# The global convergence time of stochastic gradient descent in non-convex landscapes: Sharp estimates via large deviations

arXiv: https://arxiv.org/abs/2503.16398 · HTML: https://mlsanity.com/paper/2503.16398/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Mar 2025.

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

- [A dynamic view of some anomalous phenomena in SGD](https://mlsanity.com/paper/2505.01751/) (2025)
- [Tight Long-Term Tail Decay of (Clipped) SGD in Non-Convex Optimization](https://mlsanity.com/paper/2602.05657/) (2026)
- [Large Spikes in Stochastic Gradient Descent: A Large-Deviations View](https://mlsanity.com/paper/2603.10079/) (2026)
- [Shuffling the Data, Stretching the Step-size: Sharper Bias in constant step-size SGD](https://mlsanity.com/paper/2604.10373/) (2026)
- [Multi-agent learning under uncertainty: Recurrence vs. concentration](https://mlsanity.com/paper/2512.08132/) (2025)
- [Proximal basin hopping: global optimization with guarantees](https://mlsanity.com/paper/2605.18364/) (2026)
- [Stochastic gradient descent with discontinuity across a manifold](https://mlsanity.com/paper/2608.07618/) (2026)
- [Distribution of hitting times for dissipative random dynamical systems on ℝd, with application to stochastic gradient descent](https://mlsanity.com/paper/2609.30274/) (2026)

## Abstract (arXiv)

In this paper, we examine the time it takes for stochastic gradient descent (SGD) to reach the global minimum of a general, non-convex loss function. We approach this question through the lens of randomly perturbed dynamical systems and large deviations theory, and we provide a tight characterization of the global convergence time of SGD via matching upper and lower bounds. These bounds are dominated by the most "costly" set of obstacles that the algorithm may need to overcome in order to reach a global minimizer from a given initialization, coupling in this way the global geometry of the underlying loss landscape with the statistics of the noise entering the process. Finally, motivated by applications to the training of deep neural networks, we also provide a series of refinements and extensions of our analysis for loss functions with shallow local minima.