Generalization for Least Squares Regression With Simple Spiked Covariances (2410.13991v1)

Abstract: Random matrix theory has proven to be a valuable tool in analyzing the generalization of linear models. However, the generalization properties of even two-layer neural networks trained by gradient descent remain poorly understood. To understand the generalization performance of such networks, it is crucial to characterize the spectrum of the feature matrix at the hidden layer. Recent work has made progress in this direction by describing the spectrum after a single gradient step, revealing a spiked covariance structure. Yet, the generalization error for linear models with spiked covariances has not been previously determined. This paper addresses this gap by examining two simple models exhibiting spiked covariances. We derive their generalization error in the asymptotic proportional regime. Our analysis demonstrates that the eigenvector and eigenvalue corresponding to the spike significantly influence the generalization error.

• The paper derives closed-form generalization error expressions for regression models with simple spiked covariance structures.
• It employs random matrix theory and extensions of the Stieltjes transform to quantify finite effects in both signal-only and signal-plus-noise models.
• It reveals that eigenvector alignment and eigenvalue spikes significantly affect risk in finite matrices, guiding improved feature learning in neural networks.

Generalization for Least Squares Regression with Simple Spiked Covariances

The paper "Generalization for Least Squares Regression with Simple Spiked Covariances" by Jiping Li and Rishi Sonthalia investigates the generalization error in least squares regression models with a spiked covariance structure. This research is motivated by the challenge of characterizing the spectrum of the feature matrix in neural networks and understanding how spiked covariances influence generalization.

Summary

The paper addresses the need to quantify the generalization error in models with spiked covariances. Previous efforts have identified a spiked covariance structure in two-layer neural networks post-gradient descent, making it essential to explore its effects on generalization. The primary contributions of the paper include:

1. Model Exploration: It examines two linear regression models illustrating spiked covariances. The paper extends existing models by considering how learned features interact with target functions through two specific regression targets.
2. Theoretical Derivations: The research provides closed-form expressions for the generalization error in these models. The results are theoretically grounded, building upon random matrix theory and previous work in the field.
3. Impact of Spikes: The authors demonstrate that the generalization error can be decomposed into an asymptotic risk plus a correction term influenced by the eigenvector and eigenvalue corresponding to the spike. They show significant corrections in finite matrices.

Detailed Analysis

Challenges with Spiked Covariances: The paper identifies that spiked covariance matrices introduce specific challenges in computing generalization errors. While traditional random matrix theory techniques provide insights into the limits, they often fail to capture the finite effects of spikes. The paper extends tools like the Stieltjes transform to quantify these effects.

Signal Models: Two problems are primarily addressed:

• Signal-Only Model: The target depends exclusively on the signal, ignoring the noise (bulk component).
• Signal-and-Noise Model: The model considers both the signal and noise components in the target function.

Key Findings:

• In the signal-plus-noise model, spikes affect the risk only within finite matrices, which can become negligible in asymptotic regimes.
• In the signal-only model, alignment between the spike's eigenvector and the target significantly affects risk, even in asymptotic considerations.

Implications and Future Directions

This paper's findings highlight the importance of considering finite-dimensional effects when assessing the generalization performance of models with spiked covariance structures. Theoretical advancements from this paper can inform better feature learning strategies in neural networks, emphasizing eigenvector alignment.

Future Research:

• Addressing multiple spikes and extending the analysis to multi-step neural network training offers potential advancements.
• Integrating dependencies between different matrix components, as seen in practical applications, would be an important step forward.

Overall, the paper provides significant insights into the interaction between spiked covariances and generalization performance, offering a theoretical framework that could guide future explorations of neural network training dynamics.