Out of the Ordinary: Spectrally Adapting Regression for Covariate Shift

Designing deep neural network classifiers that perform robustly on distributions differing from the available training data is an active area of machine learning research. However, out-of-distribution generalization for regression---the analogous problem for modeling continuous targets---remains relatively unexplored. To tackle this problem, we return to first principles and analyze how the closed-form solution for Ordinary Least Squares (OLS) regression is sensitive to covariate shift. We characterize the out-of-distribution risk of the OLS model in terms of the eigenspectrum decomposition of the source and target data. We then use this insight to propose a method called Spectral Adapted Regressor (SpAR) for adapting the weights of the last layer of a pre-trained neural regression model to perform better on input data originating from a different distribution. We demonstrate how this lightweight spectral adaptation procedure can improve out-of-distribution performance for synthetic and real-world datasets.