Online Covariance Matrix Estimation in Sketched Newton Methods

TL;DR

This paper introduces an online covariance estimator for sketched Newton methods that is fully online, batch-free, and enables statistical inference in streaming data scenarios.

Contribution

It proposes a novel, fully online covariance estimator for second-order sketched Newton methods, addressing a key open problem in online statistical inference.

Findings

The estimator is consistent and converges at a quantifiable rate.

It enables online statistical inference for model parameters.

Demonstrated superior performance on regression and benchmark problems.

Abstract

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.