# Online Covariance Matrix Estimation in Stochastic Gradient Descent

arXiv: https://arxiv.org/abs/2002.03979 · HTML: https://mlsanity.com/paper/2002.03979/

## Key facts

- Cited by 37 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Feb 2020.

## Built on by (most cited first, 30 of 37)

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- [Online Statistical Inference for Stochastic Optimization via Kiefer-Wolfowitz Methods](https://mlsanity.com/paper/2102.03389/) (2021)
- [Statistical Inference of Constrained Stochastic Optimization via Sketched Sequential Quadratic Programming](https://mlsanity.com/paper/2205.13687/) (2022)
- [High Confidence Level Inference is Almost Free using Parallel Stochastic Optimization](https://mlsanity.com/paper/2401.09346/) (2024)
- [Gaussian Approximation and Multiplier Bootstrap for Stochastic Gradient Descent](https://mlsanity.com/paper/2502.06719/) (2025)
- [Acceleration of stochastic gradient descent with momentum by averaging: finite-sample rates and asymptotic normality](https://mlsanity.com/paper/2305.17665/) (2023)
- [Online Statistical Inference for Contextual Bandits via Stochastic Gradient Descent](https://mlsanity.com/paper/2212.14883/) (2022)
- [Weighted Averaged Stochastic Gradient Descent: Asymptotic Normality and Optimality](https://mlsanity.com/paper/2307.06915/) (2023)
- [Online covariance estimation for stochastic gradient descent under Markovian sampling](https://mlsanity.com/paper/2308.01481/) (2023)
- [Covariance Estimators for the ROOT-SGD Algorithm in Online Learning](https://mlsanity.com/paper/2212.01259/) (2022)
- [Effectiveness of Constant Stepsize in Markovian LSA and Statistical Inference](https://mlsanity.com/paper/2312.10894/) (2023)
- [Fairness Uncertainty Quantification: How certain are you that the model is fair?](https://mlsanity.com/paper/2304.13950/) (2023)
- [Stochastic Optimization Algorithms for Instrumental Variable Regression with Streaming Data](https://mlsanity.com/paper/2405.19463/) (2024)
- [Online Covariance Estimation in Nonsmooth Stochastic Approximation](https://mlsanity.com/paper/2502.05305/) (2025)
- [Improved Central Limit Theorem and Bootstrap Approximations for Linear Stochastic Approximation](https://mlsanity.com/paper/2510.12375/) (2025)
- [SGMM: Stochastic Approximation to Generalized Method of Moments](https://mlsanity.com/paper/2308.13564/) (2023)
- [Asymptotic Time-Uniform Inference for Parameters in Averaged Stochastic Approximation](https://mlsanity.com/paper/2410.15057/) (2024)
- [Beyond Sin-Squared Error: Linear-Time Entrywise Uncertainty Quantification for Streaming PCA](https://mlsanity.com/paper/2506.12655/) (2025)
- [Statistical Estimation and Inference via Local SGD in Federated Learning](https://mlsanity.com/paper/2109.01326/) (2021)
- [Statistical Inference for Online Algorithms](https://mlsanity.com/paper/2505.17300/) (2025)
- [Statistical Inference for Linear Functionals of Online SGD in High-dimensional Linear Regression](https://mlsanity.com/paper/2302.09727/) (2023)
- [A Full Adagrad algorithm with O(Nd) operations](https://mlsanity.com/paper/2405.01908/) (2024)
- [Uncertainty quantification by block bootstrap for differentially private stochastic gradient descent](https://mlsanity.com/paper/2405.12553/) (2024)
- [Sharp Gaussian approximations for Decentralized Federated Learning](https://mlsanity.com/paper/2505.08125/) (2025)
- [Statistical Inference for Linear Functionals of Online Least-squares SGD when t gtrsim d1+δ](https://mlsanity.com/paper/2510.19734/) (2025)
- [Gaussian Approximation for Asynchronous Q-learning](https://mlsanity.com/paper/2604.07323/) (2026)
- [Statistical Inference on Gradient Flows](https://mlsanity.com/paper/2606.01257/) (2026)

## Abstract (arXiv)

The stochastic gradient descent (SGD) algorithm is widely used for parameter estimation, especially for huge data sets 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 paper 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 iterates (ASGD) 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.