Spurious Correlations in High Dimensional Regression: The Roles of Regularization, Simplicity Bias and Over-Parameterization
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| author | Bombari, Simone Mondelli, Marco |
| author_facet | Bombari, Simone Mondelli, Marco |
| contents | 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 $λ$ 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 $λ$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_01347 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spurious Correlations in High Dimensional Regression: The Roles of Regularization, Simplicity Bias and Over-Parameterization Bombari, Simone Mondelli, Marco Machine Learning 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 $λ$ 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 $λ$ 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. |
| title | Spurious Correlations in High Dimensional Regression: The Roles of Regularization, Simplicity Bias and Over-Parameterization |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2502.01347 |