In distributed optimization and distributed numerical linear algebra, we often encounter an inversion bias: if we want to compute a quantity that depends on the inverse of a sum of distributed matrices, then the sum of the inverses does not equal the inverse of the sum. An example of this occurs in distributed Newton's method, where we wish to compute (or implicitly work with) the inverse Hessian multiplied by the gradient. In this case, locally computed estimates are biased, and so taking a uniform average will not recover the correct solution. To address this, we propose determinantal averaging, a new approach for correcting the inversion bias. This approach involves reweighting the local estimates of the Newton's step proportionally to the determinant of the local Hessian estimate, and then averaging them together to obtain an improved global estimate. This method provides the first known distributed Newton step that is asymptotically consistent, i.e., it recovers the exact step in the limit as the number of distributed partitions grows to infinity. To show this, we develop new expectation identities and moment bounds for the determinant and adjugate of a random matrix. Determinantal averaging can be applied not only to Newton's method, but to computing any quantity that is a linear tranformation of a matrix inverse, e.g., taking a trace of the inverse covariance matrix, which is used in data uncertainty quantification.

提出了一种纠正分布式优化和分布式数值线性代数中的倒置偏差的新方法——行列式平均法。通过利用行列式对局部Hessian估计进行加权重新计算，然后进行平均，从而提高全局估计的准确性，其中包括应用于牛顿法和数据不确定范围内协方差矩阵的逆矩阵迹的计算等数量。该方法是第一个已知具有渐近一致性的分布式牛顿步骤的方法，为此开发了随机矩阵的行列式和伴随量的新的期望身份和矩界。

通过行列式均值分布估计逆黑塞矩阵