We consider sufficient dimension reduction for heterogeneous massive data. We show that, even in the presence of heterogeneity and nonlinear dependence, the minimizers of convex loss functions of linear regression fall into the central subspace at the population level. We suggest a distributed algorithm to perform sufficient dimension reduction, where the convex loss functions are approximated with surrogate quadratic losses. This allows to perform dimension reduction in a unified least squares framework and facilitates to transmit the gradients in our distributed algorithm. The minimizers of these surrogate quadratic losses possess a nearly oracle rate after a finite number of iterations. We conduct simulations and an application to demonstrate the effectiveness of our proposed distributed algorithm for heterogeneous massive data.

中文翻译：

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接近预言机速率的分布式降维

我们考虑对异构海量数据进行足够的降维。我们表明，即使存在异质性和非线性相关性，线性回归的凸损失函数的最小值也会落入总体水平的中心子空间。我们建议使用分布式算法来执行足够的降维，其中凸损失函数用代理二次损失近似。这允许在统一的最小二乘框架中执行降维，并有助于在我们的分布式算法中传输梯度。这些替代二次损失的最小化器在有限次数的迭代后具有接近预言机率。我们进行了模拟和应用程序，以证明我们提出的分布式算法对异构海量数据的有效性。