aiqbench analysis
This paper presents a novel primal-accelerated Newton method that achieves an $O(1/k^3)$ convergence rate for convex optimization with Lipschitz continuous Hessians using only one linear solve per iteration. By eliminating auxiliary nonlinear subproblems and supporting Hessian-free implementations, the work offers significant efficiency gains for large-scale second-order machine learning algorithms. It represents a substantive advance in the theoretical foundations of optimization methods critical to training complex AI models.
Primal Acceleration of Newton's Method
arXiv:2608.21359v1 Announce Type: cross Abstract: We develop a new direct accelerated Newton method for minimizing convex functions with Lipschitz continuous Hessian. The algorithm uses only primal variables and performs just one linear solve per iteration. With a simple predetermined choice of parameters, it achieves the global convergence rate of $O(1/k^3)$ in terms of the functional residual. To the best of our knowledge, this is the first second-order method for this problem class attaining this rate while relying solely on one linear system solve per iteration (without solving auxiliary nonlinear regularized subproblems, such as cubic regularization, performing nonlinear parameter searches, or using dual extragradient corrections). Our method can be implemented in a Hessian-free way, using an inexact linear system solver, while preserving the fast global rate. We further extend our construction to arbitrary geometry through Bregman divergence, and to composite optimization problems.
Abstract:We develop a new direct accelerated Newton method for minimizing convex functions with Lipschitz continuous Hessian. The algorithm uses only primal variables and performs just one linear solve per iteration. With a simple predetermined choice of parameters, it achieves the global convergence rate of $O(1/k^3)$ in terms of the functional residual. To the best of our knowledge, this is the first second-order method for this problem class attaining this rate while relying solely on one linear system solve per iteration (without solving auxiliary nonlinear regularized subproblems, such as cubic regularization, performing nonlinear parameter searches, or using dual extragradient corrections). Our method can be implemented in a Hessian-free way, using an inexact linear system solver, while preserving the fast global rate. We further extend our construction to arbitrary geometry through Bregman divergence, and to composite optimization problems.
Optimization and Control (math.OC); Artificial Intelligence (cs.AI); Machine Learning (cs.LG)
arXiv:2608.21359 [math.OC]
(or arXiv:2608.21359v1 [math.OC] for this version)
https://doi.org/10.48550/arXiv.2608.21359 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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Full text under the publisher's open licence · 237 words
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