Refining Covariance Matrix Estimation in Stochastic Gradient Descent Through Bias Reduction

2026-04-22

AI Summary

This work addresses the challenge of covariance estimation in online stochastic gradient descent (SGD), where conventional methods either rely on the Hessian matrix—often intractable in practice—or suffer from slow convergence. The paper proposes a novel fully online debiased covariance estimator that leverages bias-reduction techniques to achieve substantially improved estimation accuracy and convergence speed without requiring second-order derivative information. The method attains a convergence rate of $n^{(\alpha-1)/2} \sqrt{\log n}$, outperforming existing Hessian-free approaches. This advancement provides an efficient and practical tool for asymptotic statistical inference in online SGD settings.

Technology Category

• Machine Learning: Online Learning & Bandits
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We study online inference and asymptotic covariance estimation for the stochastic gradient descent (SGD) algorithm. While classical methods (such as plug-in and batch-means estimators) are available, they either require inaccessible second-order (Hessian) information or suffer from slow convergence. To address these challenges, we propose a novel, fully online de-biased covariance estimator that eliminates the need for second-order derivatives while significantly improving estimation accuracy. Our method employs a bias-reduction technique to achieve a convergence rate of $n^{(α-1)/2} \sqrt{\log n}$, outperforming existing Hessian-free alternatives.

Problem

• covariance estimation
• stochastic gradient descent
• bias reduction
• online inference
• asymptotic covariance