Benign Overfitting in Linear Regression (1906.11300v3)

Abstract: The phenomenon of benign overfitting is one of the key mysteries uncovered by deep learning methodology: deep neural networks seem to predict well, even with a perfect fit to noisy training data. Motivated by this phenomenon, we consider when a perfect fit to training data in linear regression is compatible with accurate prediction. We give a characterization of linear regression problems for which the minimum norm interpolating prediction rule has near-optimal prediction accuracy. The characterization is in terms of two notions of the effective rank of the data covariance. It shows that overparameterization is essential for benign overfitting in this setting: the number of directions in parameter space that are unimportant for prediction must significantly exceed the sample size. By studying examples of data covariance properties that this characterization shows are required for benign overfitting, we find an important role for finite-dimensional data: the accuracy of the minimum norm interpolating prediction rule approaches the best possible accuracy for a much narrower range of properties of the data distribution when the data lies in an infinite dimensional space versus when the data lies in a finite dimensional space whose dimension grows faster than the sample size.

• The paper demonstrates that the minimum norm interpolating solution in linear regression can achieve near-optimal prediction accuracy under specific effective rank conditions.
• It rigorously characterizes how overparameterization and eigenvalue decay, whether polynomial or exponential, impact the balance between noise and prediction error.
• The results bridge theory and practice by connecting benign overfitting phenomena in linear models to potential applications in deep neural networks via NTK theory.

Benign Overfitting in Linear Regression

The paper authored by Peter L. Bartlett, Philip M. Long, Gábor Lugosi, and Alexander Tsigler addresses a significant phenomenon observed in deep learning, known as benign overfitting. This occurs when complex models, such as deep neural networks, achieve zero training error while still delivering strong predictive performance on new, unseen data. This observation contravenes classical statistical learning theories, which typically posit that overfitting—achieving very low training error at the cost of high model complexity—tends to harm predictive performance.

Key Contributions

The primary focus of this paper is to investigate when overfitting can be considered benign in the context of linear regression. The authors provide a comprehensive analysis, characterizing the conditions under which the minimum norm interpolating solution—a specific way to exactly fit the training data—yields near-optimal prediction accuracy.

Effective Rank and Overparameterization

Central to the authors' characterization are two notions of effective rank tied to the covariance matrix of the data:

1. Effective Rank $r_k$: Based on the sum of eigenvalues beyond a certain index k.
2. Alternative Effective Rank $R_k$: Dependent on the squared sum of eigenvalues beyond the index k, normalized by their sum.

For overfitting to be benign, the authors argue that overparameterization is crucial. Specifically, the number of directions (i.e., dimensions in parameter space) that are unimportant for prediction must significantly exceed the sample size. This is quantified by showing that $r_k$ must be large relative to the sample size $n$, meaning the projection of the data into a subspace of low-variance directions should be spread out enough to mask the noise without severely affecting prediction accuracy.

Main Theoretical Results

The authors derive an upper and a lower bound on the excess risk of the minimum norm interpolating estimator:

• Upper Bound: The excess risk is bounded by terms involving the effective ranks and the norm of the optimal parameter, $\|\theta^*\|^2$, and the largest eigenvalue of the covariance matrix, $\| \Sigma \|$.
• Lower Bound: In contrast, they establish that when the effective rank conditions are not met, the excess risk will be proportional to a constant, emphasizing that severe overfitting cannot be benign without appropriate data covariance properties.

Numerical Implications and Examples

The authors elucidate these results using various sequences and patterns of eigenvalues of the covariance matrix:

1. Polynomial Decay: Covariances with polynomial decay rates in their eigenvalues may lead to varying results. Specifically, for eigenvalues decaying as $k^{-\alpha}$, benign overfitting occurs only when $\alpha = 1$ and there is an additional logarithmic decay term.
2. Exponential Plus Constant: When adding a small isotropic noise component to a high-dimensional but finite space, benign overfitting occurs only if both the dimension is large relative to the sample size and the noise level is appropriately tuned.

Implications for Deep Learning

The results provide insights into why deep neural networks might exhibit benign overfitting. The neural tangent kernel (NTK) theory, which approximates wide neural networks as nearly linear models, can potentially exhibit suitable covariance eigenvalue distributions. This theory aligns with the findings that slowly decaying eigenvalues in high-dimensional spaces allow for benign overfitting, as observed in deep learning models.

Conclusion and Future Work

This work illuminates the conditions under which benign overfitting is possible in linear models, providing a bridge to understand similar phenomena in deep neural networks. Future work could extend these results to non-linear settings or explore other loss functions. Moreover, testing these theoretical findings empirically in real-world deep learning models remains a critical task.

This paper advances our understanding of overfitting in high-dimensional settings and suggests that the high-dimensional, weakly informative parameter spaces inherent to modern deep learning architectures might naturally accommodate benign overfitting.