# The global rate of convergence for optimal tensor methods in smooth convex optimization

arXiv: https://arxiv.org/abs/1809.00382 · HTML: https://mlsanity.com/paper/1809.00382/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Sep 2018.

## Built on by (most cited first, 10 of 10)

- [The First Optimal Acceleration of High-Order Methods in Smooth Convex Optimization](https://mlsanity.com/paper/2205.09647/) (2022)
- [Extra-Newton: A First Approach to Noise-Adaptive Accelerated Second-Order Methods](https://mlsanity.com/paper/2211.01832/) (2022)
- [Near-Optimal Lower Bounds For Convex Optimization For All Orders of Smoothness](https://mlsanity.com/paper/2112.01118/) (2021)
- [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)
- [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)
- [Halpern Iteration Achieves tildeO(ε-1/p) pth-Order Oracle Complexity for Monotone Variational Inequalities](https://mlsanity.com/paper/2608.08463/) (2026)
- [Optimal High-Order Methods for Solving Monotone Variational Inequalities](https://mlsanity.com/paper/2609.23557/) (2026)

## Abstract (arXiv)

We consider convex optimization problems with the objective function having Lipshitz-continuous p-th order derivative, where p≥ 1. We propose a new tensor method, which closes the gap between the lower Oleft(varepsilon^-frac23p+1 right) and upper Oleft(varepsilon^-frac1p+1 right) iteration complexity bounds for this class of optimization problems. We also consider uniformly convex functions, and show how the proposed method can be accelerated under this additional assumption. Moreover, we introduce a p-th order condition number which naturally arises in the complexity analysis of tensor methods under this assumption. Finally, we make a numerical study of the proposed optimal method and show that in practice it is faster than the best known accelerated tensor method. We also compare the performance of tensor methods for p=2 and p=3 and show that the 3rd-order method is superior to the 2nd-order method in practice.