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# Polynomial-Time Sum-of-Squares Can Robustly Estimate Mean and Covariance of Gaussians Optimally

*Proceedings of The 33rd International Conference on Algorithmic Learning Theory*, PMLR 167:638-667, 2022.

#### Abstract

In this work, we revisit the problem of estimating the mean and covariance of an unknown $d$-dimensional Gaussian distribution in the presence of an $\varepsilon$-fraction of adversarial outliers. The work of Diakonikolas et al. (2016) gave a polynomial time algorithm for this task with optimal $\tilde{O}(\varepsilon)$ error using $n = \textrm{poly}(d, 1/\varepsilon)$ samples. On the other hand, Kothari and Steurer (2017) introduced a general framework for robust moment estimation via a canonical sum-of-squares relaxation that succeeds for the more general class of \emph{certifiably subgaussian} and \emph{certifiably hypercontractive} (Bakshi and Kothari, 2020) distributions. When specialized to Gaussians, this algorithm obtains the same $\tilde{O}(\varepsilon)$ error guarantee as Diakonikolas et al. (2016) but incurs a super-polynomial sample complexity ($n = d^{O(\log 1/\varepsilon)}$) and running time ($n^{O(\log(1/\varepsilon))}$). This cost appears inherent to their analysis as it relies only on sum-of-squares certificates of upper bounds on directional moments while the analysis in Diakonikolas et al. (2016) relies on \emph{lower bounds} on directional moments inferred from algebraic relationships between moments of Gaussian distributions. We give a new, simple analysis of the \emph{same} canonical sum-of-squares relaxation used in Kothari and Steurer (2017) and Bakshi and Kothari (2020) and show that for Gaussian distributions, their algorithm achieves the same error, sample complexity and running time guarantees as of the specialized algorithm in Diakonikolas et al. (2016). Our key innovation is a new argument that allows using moment lower bounds without having sum-of-squares certificates for them. We believe that our proof technique will likely be useful in designing new robust estimation algorithms.