Empirical risk minimization, where the underlying loss function depends on a pair of data points, covers a wide range of application areas in statistics including pairwise ranking and survival analysis. The common empirical risk estimator obtained by averaging values of a loss function over all possible pairs of observations is essentially a U-statistic. One well-known problem with minimizing U-statistic type empirical risks, is that the computational complexity of U-statistics increases quadratically with the sample size. When faced with big data, this poses computational challenges as the colossal number of observation pairs virtually prohibits centralized computing to be performed on a single machine. This paper addresses this problem by developing two computationally and statistically efficient methods based on the divide-and-conquer strategy on a decentralized computing system, whereby the data are distributed among machines to perform the tasks. One of these methods is based on a surrogate of the empirical risk, while the other method extends the one-step updating scheme in classical M-estimation to the case of pairwise loss. We show that the proposed estimators are as asymptotically efficient as the benchmark global U-estimator obtained under centralized computing. As well, we introduce two distributed iterative algorithms to facilitate the implementation of the proposed methods, and conduct extensive numerical experiments to demonstrate their merit.

Input uncertainty is an aspect of simulation model risk that arises when the driving input distributions are derived or “fit” to real-world, historical data. Although there has been significant progress on quantifying and hedging against input uncertainty, there has been no direct attempt to reduce it via better input modeling. The meaning of “better” depends on the context and the objective: Our context is when (a) there are one or more families of parametric distributions that are plausible choices; (b) the real-world historical data are not expected to perfectly conform to any of them; and (c) our primary goal is to obtain higher-fidelity simulation output rather than to discover the “true” distribution. In this paper, we show that frequentist model averaging can be an effective way to create input models that better represent the true, unknown input distribution, thereby reducing model risk. Input model averaging builds from standard input modeling practice, is not computationally burdensome, requires no change in how the simulation is executed nor any follow-up experiments, and is available on the Comprehensive R Archive Network (CRAN). We provide theoretical and empirical support for our approach.