Out-of-Distribution Generalization in Kernel Regression

In real word applications, the data generating process for training a machine learning model often differs from what the model encounters in the test stage. Understanding how and whether machine learning models generalize under such distributional shifts remains a theoretical challenge. Here, we study generalization in kernel regression when the training and test distributions are different using the replica method from statistical physics. We derive an analytical formula for the out-of-distribution generalization error applicable to any kernel and real datasets. We identify an overlap matrix that quantifies the mismatch between distributions for a given kernel as a key determinant of generalization performance under distribution shift. Using our analytical expressions we elucidate various generalization phenomena including possible improvement in generalization when there is a mismatch. We develop procedures for optimizing training and test distributions for a given data budget to find best and worst case generalizations under the shift. We present applications of our theory to real and synthetic datasets and for many kernels. We compare results of our theory applied to Neural Tangent Kernel with simulations of wide networks and show agreement. We analyze linear regression in further depth.