We give the first polynomial time and sample $(\epsilon, \delta)$-differentially private (DP) algorithm to estimate the mean, covariance and higher moments in the presence of a constant fraction of adversarial outliers. Our algorithm succeeds for families of distributions that satisfy two well-studied properties in prior works on robust estimation: certifiable subgaussianity of directional moments and certifiable hypercontractivity of degree 2 polynomials. Our recovery guarantees hold in the "right affine-invariant norms": Mahalanobis distance for mean, multiplicative spectral and relative Frobenius distance guarantees for covariance and injective norms for higher moments. Prior works obtained private robust algorithms for mean estimation of subgaussian distributions with bounded covariance. For covariance estimation, ours is the first efficient algorithm (even in the absence of outliers) that succeeds without any condition-number assumptions. Our algorithms arise from a new framework that provides a general blueprint for modifying convex relaxations for robust estimation to satisfy strong worst-case stability guarantees in the appropriate parameter norms whenever the algorithms produce witnesses of correctness in their run. We verify such guarantees for a modification of standard sum-of-squares (SoS) semidefinite programming relaxations for robust estimation. Our privacy guarantees are obtained by combining stability guarantees with a new "estimate dependent" noise injection mechanism in which noise scales with the eigenvalues of the estimated covariance. We believe this framework will be useful more generally in obtaining DP counterparts of robust estimators. Independently of our work, Ashtiani and Liaw [AL21] also obtained a polynomial time and sample private robust estimation algorithm for Gaussian distributions.
翻译:我们给出了第一个多元值时间和样本 $(\\ epsilon,\ delta) 美元, 不同的私人( DP) 算法, 以估计平均、 多复制光谱和相对Frobenius 距离距离的距离, 以在对角离子体的固定部分出现时。 我们的算法成功给那些在先前的强力估算工作中满足了两个经充分研究的属性的分布家庭: 方向时刻的可证实的亚毛逊尼基亚性, 和21度多尼基亚的可确证的超缩缩缩缩。 我们的恢复保证在“ 直立光谱和相对Frobenius 距离的距离保障” 中, 以平均值为基础计算。 我们的定量算法是第一个高效的算法( 甚至在没有外部数据的情况下), 在没有任何条件- 假设的情况下成功。 我们的算法来自一个新的框架, 它提供了一个总体蓝图, 来修改稳定度的直立面光线光线光谱, 当我们的安全度的精确度调整时, 校准的精确的校准的校准的校准的校正的校正的校正的校正的校正的校正的校正的校正的校正的校准的校正的校正的校正的校正的校正的校正的校正的校正的校正的校正的校正的校正的校正的校正的校正的校正的校正的校正的校正的校正的校正的校正的校正的校正的校正的校正的校正的校正的校正的校正的校正的校正的校正的校正的校正的校正的校正的校正的校正的校正的校正的校正的校正的校正机制将的校正的校正的校正的校正的校正的校正的校正的校正的校正的校正的校正的校正的校正的校正的校正的校正的校正的校正的校正的校正的校正的校正的校正的校正的校正的校正的校正的校正的校正的校正的校