Precise analysis of ridge interpolators under heavy correlations -- a Random Duality Theory view
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866929385209069568 |
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| author | Stojnic, Mihailo |
| author_facet | Stojnic, Mihailo |
| contents | We consider fully row/column-correlated linear regression models and study several classical estimators (including minimum norm interpolators (GLS), ordinary least squares (LS), and ridge regressors). We show that \emph{Random Duality Theory} (RDT) can be utilized to obtain precise closed form characterizations of all estimators related optimizing quantities of interest, including the \emph{prediction risk} (testing or generalization error). On a qualitative level out results recover the risk's well known non-monotonic (so-called double-descent) behavior as the number of features/sample size ratio increases. On a quantitative level, our closed form results show how the risk explicitly depends on all key model parameters, including the problem dimensions and covariance matrices. Moreover, a special case of our results, obtained when intra-sample (or time-series) correlations are not present, precisely match the corresponding ones obtained via spectral methods in [6,16,17,24]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_09199 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Precise analysis of ridge interpolators under heavy correlations -- a Random Duality Theory view Stojnic, Mihailo Machine Learning Information Theory Statistics Theory We consider fully row/column-correlated linear regression models and study several classical estimators (including minimum norm interpolators (GLS), ordinary least squares (LS), and ridge regressors). We show that \emph{Random Duality Theory} (RDT) can be utilized to obtain precise closed form characterizations of all estimators related optimizing quantities of interest, including the \emph{prediction risk} (testing or generalization error). On a qualitative level out results recover the risk's well known non-monotonic (so-called double-descent) behavior as the number of features/sample size ratio increases. On a quantitative level, our closed form results show how the risk explicitly depends on all key model parameters, including the problem dimensions and covariance matrices. Moreover, a special case of our results, obtained when intra-sample (or time-series) correlations are not present, precisely match the corresponding ones obtained via spectral methods in [6,16,17,24]. |
| title | Precise analysis of ridge interpolators under heavy correlations -- a Random Duality Theory view |
| topic | Machine Learning Information Theory Statistics Theory |
| url | https://arxiv.org/abs/2406.09199 |