# Primal Acceleration of Newton's Method

arXiv: https://arxiv.org/abs/2608.21359 · HTML: https://mlsanity.com/paper/2608.21359/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 21 of its 39 references that are in the map.
- First posted on arXiv 21 Aug 2026.

## Builds on (most cited first, 21 of 21)

- [Generalized Self-Concordant Functions: A Recipe for Newton-Type Methods](https://mlsanity.com/paper/1703.04599/) (2017)
- [Acceleration with a Ball Optimization Oracle](https://mlsanity.com/paper/2003.08078/) (2020)
- [Oracle Complexity of Second-Order Methods for Smooth Convex Optimization](https://mlsanity.com/paper/1705.07260/) (2017)
- [Optimal and Adaptive Monteiro-Svaiter Acceleration](https://mlsanity.com/paper/2205.15371/) (2022)
- [Regularized Newton Method with Global O(1/k2) Convergence](https://mlsanity.com/paper/2112.02089/) (2021)
- [The First Optimal Acceleration of High-Order Methods in Smooth Convex Optimization](https://mlsanity.com/paper/2205.09647/) (2022)
- [Super-Universal Regularized Newton Method](https://mlsanity.com/paper/2208.05888/) (2022)
- [Non-asymptotic Superlinear Convergence of Standard Quasi-Newton Methods](https://mlsanity.com/paper/2003.13607/) (2020)
- [Gradient Regularization of Newton Method with Bregman Distances](https://mlsanity.com/paper/2112.02952/) (2021)
- [New Results on Superlinear Convergence of Classical Quasi-Newton Methods](https://mlsanity.com/paper/2004.14866/) (2020)
- [Lower Bounds for Higher-Order Convex Optimization](https://mlsanity.com/paper/1710.10329/) (2017)
- [Minimizing Uniformly Convex Functions by Cubic Regularization of Newton Method](https://mlsanity.com/paper/1905.02671/) (2019)
- [Extra-Newton: A First Approach to Noise-Adaptive Accelerated Second-Order Methods](https://mlsanity.com/paper/2211.01832/) (2022)
- [Minimizing Quasi-Self-Concordant Functions by Gradient Regularization of Newton Method](https://mlsanity.com/paper/2308.14742/) (2023)
- [Tensor Methods for Minimizing Convex Functions with H"older Continuous Higher-Order Derivatives](https://mlsanity.com/paper/1904.12559/) (2019)
- [Contracting Proximal Methods for Smooth Convex Optimization](https://mlsanity.com/paper/1912.07972/) (2019)
- [Advancing the lower bounds: An accelerated, stochastic, second-order method with optimal adaptation to inexactness](https://mlsanity.com/paper/2309.01570/) (2023)
- [Non-Euclidean High-Order Smooth Convex Optimization](https://mlsanity.com/paper/2411.08987/) (2024)
- [Gradient-Normalized Smoothness for Optimization with Approximate Hessians](https://mlsanity.com/paper/2506.13710/) (2025)
- [Optimal Convex Optimization with Inexact Second-Order Oracles](https://mlsanity.com/paper/2607.24520/) (2026)
- [Improved global performance guarantees of second-order methods in convex minimization](https://mlsanity.com/paper/2408.11022/) (2024)

## Abstract (arXiv)

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/k3) 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.