# Distribution of hitting times for dissipative random dynamical systems on ℝd, with application to stochastic gradient descent

arXiv: https://arxiv.org/abs/2609.30274 · HTML: https://mlsanity.com/paper/2609.30274/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 37 references that are in the map.
- First posted on arXiv 11 Aug 2026.

## Builds on (most cited first, 13 of 13)

- [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)
- [Exploring Generalization in Deep Learning](https://mlsanity.com/paper/1706.08947/) (2017)
- [Loss Surfaces, Mode Connectivity, and Fast Ensembling of DNNs](https://mlsanity.com/paper/1802.10026/) (2018)
- [Essentially No Barriers in Neural Network Energy Landscape](https://mlsanity.com/paper/1803.00885/) (2018)
- [Loss landscapes and optimization in over-parameterized non-linear systems and neural networks](https://mlsanity.com/paper/2003.00307/) (2020)
- [Fit without fear: remarkable mathematical phenomena of deep learning through the prism of interpolation](https://mlsanity.com/paper/2105.14368/) (2021)
- [Deep Learning is Not So Mysterious or Different](https://mlsanity.com/paper/2503.02113/) (2025)
- [What is the long-run distribution of stochastic gradient descent? A large deviations analysis](https://mlsanity.com/paper/2406.09241/) (2024)
- [On Stationary-Point Hitting Time and Ergodicity of Stochastic Gradient Langevin Dynamics](https://mlsanity.com/paper/1904.13016/) (2019)
- [A PAC-Bayesian Link Between Generalisation and Flat Minima](https://mlsanity.com/paper/2402.08508/) (2024)
- [Training of Deep Neural Networks based on Distance Measures using RMSProp](https://mlsanity.com/paper/1708.01911/) (2017)
- [The global convergence time of stochastic gradient descent in non-convex landscapes: Sharp estimates via large deviations](https://mlsanity.com/paper/2503.16398/) (2025)

## Abstract (arXiv)

Machine Learning and more specifically Deep Learning involves solving large scale nonconvex optimization problems. Several algorithms have been proposed in the literature, that seem to achieve satisfactory practical efficiency for difficult instances, the Stochastic Gradient Method being the most rudimentary, while still outperforming more recent algorithms at a number of learning tasks. A major open question about the current methods used in deep learning is to understand their convergence properties. Following a line of previous works about the long time behavior of gradient-type algorithms, %and in particular the recent contributions from Azizian et al., we present a new approach for studying the asymptotic properties of a wide family of methods from an ergodic theoretical viewpoint. Our main results include a study of the expected time for a stochastic optimisation algorithm to reach a certain small neighborhood of a minimizer and show that this reaching time distributes exponentially around its average, which is given by the inverse of the stationary measure of the target. The assumptions on the Stochastic Gradient noise include the Gaussian and the Sub-Exponential assumptions.