# Universal heavy-ball method for nonconvex optimization under H"older continuous Hessians

arXiv: https://arxiv.org/abs/2303.01073 · HTML: https://mlsanity.com/paper/2303.01073/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Mar 2023.

## Built on by (most cited first, 4 of 4)

- [Improved Complexity for Smooth Nonconvex Optimization: A Two-Level Online Learning Approach with Quasi-Newton Methods](https://mlsanity.com/paper/2412.02175/) (2024)
- [Improving Stochastic Cubic Newton with Momentum](https://mlsanity.com/paper/2410.19644/) (2024)
- [Improving Online-to-Nonconvex Conversion for Smooth Optimization via Double Optimism](https://mlsanity.com/paper/2510.03167/) (2025)
- [A Parameter-Free First-Order Algorithm for Non-Convex Optimization with tildemkern1mu O(ε-5/3) Global Rate](https://mlsanity.com/paper/2605.02127/) (2026)

## Abstract (arXiv)

We propose a new first-order method for minimizing nonconvex functions with Lipschitz continuous gradients and H"older continuous Hessians. The proposed algorithm is a heavy-ball method equipped with two particular restart mechanisms. It finds a solution where the gradient norm is less than ε in O(Hnu^frac12 + 2 nu ε^- frac4 + 3 nu2 + 2 nu) function and gradient evaluations, where nu ∈ [0, 1] and Hnu are the H"older exponent and constant, respectively. Our algorithm is nu-independent and thus universal; it automatically achieves the above complexity bound with the optimal nu ∈ [0, 1] without knowledge of Hnu. In addition, the algorithm does not require other problem-dependent parameters as input, including the gradient's Lipschitz constant or the target accuracy ε. Numerical results illustrate that the proposed method is promising.