Online Covariance Matrix Estimation in Sketched Newton Methods
This work addresses the problem of efficient online parameter estimation for users of second-order methods, particularly in the context of streaming data, providing an incremental improvement in the field of online algorithms.
The authors tackled the problem of online covariance matrix estimation in sketched Newton methods, proposing a fully online estimator that achieves consistency and convergence without requiring matrix factorization, and demonstrated its superior performance on regression and benchmark problems. The estimator enables online statistical inference for model parameters with sketched Newton methods.
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.