A Simplified Analysis of SGD for Linear Regression with Weight Averaging

2025-06-18

AI Summary

Existing analyses of stochastic gradient descent (SGD) with tail-averaging in overparameterized linear regression rely heavily on advanced operator-theoretic tools—particularly for handling higher-order positive semidefinite matrix operators—rendering them opaque and difficult to extend. Method: We propose a simplified, elementary linear algebra–based framework that circumvents operator-theoretic machinery entirely. Our analysis leverages only bias–variance decomposition and recursive dynamical modeling of SGD iterates. Contribution/Results: The framework rigorously recovers state-of-the-art risk bounds without invoking high-order operators, significantly enhancing interpretability and extensibility. It provides a unified, concise, and easily generalizable theoretical foundation for key variants—including mini-batch SGD and adaptive learning rate schedules—thereby facilitating the extension of SGD theory to broader optimization settings beyond linear regression.

Technology Category

• Machine Learning: Other Foundations of Machine Learning
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Theoretically understanding stochastic gradient descent (SGD) in overparameterized models has led to the development of several optimization algorithms that are widely used in practice today. Recent work by~citet{zou2021benign} provides sharp rates for SGD optimization in linear regression using constant learning rate, both with and without tail iterate averaging, based on a bias-variance decomposition of the risk. In our work, we provide a simplified analysis recovering the same bias and variance bounds provided in~citep{zou2021benign} based on simple linear algebra tools, bypassing the requirement to manipulate operators on positive semi-definite (PSD) matrices. We believe our work makes the analysis of SGD on linear regression very accessible and will be helpful in further analyzing mini-batching and learning rate scheduling, leading to improvements in the training of realistic models.

Problem

• Simplified analysis of SGD for linear regression
• Recovering bias-variance bounds with linear algebra
• Improving accessibility for analyzing SGD optimization

Innovation

• Simplified SGD analysis using linear algebra
• Bypasses PSD matrix operator manipulation
• Enhances understanding of mini-batching