Refining Covariance Matrix Estimation in Stochastic Gradient Descent through Bias Reduction

Ziyang Wei, Wanrong Zhu, Jingyang Lyu, Wei Biao Wu

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 $n^{(\alpha-1)/2}\sqrt{\log n}$, outperforming existing Hessian-free alternatives. We provide theoretical guarantees for consistency and validate the estimator’s superior finite-sample performance through extensive simulations.

Submission Number: 1526