# Convergence of Stochastic Gradient Langevin Dynamics in the Lazy Training Regime

arXiv: https://arxiv.org/abs/2510.21245 · HTML: https://mlsanity.com/paper/2510.21245/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 19 references that are in the map.
- First posted on arXiv 24 Oct 2025.

## Builds on (most cited first, 16 of 16)

- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Reconciling modern machine learning practice and the bias-variance trade-off](https://mlsanity.com/paper/1812.11118/) (2018)
- [On Lazy Training in Differentiable Programming](https://mlsanity.com/paper/1812.07956/) (2018)
- [Gradient Descent Provably Optimizes Over-parameterized Neural Networks](https://mlsanity.com/paper/1810.02054/) (2018)
- [Convex Optimization: Algorithms and Complexity](https://mlsanity.com/paper/1405.4980/) (2014)
- [Gradient Descent Finds Global Minima of Deep Neural Networks](https://mlsanity.com/paper/1811.03804/) (2018)
- [Benign Overfitting in Linear Regression](https://mlsanity.com/paper/1906.11300/) (2019)
- [Learning Overparameterized Neural Networks via Stochastic Gradient Descent on Structured Data](https://mlsanity.com/paper/1808.01204/) (2018)
- [A Tail-Index Analysis of Stochastic Gradient Noise in Deep Neural Networks](https://mlsanity.com/paper/1901.06053/) (2019)
- [Towards moderate overparameterization: global convergence guarantees for training shallow neural networks](https://mlsanity.com/paper/1902.04674/) (2019)
- [Stochastic modified equations and adaptive stochastic gradient algorithms](https://mlsanity.com/paper/1511.06251/) (2015)
- [The Heavy-Tail Phenomenon in SGD](https://mlsanity.com/paper/2006.04740/) (2020)
- [On the SDEs and Scaling Rules for Adaptive Gradient Algorithms](https://mlsanity.com/paper/2205.10287/) (2022)
- [Beyond Linearization: On Quadratic and Higher-Order Approximation of Wide Neural Networks](https://mlsanity.com/paper/1910.01619/) (2019)
- [Convergence of Gradient Descent for Recurrent Neural Networks: A Nonasymptotic Analysis](https://mlsanity.com/paper/2402.12241/) (2024)
- [Convergence of continuous-time stochastic gradient descent with applications to deep neural networks](https://mlsanity.com/paper/2409.07401/) (2024)

## Abstract (arXiv)

Continuous-time models provide important insights into the training dynamics of optimization algorithms in deep learning. In this work, we establish a non-asymptotic convergence analysis of stochastic gradient Langevin dynamics (SGLD), which is an Ito stochastic differential equation (SDE) approximation of stochastic gradient descent in continuous time, in the lazy training regime. We show that, under regularity conditions on the Hessian of the loss function, SGLD with multiplicative and state-dependent noise (i) yields a non-degenerate kernel throughout the training process with high probability, and (ii) achieves exponential convergence to the empirical risk minimizer in expectation, and we establish finite-time and finite-width bounds on the optimality gap. We corroborate our theoretical findings with numerical examples in the regression setting.