# Online Covariance Matrix Estimation in Sketched Newton Methods

arXiv: https://arxiv.org/abs/2502.07114 · HTML: https://mlsanity.com/paper/2502.07114/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Feb 2025.

## Built on by (most cited first, 4 of 4)

- [High Probability Complexity Bounds of Trust-Region Stochastic Sequential Quadratic Programming with Heavy-Tailed Noise](https://mlsanity.com/paper/2503.19091/) (2025)
- [Online Statistical Inference of Constrained Stochastic Optimization via Random Scaling](https://mlsanity.com/paper/2505.18327/) (2025)
- [TRSVR: An Adaptive Stochastic Trust-Region Method with Variance Reduction](https://mlsanity.com/paper/2601.14647/) (2026)
- [Statistical Inference for Rank Allocation in Low-Rank Adaptation](https://mlsanity.com/paper/2607.20205/) (2026)

## Abstract (arXiv)

Given the ubiquity of streaming data, online algorithms have been widely used for parameter estimation, with second-order methods particularly standing out for their efficiency and robustness. In this paper, we study an online sketched Newton method that leverages a randomized sketching technique to perform an approximate Newton step in each iteration, thereby eliminating the computational bottleneck of second-order methods. While existing studies have established the asymptotic normality of sketched Newton methods, a consistent estimator of the limiting covariance matrix remains an open problem. We propose a fully online covariance matrix estimator that is constructed entirely from the Newton iterates and requires no matrix factorization. Compared to covariance estimators for first-order online methods, our estimator for second-order methods is batch-free. We establish the consistency and convergence rate of our estimator, and coupled with asymptotic normality results, we can then perform online statistical inference for the model parameters based on sketched Newton methods. We also discuss the extension of our estimator to constrained problems, and demonstrate its superior performance on regression problems as well as benchmark problems in the CUTEst set.