On the Optimality of Averaging in Distributed Statistical Learning
Jonathan D. Rosenblatt, Boaz Nadler · 2015
A common approach to statistical learning with big-data is to randomly split it among m machines and learn the parameter of interest by averag-ing the m individual estimates. In this paper, focusing on empirical risk minimization, or equivalently M-estimation, we study the statistical error incurred by this strategy. We consider two large-sample settings: First, a classical setting where the number of parameters p is xed, and the num-ber of samples per machine n! 1. Second, a high-dimensional regime where both p; n! 1 with p=n! 2 (0; 1). For both regimes and under suitable assumptions, we present asymptotically exact expressions for this es-timation error. In the xed-p setting, under suitable assumptions, we prove that to leading order averaging is as accurate as the centralized solution. We also derive the second order error terms, and show that these can be non-negligible, notably for non-linear models. The high-dimensional setting, in contrast, exhibits a qualitatively different behavior: data splitting incurs a rst-order accuracy loss, which to leading order increases linearly with the number of machines. The dependence of our error approximations on the number of machines traces an interesting accuracy-complexity tradeoff, al-lowing the practitioner an informed choice on the number of machines to deploy. Finally, we conrm our theoretical analysis with several simulations. 1 1.