First-order Newton-type Estimator for Distributed Estimation and Inference

Abstract: This paper studies distributed estimation and inference for a general statistical problem with a convex loss that could be non-differentiable. For the purpose of efficient computation, we restrict ourselves to stochastic first-order optimization, which enjoys low per-iteration complexity. To motivate the proposed method, we first investigate the theoretical properties of a straightforward Divide-and-Conquer Stochastic Gradient Descent (DC-SGD) approach. Our theory shows that there is a restriction on the number of machines and this restriction becomes more stringent when the dimension $p$ is large. To overcome this limitation, this paper proposes a new multi-round distributed estimation procedure that approximates the Newton step only using stochastic subgradient. The key component in our method is the proposal of a computationally efficient estimator of $\Sigma^{-1} w$, where $\Sigma$ is the population Hessian matrix and $w$ is any given vector. Instead of estimating $\Sigma$ (or $\Sigma<sup>{-1}$) that usually requires the second-order differentiability of the loss, the proposed First-Order Newton-type Estimator (FONE) directly estimates the vector of interest $\Sigma^{-1} w$ as a whole and is applicable to non-differentiable losses. Our estimator also facilitates the inference for the empirical risk minimizer. It turns out that the key term in the limiting covariance has the form of $\Sigma^{-1} w$, which can be estimated by FONE.