Theory of Agreement-on-the-Line in Linear Models and Gaussian Data

Christina Baek, Aditi Raghunathan, J Zico Kolter

Abstract: Past work has observed an ``agreement-on-the-line'' phenomena in deep networks: when the in-distribution (ID) versus out-of-distribution (OOD) accuracy is strongly linearly correlated for a set of classifiers, the same linear trend is observed for the ID versus OOD agreement rate between pairs of classifiers. Theoretically, however, it has been unclear why this phenomena occurs, and whether it was limited to deep networks alone. In this work, we demonstrate that agreement-on-the-line \emph{also} appears in linear models and Gaussian data under additional constraints on the distribution shift beyond those required to observe strong linear trends in accuracy alone. We prove that agreement-on-the-line occurs because models trained from random initialization tend to observe sufficiently uncorrelated errors under distribution shift. We characterize the types of distribution shifts where models behave in this manner. Furthermore, we prove that the correlation of ID versus OOD agreement tends to be at most as good as accuracy. Our conclusions transfer to real-world settings such as CLIP linear probing.

Submission Number: 1518