We design an $(\varepsilon, \delta)$-differentially private algorithm to estimate the mean of a $d$-variate distribution, with unknown covariance $\Sigma$, that is adaptive to $\Sigma$. To within polylogarithmic factors, the estimator achieves optimal rates of convergence with respect to the induced Mahalanobis norm $||\cdot||_\Sigma$, takes time $\tilde{O}(n d^2)$ to compute, has near linear sample complexity for sub-Gaussian distributions, allows $\Sigma$ to be degenerate or low rank, and adaptively extends beyond sub-Gaussianity. Prior to this work, other methods required exponential computation time or the superlinear scaling $n = \Omega(d^{3/2})$ to achieve non-trivial error with respect to the norm $||\cdot||_\Sigma$.
翻译:我们设计了一个$( varepsilon,\ ddelta) 的私人算法, 来估计一个以美元变差分布的平均值( 未知的共差 $\ sigma $ ), 以适应于 $\ sigma $ 。 在多元数系数中, 估测器在引致的 Mahalanobis 规范方面达到最佳的趋同率 $ çcdot\\ grama $, 计算需要时间 $\ tilde{ O} (n d ⁇ 2), 并且具有接近 线性样本的复杂度, 使 $\ Sigma 的分布退化或低等级, 并且适应性扩展到 亚加西 度之外。 在这项工作之前, 其他方法需要指数计算时间或超线缩放 $ = \ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\