mlsanityData to 7 Oct 2026

Online Covariance Matrix Estimation in Stochastic Gradient Descent

arXiv 2002.03979 ↗First posted 10 Feb 2020stat.ML cs.LGcited by 37

Not on the map. Why?

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.

Key facts. Cited by 37 papers in the map (data through 7 Oct 2026). First posted on arXiv 10 Feb 2020.

Builds on

No reference list yet.

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 4; 2022: 4; 2023: 8; 2024: 8; 2025: 8; 2026: 5.

37 papers.

Sort:

  1. 2021A Statistical Analysis of Polyak-Ruppert Averaged Q-learningcited by 29
  2. 2021Fast and Robust Online Inference with Stochastic Gradient Descent via Random Scalingcited by 28
  3. 2024Gaussian Approximation and Multiplier Bootstrap for Polyak-Ruppert Averaged Linear Stochastic Approximation with Applications to TD Learningcited by 19
  4. 2023Online Statistical Inference for Nonlinear Stochastic Approximation with Markovian Datacited by 18
  5. 2021Online Statistical Inference for Stochastic Optimization via Kiefer-Wolfowitz Methodscited by 14
  6. 2022Statistical Inference of Constrained Stochastic Optimization via Sketched Sequential Quadratic Programmingcited by 10
  7. 2024High Confidence Level Inference is Almost Free using Parallel Stochastic Optimizationcited by 9
  8. 2025Gaussian Approximation and Multiplier Bootstrap for Stochastic Gradient Descentcited by 8
  9. 2022Online Statistical Inference for Contextual Bandits via Stochastic Gradient Descentcited by 7
  10. 2023Weighted Averaged Stochastic Gradient Descent: Asymptotic Normality and Optimalitycited by 7
  11. 2023Online covariance estimation for stochastic gradient descent under Markovian samplingcited by 7
  12. 2023Acceleration of stochastic gradient descent with momentum by averaging: finite-sample rates and asymptotic normalitycited by 6
25 more
  1. 2022Covariance Estimators for the ROOT-SGD Algorithm in Online Learningcited by 5
  2. 2023Effectiveness of Constant Stepsize in Markovian LSA and Statistical Inferencecited by 5
  3. 2023Fairness Uncertainty Quantification: How certain are you that the model is fair?cited by 4
  4. 2024Stochastic Optimization Algorithms for Instrumental Variable Regression with Streaming Datacited by 4
  5. 2025Online Covariance Estimation in Nonsmooth Stochastic Approximationcited by 4
  6. 2023SGMM: Stochastic Approximation to Generalized Method of Momentscited by 3
  7. 2024Asymptotic Time-Uniform Inference for Parameters in Averaged Stochastic Approximationcited by 3
  8. 2021Statistical Estimation and Inference via Local SGD in Federated Learningcited by 2
  9. 2025Statistical Inference for Online Algorithmscited by 2
  10. 2025Beyond Sin-Squared Error: Linear-Time Entrywise Uncertainty Quantification for Streaming PCAcited by 2
  11. 2025Improved Central Limit Theorem and Bootstrap Approximations for Linear Stochastic Approximationcited by 2
  12. 2023Statistical Inference for Linear Functionals of Online SGD in High-dimensional Linear Regressioncited by 1
  13. 2024Uncertainty quantification by block bootstrap for differentially private stochastic gradient descentcited by 1
  14. 2025Sharp Gaussian approximations for Decentralized Federated Learningcited by 1
  15. 2025Statistical Inference for Linear Functionals of Online Least-squares SGD when t gtrsim d1+δcited by 1
  16. 2026Gaussian Approximation for Asynchronous Q-learningcited by 1
  17. 2022Parameter-Parallel Distributed Variational Quantum Algorithmcited by 0
  18. 2024A Selective Review on Statistical Methods for Massive Data Computation: Distributed Computing, Subsampling, and Minibatch Techniquescited by 0
  19. 2024A Full Adagrad algorithm with O(Nd) operationscited by 0
  20. 2024Enhancing Stochastic Optimization for Statistical Efficiency Using ROOT-SGD with Diminishing Stepsizecited by 0
  21. 2025Central Limit Theorems for Stochastic Gradient Descent Quantile Estimatorscited by 0
  22. 2026Sharp asymptotic theory for Q-learning with LDTZ learning rate and its generalizationcited by 0
  23. 2026Online Covariance Estimation in Averaged SGD: Improved Batch-Mean Rates and Minimax Optimality via Trajectory Regressioncited by 0
  24. 2026Refining Covariance Matrix Estimation in Stochastic Gradient Descent Through Bias Reductioncited by 0
  25. 2026Statistical Inference on Gradient Flowscited by 0