First-Order Newton-Type Estimator for Distributed Estimation and Inference

This article studies distributed estimation and inference for a general statistical problem with a convex loss that could be nondifferentiable. 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 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 article 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 Σ−1w , where Σ is the population Hessian matrix and w is any given vector. Instead of estimating Σ (or Σ−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 Σ−1w as a whole and is applicable to nondifferentiable 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 Σ−1w , which can be estimated by FONE.