Abstract: Learning models have been shown to rely on spurious correlations between non-predictive features and the associated labels in the training data, with negative implications on robustness, bias and fairness.In this work, we provide a statistical characterization of this phenomenon for high-dimensional regression, when the data contains a predictive *core* feature $x$ and a *spurious* feature $y$. Specifically, we quantify the amount of spurious correlations $\mathcal C$ learned via linear regression, in terms of the data covariance and the strength $\lambda$ of the ridge regularization.As a consequence, we first capture the simplicity of $y$ through the spectrum of its covariance, and its correlation with $x$ through the Schur complement of the full data covariance. Next, we prove a trade-off between $\mathcal C$ and the in-distribution test loss $\mathcal L$, by showing that the value of $\lambda$ that minimizes $\mathcal L$ lies in an interval where $\mathcal C$ is increasing. Finally, we investigate the effects of over-parameterization via the random features model, by showing its equivalence to regularized linear regression.Our theoretical results are supported by numerical experiments on Gaussian, Color-MNIST, and CIFAR-10 datasets.

Lay Summary:

Machine learning models often learn misleading patterns from their training data. For example, a neural network might think that a photo of a person swimming in the ocean is a boat, simply because the network (when training) used to see water only as the background of boats. These patterns are known as spurious correlations, and can negatively impact the fairness, accuracy, and reliability of AI models. In this study, we theoretically explore why and how these misleading correlations happen. We look at the mathematically tractable setting of linear regression, and we provide insights that could be statistically relevant also for more complex models. For example, we attempt to formalize the known fact that neural networks prefer to learn "easy" patterns: intuitively, a blue background is easier to recognize than a heterogeneously-shaped object (e.g., a boat). Furthermore we see that, unfortunately, there is sometimes a trade-off between maximizing the accuracy of a model, and minimizing the amount of spurious correlations. Finally, we extend our theoretical results to the case where models become larger and larger, as in the setting of modernly used deep neural networks.