Paper Overview
Research area: Machine Learning Author: Nikita Doikov Published: 2026-08-21 arXiv: 2608.21359
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 generalize the construction to arbitrary geometries (via Bregman divergences) and to composite optimization problems.
Key Highlights
- Direct acceleration of Newton's method, primal variables only
- One linear solve per iteration
- Global \(O(1/k^3)\) rate in functional residual with predetermined parameters
- Hessian-free implementation possible with inexact linear solvers
- Extensions to Bregman divergence geometries and composite optimization