# Near-optimal method for highly smooth convex optimization

arXiv: https://arxiv.org/abs/1812.08026 · HTML: https://mlsanity.com/paper/1812.08026/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Dec 2018.

## Built on by (most cited first, 16 of 16)

- [The First Optimal Acceleration of High-Order Methods in Smooth Convex Optimization](https://mlsanity.com/paper/2205.09647/) (2022)
- [Acceleration via Fractal Learning Rate Schedules](https://mlsanity.com/paper/2103.01338/) (2021)
- [Near-Optimal Lower Bounds For Convex Optimization For All Orders of Smoothness](https://mlsanity.com/paper/2112.01118/) (2021)
- [Debiasing Conditional Stochastic Optimization](https://mlsanity.com/paper/2304.10613/) (2023)
- [Proximal Oracles for Optimization and Sampling](https://mlsanity.com/paper/2404.02239/) (2024)
- [Accelerated Quasi-Newton Proximal Extragradient: Faster Rate for Smooth Convex Optimization](https://mlsanity.com/paper/2306.02212/) (2023)
- [Understanding Riemannian Acceleration via a Proximal Extragradient Framework](https://mlsanity.com/paper/2111.02763/) (2021)
- [Higher-Order Newton Methods with Polynomial Work per Iteration](https://mlsanity.com/paper/2311.06374/) (2023)
- [A Whole New Ball Game: A Primal Accelerated Method for Matrix Games and Minimizing the Maximum of Smooth Functions](https://mlsanity.com/paper/2311.10886/) (2023)
- [Tight Lower Bounds under Asymmetric High-Order H"older Smoothness and Uniform Convexity](https://mlsanity.com/paper/2409.10773/) (2024)
- [Non-Euclidean High-Order Smooth Convex Optimization](https://mlsanity.com/paper/2411.08987/) (2024)
- [Faster Acceleration for Steepest Descent](https://mlsanity.com/paper/2409.19200/) (2024)
- [Halpern Iteration Achieves tildeO(ε-1/p) pth-Order Oracle Complexity for Monotone Variational Inequalities](https://mlsanity.com/paper/2608.08463/) (2026)
- [Solving Convex-Concave Problems with tildeO(ε-4/7) Second-Order Oracle Complexity](https://mlsanity.com/paper/2506.08362/) (2025)
- [Simple Stepsize for Quasi-Newton Methods with Global Convergence Guarantees](https://mlsanity.com/paper/2508.19712/) (2025)
- [Optimal High-Order Methods for Solving Monotone Variational Inequalities](https://mlsanity.com/paper/2609.23557/) (2026)

## Abstract (arXiv)

We propose a near-optimal method for highly smooth convex optimization. More precisely, in the oracle model where one obtains the pth order Taylor expansion of a function at the query point, we propose a method with rate of convergence tildeO(1/k^frac 3p +12) after k queries to the oracle for any convex function whose pth order derivative is Lipschitz.