Optimal Excess Risk Bounds for Empirical Risk Minimization on $p$-Norm Linear Regression
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866909225659138048 |
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| author | Hanchi, Ayoub El Erdogdu, Murat A. |
| author_facet | Hanchi, Ayoub El Erdogdu, Murat A. |
| contents | We study the performance of empirical risk minimization on the $p$-norm linear regression problem for $p \in (1, \infty)$. We show that, in the realizable case, under no moment assumptions, and up to a distribution-dependent constant, $O(d)$ samples are enough to exactly recover the target. Otherwise, for $p \in [2, \infty)$, and under weak moment assumptions on the target and the covariates, we prove a high probability excess risk bound on the empirical risk minimizer whose leading term matches, up to a constant that depends only on $p$, the asymptotically exact rate. We extend this result to the case $p \in (1, 2)$ under mild assumptions that guarantee the existence of the Hessian of the risk at its minimizer. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_12437 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Optimal Excess Risk Bounds for Empirical Risk Minimization on $p$-Norm Linear Regression Hanchi, Ayoub El Erdogdu, Murat A. Statistics Theory Machine Learning We study the performance of empirical risk minimization on the $p$-norm linear regression problem for $p \in (1, \infty)$. We show that, in the realizable case, under no moment assumptions, and up to a distribution-dependent constant, $O(d)$ samples are enough to exactly recover the target. Otherwise, for $p \in [2, \infty)$, and under weak moment assumptions on the target and the covariates, we prove a high probability excess risk bound on the empirical risk minimizer whose leading term matches, up to a constant that depends only on $p$, the asymptotically exact rate. We extend this result to the case $p \in (1, 2)$ under mild assumptions that guarantee the existence of the Hessian of the risk at its minimizer. |
| title | Optimal Excess Risk Bounds for Empirical Risk Minimization on $p$-Norm Linear Regression |
| topic | Statistics Theory Machine Learning |
| url | https://arxiv.org/abs/2310.12437 |