Inference of Online Newton Methods with Nesterov's Accelerated Sketching

2026-04-25

AI Summary

This work addresses the challenge of achieving robust uncertainty quantification in online learning, where first-order methods are computationally efficient yet sensitive to ill-conditioning and heterogeneous noise, while classical Newton methods suffer from prohibitive $O(d^3)$ complexity. The authors propose an online Newton method based on averaged Hessian information, introducing Nesterov acceleration into a sketch-based solver to approximate Newton directions for the first time. This approach achieves robust uncertainty estimation with only $O(d^2)$ per-iteration computational cost. Theoretical contributions include almost sure convergence of the iterates, asymptotic normality of the last iterate, and the first fully online non-asymptotic covariance estimator. Experiments demonstrate that the method significantly outperforms existing algorithms on regression tasks, delivering more accurate and robust uncertainty estimates at a computational cost comparable to first-order methods.

Technology Category

• Machine Learning: Online Learning & Bandits
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Learning to Search

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Social networks and social learning
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

Reliable decision-making with streaming data requires principled uncertainty quantification of online methods. While first-order methods enable efficient iterate updates, their inference procedures still require updating proper (covariance) matrices, incurring $O(d^2)$ time and memory complexity, and are sensitive to ill-conditioning and noise heterogeneity of the problem. This costly inference task offers an opportunity for more robust second-order methods, which are, however, bottlenecked by solving Newton systems with $O(d^3)$ complexity. In this paper, we address this gap by studying an online Newton method with Hessian averaging, where the Newton direction at each step is approximately computed using a sketch-and-project solver with Nesterov's acceleration, matching $O(d^2)$ complexity of first-order methods. For the proposed method, we quantify its uncertainty arising from both random data and randomized computation. Under standard smoothness and moment conditions, we establish global almost-sure convergence, prove asymptotic normality of the last iterate with a limiting covariance characterized by a Lyapunov equation, and develop a fully online covariance estimator with non-asymptotic convergence guarantees. We also connect the resulting uncertainty quantification to that of exact and sketched Newton methods without Nesterov's acceleration. Extensive experiments on regression models demonstrate the superiority of the proposed method for online inference.

Problem

• online inference
• Newton methods
• uncertainty quantification
• sketching
• Nesterov acceleration

Innovation

• online Newton method
• Nesterov acceleration
• sketch-and-project
• uncertainty quantification
• asymptotic normality