# A nonlinear conjugate gradient method with complexity guarantees and its application to nonconvex regression

arXiv: https://arxiv.org/abs/2201.08568 · HTML: https://mlsanity.com/paper/2201.08568/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Jan 2022.

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

- [Overcoming Saturation in Density Ratio Estimation by Iterated Regularization](https://mlsanity.com/paper/2402.13891/) (2024)
- [Convergence Conditions for Stochastic Line Search Based Optimization of Over-parametrized Models](https://mlsanity.com/paper/2408.03199/) (2024)
- [Effectively Leveraging Momentum Terms in Stochastic Line Search Frameworks for Fast Optimization of Finite-Sum Problems](https://mlsanity.com/paper/2411.07102/) (2024)
- [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)

Nonlinear conjugate gradients are among the most popular techniques for solving continuous optimization problems. Although these schemes have long been studied from a global convergence standpoint, their worst-case complexity properties have yet to be fully understood, especially in the nonconvex setting. In particular, it is unclear whether nonlinear conjugate gradient methods possess better guarantees than first-order methods such as gradient descent. Meanwhile, recent experiments have shown impressive performance of standard nonlinear conjugate gradient techniques on certain nonconvex problems, even when compared with methods endowed with the best known complexity guarantees. In this paper, we propose a nonlinear conjugate gradient scheme based on a simple line-search paradigm and a modified restart condition. These two ingredients allow for monitoring the properties of the search directions, which is instrumental in obtaining complexity guarantees. Our complexity results illustrate the possible discrepancy between nonlinear conjugate gradient methods and classical gradient descent. A numerical investigation on nonconvex robust regression problems as well as a standard benchmark illustrate that the restarting condition can track the behavior of a standard implementation.