# Neural network optimization strategies and the topography of the loss landscape

arXiv: https://arxiv.org/abs/2602.21276 · HTML: https://mlsanity.com/paper/2602.21276/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 39 references that are in the map.
- First posted on arXiv 24 Feb 2026.

## Builds on (most cited first, 17 of 17)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [On Large-Batch Training for Deep Learning: Generalization Gap and Sharp Minima](https://mlsanity.com/paper/1609.04836/) (2016)
- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [Visualizing the Loss Landscape of Neural Nets](https://mlsanity.com/paper/1712.09913/) (2017)
- [Loss Surfaces, Mode Connectivity, and Fast Ensembling of DNNs](https://mlsanity.com/paper/1802.10026/) (2018)
- [Entropy-SGD: Biasing Gradient Descent Into Wide Valleys](https://mlsanity.com/paper/1611.01838/) (2016)
- [Essentially No Barriers in Neural Network Energy Landscape](https://mlsanity.com/paper/1803.00885/) (2018)
- [An Investigation into Neural Net Optimization via Hessian Eigenvalue Density](https://mlsanity.com/paper/1901.10159/) (2019)
- [Three Factors Influencing Minima in SGD](https://mlsanity.com/paper/1711.04623/) (2017)
- [The Marginal Value of Adaptive Gradient Methods in Machine Learning](https://mlsanity.com/paper/1705.08292/) (2017)
- [Qualitatively characterizing neural network optimization problems](https://mlsanity.com/paper/1412.6544/) (2014)
- [A Simple Way to Initialize Recurrent Networks of Rectified Linear Units](https://mlsanity.com/paper/1504.00941/) (2015)
- [A high-bias, low-variance introduction to Machine Learning for physicists](https://mlsanity.com/paper/1803.08823/) (2018)
- [Hessian-based Analysis of Large Batch Training and Robustness to Adversaries](https://mlsanity.com/paper/1802.08241/) (2018)
- [SGD Converges to Global Minimum in Deep Learning via Star-convex Path](https://mlsanity.com/paper/1901.00451/) (2019)
- [How noise affects the Hessian spectrum in overparameterized neural networks](https://mlsanity.com/paper/1910.00195/) (2019)
- [Do Deep Neural Network Solutions Form a Star Domain?](https://mlsanity.com/paper/2403.07968/) (2024)

## Abstract (arXiv)

Neural networks are trained by optimizing multi-dimensional sets of fitting parameters on non-convex loss landscapes. Low-loss regions of the landscapes correspond to the parameter sets that perform well on the training data. A key issue in machine learning is the performance of trained neural networks on previously unseen test data. Here, we investigate neural network training by stochastic gradient descent (SGD) - a non-convex global optimization algorithm which relies only on the gradient of the objective function. We contrast SGD solutions with those obtained via a non-stochastic quasi-Newton method, which utilizes curvature information to determine step direction and Golden Section Search to choose step size. We use several computational tools to investigate neural network parameters obtained by these two optimization methods, including kernel Principal Component Analysis and a novel, general-purpose algorithm for finding low-height paths between pairs of points on loss or energy landscapes, FourierPathFinder. We find that the choice of the optimizer profoundly affects the nature of the resulting solutions. SGD solutions tend to be separated by lower barriers than quasi-Newton solutions, even if both sets of solutions are regularized by early stopping to ensure adequate performance on test data. When allowed to fit extensively on the training data, quasi-Newton solutions occupy deeper minima on the loss landscapes that are not reached by SGD. These solutions are less generalizable to the test data however. Overall, SGD explores smooth basins of attraction, while quasi-Newton optimization is capable of finding deeper, more isolated minima that are more spread out in the parameter space. Our findings help understand both the topography of the loss landscapes and the fundamental role of landscape exploration strategies in creating robust, transferrable neural network models.