While including pairwise interactions in a regression model can better approximate response surface, fitting such an interaction model is a well-known difficult problem. In particular, analyzing contemporary high-dimensional datasets often leads to extremely large-scale interaction modeling problem, where the challenge is posed to identify important interactions among millions or even billions of candidate interactions. While several methods have recently been proposed to tackle this challenge, they are mostly designed by (1) assuming the hierarchy assumption among the important interactions and (or) (2) focusing on the case in linear models with interactions and (sub)Gaussian errors. In practice, however, neither of these two building blocks has to hold. In this paper, we propose an interaction modeling framework in generalized linear models (GLMs) which is free of any assumptions on hierarchy. We develop a non-trivial extension of the reluctance interaction selection principle to the GLMs setting, where a main effect is preferred over an interaction if all else is equal. Our proposed method is easy to implement, and is highly scalable to large-scale datasets. Theoretically, we demonstrate that it possesses screening consistency under high-dimensional setting. Numerical studies on simulated datasets and a real dataset show that the proposed method does not sacrifice statistical performance in the presence of significant computational gain.

翻译：在回归模型中包含两两交互可以更好地逼近响应曲面，但拟合此类交互模型是一个众所周知的难题。特别是，分析当代高维数据集往往会导致超大规模的交互建模问题，其挑战在于从数百万甚至数十亿候选交互中识别出重要交互。尽管最近已有多种方法被提出以应对这一挑战，但它们大多是基于以下两种设计思路：(1) 假设重要交互之间存在层级结构；(2) 仅针对线性模型及(次)高斯误差下的交互建模情形。然而在实践中，这两个基本假设并非必然成立。本文提出了一种在广义线性模型（GLMs）中无需任何层级假设的交互建模框架。我们将惰性交互选择原理非平凡地推广至GLM场景：当其他条件相同时，主效应优先于交互效应。所提方法易于实现，且具有面向大规模数据集的高度可扩展性。理论上，我们证明了该方法在高维设置下具有筛选一致性。基于模拟数据集和真实数据集的数值实验表明，所提方法在显著提升计算效率的同时，并未牺牲统计性能。