Benign Overfitting in Out-of-Distribution Generalization of Linear Models

ICLR 2025 Conference Submission12693 Authors

Keywords: Over-parameterization, benign overfitting, OOD generalization, principal component regression, minimum norm interpolation, ridge regression

TL;DR: We provide non-asymptotic guarantees of over-parameterized ridge regression under general covariate shift.

Abstract: Benign overfitting refers to the phenomenon where a over-parameterized model fits the training data perfectly, including noise in the data, but still generalizes well to the unseen test data. While prior work provide a solid theoretical understanding of this phenomenon under the in-distribution setup, modern machine learning often operates in a more challenging Out-of-Distribution (OOD) regime, where the target (test) distribution can be rather different from the source (training) distribution. In this work, we take an initial step towards understanding benign overfitting in the OOD regime by focusing on the basic setup of over-parameterized linear models under covariate shift. We provide non-asymptotic guarantees proving that, when the target covariance satisfies certain structural conditions, benign overfitting occurs in standard ridge regression even under the OOD regime. We identify a number of key quantities relating source and target covariance, which govern the performance of OOD generalization. Our result is sharp, which provably recovers prior in-distribution benign overfitting guarantee (Tsigler & Bartlett, 2023), as well as under-parameterized OOD guarantee (Ge et al., 2024) when specializing to each setup. Moreover, we also present theoretical results for a more general family of target covariance matrix, where standard ridge regression only achieves a slow statistical rate of $\mathcal{O}(1/\sqrt{n})$ for the excess risk, while Principal Component Regression (PCR) is guaranteed to achieve the fast rate $\mathcal{O}(1/n)$, where $n$ is the number of samples.

Primary Area: learning theory

Code Of Ethics: I acknowledge that I and all co-authors of this work have read and commit to adhering to the ICLR Code of Ethics.

Submission Guidelines: I certify that this submission complies with the submission instructions as described on https://iclr.cc/Conferences/2025/AuthorGuide.

Reciprocal Reviewing: I understand the reciprocal reviewing requirement as described on https://iclr.cc/Conferences/2025/CallForPapers. If none of the authors are registered as a reviewer, it may result in a desk rejection at the discretion of the program chairs. To request an exception, please complete this form at https://forms.gle/Huojr6VjkFxiQsUp6.

Anonymous Url: I certify that there is no URL (e.g., github page) that could be used to find authors’ identity.

No Acknowledgement Section: I certify that there is no acknowledgement section in this submission for double blind review.

Submission Number: 12693