Towards Data-Algorithm Dependent Generalization: a Case Study on Overparameterized Linear Regression

TL;DR

This paper investigates the generalization behavior of overparameterized linear regression models, emphasizing the importance of data and algorithm interplay, and introduces a new data-algorithm compatibility concept to improve understanding.

Contribution

It introduces the data-algorithm compatibility notion and provides a trajectory-based analysis for overparameterized linear regression, highlighting the role of early stopping.

Findings

Early stopping improves generalization under weaker conditions.

Traditional last-iterate analysis can be overly restrictive.

Data-algorithm compatibility offers a more comprehensive understanding.

Abstract

One of the major open problems in machine learning is to characterize generalization in the overparameterized regime, where most traditional generalization bounds become inconsistent even for overparameterized linear regression. In many scenarios, this failure can be attributed to obscuring the crucial interplay between the training algorithm and the underlying data distribution. This paper demonstrate that the generalization behavior of overparameterized model should be analyzed in a both data-relevant and algorithm-relevant manner. To make a formal characterization, We introduce a notion called data-algorithm compatibility, which considers the generalization behavior of the entire data-dependent training trajectory, instead of traditional last-iterate analysis. We validate our claim by studying the setting of solving overparameterized linear regression with gradient descent. Specifically, we perform a data-dependent trajectory analysis and derive a sufficient condition for compatibility in such a setting. Our theoretical results demonstrate that if we take early stopping iterates into consideration, generalization can hold with significantly weaker restrictions on the problem instance than the previous last-iterate analysis.