# Trust-Region Algorithms for Training Responses: Machine Learning Methods Using Indefinite Hessian Approximations

arXiv: https://arxiv.org/abs/1807.00251 · HTML: https://mlsanity.com/paper/1807.00251/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Jul 2018.

## Built on by (most cited first, 8 of 8)

- [Globally Convergent Multilevel Training of Deep Residual Networks](https://mlsanity.com/paper/2107.07572/) (2021)
- [Two trust region type algorithms for solving nonconvex-strongly concave minimax problems](https://mlsanity.com/paper/2402.09807/) (2024)
- [Symmetric Rank-One Quasi-Newton Methods for Deep Learning Using Cubic Regularization](https://mlsanity.com/paper/2502.12298/) (2025)
- [Using a New Nonlinear Gradient Method for Solving Large Scale Convex Optimization Problems with an Application on Arabic Medical Text](https://mlsanity.com/paper/2106.04383/) (2021)
- [A Novel Fast Exact Subproblem Solver for Stochastic Quasi-Newton Cubic Regularized Optimization](https://mlsanity.com/paper/2204.09116/) (2022)
- [On the efficiency of Stochastic Quasi-Newton Methods for Deep Learning](https://mlsanity.com/paper/2205.09121/) (2022)
- [Gradient Descent and the Power Method: Exploiting their connection to find the leftmost eigen-pair and escape saddle points](https://mlsanity.com/paper/2211.00866/) (2022)
- [Two-level domain-decomposition AdaGrad method for scalable training of graph neural networks](https://mlsanity.com/paper/2608.22575/) (2026)

## Abstract (arXiv)

Machine learning (ML) problems are often posed as highly nonlinear and nonconvex unconstrained optimization problems. Methods for solving ML problems based on stochastic gradient descent are easily scaled for very large problems but may involve fine-tuning many hyper-parameters. Quasi-Newton approaches based on the limited-memory Broyden-Fletcher-Goldfarb-Shanno (BFGS) update typically do not require manually tuning hyper-parameters but suffer from approximating a potentially indefinite Hessian with a positive-definite matrix. Hessian-free methods leverage the ability to perform Hessian-vector multiplication without needing the entire Hessian matrix, but each iteration's complexity is significantly greater than quasi-Newton methods. In this paper we propose an alternative approach for solving ML problems based on a quasi-Newton trust-region framework for solving large-scale optimization problems that allow for indefinite Hessian approximations. Numerical experiments on a standard testing data set show that with a fixed computational time budget, the proposed methods achieve better results than the traditional limited-memory BFGS and the Hessian-free methods.