Online Covariance Matrix Estimation in Stochastic Gradient Descent

Abstract

The stochastic gradient descent (SGD) algorithm is widely used for parameter estimation, especially for huge datasets and online learning. While this recursive algorithm is popular for computation and memory efficiency, quantifying variability and randomness of the solutions has been rarely studied. This article aims at conducting statistical inference of SGD-based estimates in an online setting. In particular, we propose a fully online estimator for the covariance matrix of averaged SGD (ASGD) iterates only using the iterates from SGD. We formally establish our online estimator’s consistency and show that the convergence rate is comparable to offline counterparts. Based on the classic asymptotic normality results of ASGD, we construct asymptotically valid confidence intervals for model parameters. Upon receiving new observations, we can quickly update the covariance matrix estimate and the confidence intervals. This approach fits in an online setting and takes full advantage of SGD: efficiency in computation and memory.

KEYWORDS:

• Asymptotic normality
• Averaging stochastic gradient descent
• Recursive
• Statistical inference

Acknowledgments

We thank to the anonymous reviewers and editors for the constructive feedbacks that significantly improved our article. Wanrong Zhu and Wei Biao Wu thank to the support from NSF via NSF-DMS-1916351 and NSF-DMS-2027723. Xi Chen thank to the support from NSF via IIS-1845444.

Additional information

Funding

We thank to the anonymous reviewers and editors for the constructive feedbacks that significantly improved our article. Wanrong Zhu and Wei Biao Wu thank to the support from NSF via NSF-DMS-1916351 and NSF-DMS-2027723. Xi Chen thank to the support from NSF via IIS-1845444.