Generalisation and benign over-fitting for linear regression onto random functional covariates
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866911111175995392 |
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| author | Jones, Andrew Whiteley, Nick |
| author_facet | Jones, Andrew Whiteley, Nick |
| contents | We study theoretical predictive performance of ridge and ridge-less least-squares regression when covariate vectors arise from evaluating $p$ random, means-square continuous functions over a latent metric space at $n$ random and unobserved locations, subject to additive noise. This leads us away from the standard assumption of i.i.d. data to a setting in which the $n$ covariate vectors are exchangeable but not independent in general. Under an assumption of independence across dimensions, $4$-th order moment, and other regularity conditions, we obtain probabilistic bounds on a notion of predictive excess risk adapted to our random functional covariate setting, making use of recent results of Barzilai and Shamir. We derive convergence rates in regimes where $p$ grows suitably fast relative to $n$, illustrating interplay between ingredients of the model in determining convergence behaviour and the role of additive covariate noise in benign-overfitting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_13895 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Generalisation and benign over-fitting for linear regression onto random functional covariates Jones, Andrew Whiteley, Nick Machine Learning We study theoretical predictive performance of ridge and ridge-less least-squares regression when covariate vectors arise from evaluating $p$ random, means-square continuous functions over a latent metric space at $n$ random and unobserved locations, subject to additive noise. This leads us away from the standard assumption of i.i.d. data to a setting in which the $n$ covariate vectors are exchangeable but not independent in general. Under an assumption of independence across dimensions, $4$-th order moment, and other regularity conditions, we obtain probabilistic bounds on a notion of predictive excess risk adapted to our random functional covariate setting, making use of recent results of Barzilai and Shamir. We derive convergence rates in regimes where $p$ grows suitably fast relative to $n$, illustrating interplay between ingredients of the model in determining convergence behaviour and the role of additive covariate noise in benign-overfitting. |
| title | Generalisation and benign over-fitting for linear regression onto random functional covariates |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2508.13895 |