Distributed Monitoring of the R ² Statistic for Linear Regression

Abstract

The problem of monitoring a multivariate linear regression model is relevant in studying the evolving relationship between a set of input variables (features) and one or more dependent target variables. This problem becomes challenging for large scale data in a distributed computing environment when only a subset of instances is available at individual nodes and the local data changes frequently. Data centralization and periodic model recomputation can add high overhead to tasks like anomaly detection in such dynamic settings. Therefore, the goal is to develop techniques for monitoring and updating the model over the union of all nodes' data in a communication-efficient fashion. Correctness guarantees on such techniques are also often highly desirable, especially in safety-critical application scenarios. In this paper we develop DReMo -a distributed algorithm with very low resource overhead, for monitoring the quality of a regression model in terms of its coefficient of determination (R 2 statistic). When the nodes collectively determine that R 2 has dropped below a fixed threshold, the linear regression model is recomputed via a network-wide convergecast and the updated model is broadcast back to all nodes. We show empirically, using both synthetic and real data, that our proposed method is highly communication-efficient and scalable, and also provide theoretical guarantees on correctness.

Monitoring Least Squares Models of Distributed Streams

“…Our main performance metric is communication, measured in normalized messages -the average messages sent per round by each node [3]. Note that communication of the naive algorithm is always 1.…”

“…In [36], a heuristic is applied, and the nodes do not broadcast if the newly arriving data conforms with the current model up to some tolerance. In [4] distributed monitoring was applied to monitor the prediction error (Section 3.1) and quadratic fit error R 2 [3], but not the error in the model itself.…”

“…Recent approaches monitor the prediction error |y − Xβ0|, where X, y are the current observations [36,4], the model's R 2 fit [3], or prediction error between divergent local models and the hypothetical global model [12].…”

“…The monitoring approach has received little attention, possibly because the least squares solutions involve matrix inversion, which is difficult to analyze (Section 2.1). The few existing techniques monitor the model's prediction error or R 2 fit [4,3], or a univariate model where the problem reduces to monitoring a ratio [8]. This is not always sufficient: in some settings we are interested in the model's coefficients (e.g., analysis), and in others (e.g., interpolation) we lack the ground truth to measure prediction error.…”