The Value of Out-of-Distribution Data

Generalization error always improves with more in-distribution data. However, it is an open question what happens as we add out-of-distribution (OOD) data. Intuitively, if the OOD data is quite different, it seems more data would harm generalization error, though if the OOD data are sufficiently similar, much empirical evidence suggests that OOD data can actually improve generalization error. We show a counter-intuitive phenomenon: the generalization error of a task can be a non-monotonic function of the amount of OOD data. Specifically, we prove that generalization error can improve with small amounts of OOD data, and then get worse than no OOD data with larger amounts. In other words, there is value in training on small amounts of OOD data. We analytically demonstrate these results via Fisher's Linear Discriminant on synthetic datasets, and empirically demonstrate them via deep networks on computer vision benchmarks such as MNIST, CIFAR-10, CINIC-10, PACS and DomainNet. In the idealistic setting where we know which samples are OOD, we show that these non-monotonic trends can be exploited using an appropriately weighted objective of the target and OOD empirical risk. While its practical utility is limited, this does suggest that if we can detect OOD samples, then there may be ways to benefit from them. When we do not know which samples are OOD, we show how a number of go-to strategies such as data-augmentation, hyper-parameter optimization and pre-training are not enough to ensure that the target generalization error does not deteriorate with the number of OOD samples in the dataset.