1/(2k)}). This estimator can be computed in polynomial time. Previously, such results where only known for distributions that have 2k moments that are \emph{certifiably bounded} in sum-of-squares. For the class of distributions we consider, all previous estimators either require exponential time or incur error depending on the dimension. Our algorithms are based on a generalization of the filtering technique [DK22]. We show how this machinery can be combined with Huber-loss-based approach to work with projections of the noise onto some balls. These projections have nice behaviour, and it allows us to use sum-of-squares proofs to obtain algorithmic guarantees even for distributions without first moment. We believe that this approach may find other application in future works.

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