Spurious Correlations in High Dimensional Regression: The Roles of Regularization, Simplicity Bias and Over-Parameterization

• URL: http://arxiv.org/abs/2502.01347v1
• Date: Mon, 03 Feb 2025 13:38:42 GMT
• Title: Spurious Correlations in High Dimensional Regression: The Roles of Regularization, Simplicity Bias and Over-Parameterization
• Authors: Simone Bombari, Marco Mondelli,
• Abstract summary: Learning models have been shown to rely on spurious correlations between non-predictive features and the associated labels in the training data. We quantify the amount of spurious correlations $C$ learned via linear regression, in terms of the data covariance and the strength $lambda$ of the ridge regularization.
• Score: 19.261178173399784
• License:
• 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 $C$ 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 $C$ and the in-distribution test loss $L$, by showing that the value of $\lambda$ that minimizes $L$ lies in an interval where $C$ 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.

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