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 pioneering work of [DKK+16] 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, [KS17b] introduced a general framework for robust moment estimation via a canonical sum-of-squares relaxation that succeeds for the more general class of certifiably subgaussian and certifiably hypercontractive [BK20] distributions. When specialized to Gaussians, this algorithm obtains the same $\tilde{O}(\varepsilon)$ error guarantee as [DKK+16] 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 [DKK+16] relies on lower bounds on directional moments inferred from algebraic relationships between moments of Gaussian distributions. We give a new, simple analysis of the same canonical sum-of-squares relaxation used in [KS17b, BK20] and show that for Gaussian distributions, their algorithm achieves the same error, sample complexity and running time guarantees as of the specialized algorithm in [DKK+16]. 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 developing further robust estimation algorithms.
翻译:在这项工作中, 我们重新审视了估算一个未知的 $dd- mode Gausian 的平均值和差值分布的问题。 在另一方面, [ KS17b] 引入了一个总框架, 用于在对角外出方折合$\ varepsilon 的折叠。 [ DKK+16] 的开创性工作为这一任务提供了一个多式时间算法, 其最佳的 $\ tilde{ O} (\ varepsilon) $ =\ textrrm{ polly} (d, 1/\ varepreslllon) 的差错 。 另一方面, [KS17bb] 引入了一个总体框架, 通过对对对对对对对对对对对对面外的对价的折叠叠合值的折叠叠合。 在对 Gabs descregies developteal developments a lax, lax the missional livers a rmals a rmals a liversal liversal liversal liversals a liversal liversal liversals a liversals a liversals a liverss a lits. lishs a lishs a liguts.