Benign Overfitting in Out-of-Distribution Generalization of Linear Models (2412.14474v1)

Abstract: Benign overfitting refers to the phenomenon where an 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 provides some 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 benign overfitting occurs in standard ridge regression, even under the OOD regime when the target covariance satisfies certain structural conditions. We identify several vital quantities relating to 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 and 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 $O(1/\sqrt{n})$ for the excess risk, while Principal Component Regression (PCR) is guaranteed to achieve the fast rate $O(1/n)$, where $n$ is the number of samples.

• The paper establishes non-asymptotic risk bounds for ridge regression, revealing conditions under which benign overfitting occurs despite covariate shifts.
• It shows that when minor component shifts are limited, ridge regression achieves error rates comparable to in-distribution scenarios, ensuring reliable OOD generalization.
• The paper also identifies Principal Component Regression as a robust alternative for cases with significant minor covariance shifts, enabling faster convergence.

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

The research presented focuses on understanding the phenomenon of benign overfitting within the context of Out-of-Distribution (OOD) generalization in over-parameterized linear models, with a primary application to scenarios under covariate shift. This paper provides valuable non-asymptotic guarantees that confirm benign overfitting can occur in standard ridge regression when certain structural conditions of covariate shift are met. Moreover, this work extends its analysis to include Principal Component Regression (PCR) as an effective approach under large covariate shifts in minor directions.

Conceptual Framework

Benign overfitting is a phenomenon where an over-parameterized model fits the training data closely, including noise, yet generalizes effectively to test data. Common in modern machine learning, this behavior is particularly relevant under the OOD regime, where the test distribution may differ substantially from the training distribution. The paper explores these patterns specifically, by analyzing over-parameterized linear models subjected to covariate shifts.

Key Contributions

1. Theoretical Insights into OOD Benign Overfitting: The paper establishes non-asymptotic excess risk bounds for ridge regression within an OOD context. It highlights that benign overfitting is maintained under covariate shifts, provided that the perturbation in the source and target distribution's covariance structure is suitably constrained. The analysis characterizes shifts in major and minor eigenspaces differently, showing that changes in minor directions need only a bounded overall magnitude for benign overfitting to persist.
2. Phenomenon Under Different Conditions: Sharp instance-dependent bounds demonstrate that, when the target distribution's minor components are small relative to the source, ridge regression can achieve similar error rates as the in-distribution scenario. This is particularly crucial for cases where data resides on a low-dimensional manifold, a prevalent condition in contemporary data-driven domains such as image and language data.
3. Handling Large Covariate Shifts: The authors identify scenarios where ridge regression struggles, especially when there is a substantial variance in the target distribution's minor components, yet the work elegantly navigates these issues by proposing Principal Component Regression (PCR). PCR is demonstrated to outperform traditionally naive approaches, achieving fast rates of convergence particularly when shifts in the minor covariance components are significant.
4. Simulation and Empirical Validation: Through simulation, the outcomes bolster theoretical claims by illustrating how ridge regression performs under various magnitudes of covariate shifts and confirming the superiority of PCR under predominant conditions of major-shift.

Practical and Theoretical Implications

The implications of this paper are profound, underscoring the complexities of managing generalization in OOD environments with over-parameterized models. Practically, it guides the development of more robust algorithms for real-world applications where distribution shifts are inevitable, such as cross-domain machine learning tasks. Theoretically, it opens a dialogue on the nature of overfitting and generalization in high-dimensional spaces, challenging traditional paradigms of learning theory by showing that interpolation of noise does not necessarily preclude generalization capability.

Future Directions

Future work could explore developing understanding across broader model classes, like those involving deep non-linear architectures, beyond the linear regime analyzed here. Additionally, further exploration into alternative regularization strategies coupled with large-scale empirical validation across diverse datasets and covariance structures would enhance the applicability and thoroughness of these findings.

In conclusion, this research significantly advances the theoretical understanding of benign overfitting within OOD frameworks, providing crucial insights for the deployment of robust learning algorithms in high-dimensional data landscapes, deeply influencing both current practices and future explorations in the field.