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

In this work, we study the problem of robustly estimating the mean/location parameter of distributions without moment bounds. For a large class of distributions satisfying natural symmetry constraints we give a sequence of algorithms that can efficiently estimate its location without incurring dimension-dependent factors in the error. Concretely, suppose an adversary can arbitrarily corrupt an $\varepsilon$-fraction of the observed samples. For every $k \in \mathbb{N}$, we design an estimator using time and samples $\tilde{O}({d^k})$ such that the dependence of the error on the corruption level $\varepsilon$ is an additive factor of $O(\varepsilon^{1-\frac{1}{2k}})$. The dependence on other problem parameters is also nearly optimal. Our class contains products of arbitrary symmetric one-dimensional distributions as well as elliptical distributions, a vast generalization of the Gaussian distribution. Examples include product Cauchy distributions and multi-variate $t$-distributions. In particular, even the first moment might not exist. We provide the first efficient algorithms for this class of distributions. Previously, such results where only known under boundedness assumptions on the moments of the distribution and in particular, are provably impossible in the absence of symmetry [KSS18, CTBJ22]. 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. Moreover, we show how sum-of-squares proofs can be used to obtain algorithmic guarantees even for distributions without first moment. We believe that this approach may find other application in future works.