# Geometric ergodicity of SGLD via reflection coupling

arXiv: https://arxiv.org/abs/2301.06769 · HTML: https://mlsanity.com/paper/2301.06769/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 33 references that are in the map.
- First posted on arXiv 17 Jan 2023.

## Builds on (most cited first, 8 of 8)

- [Generalization Bounds of SGLD for Non-convex Learning: Two Theoretical Viewpoints](https://mlsanity.com/paper/1707.05947/) (2017)
- [The promises and pitfalls of Stochastic Gradient Langevin Dynamics](https://mlsanity.com/paper/1811.10072/) (2018)
- [Nonasymptotic estimates for Stochastic Gradient Langevin Dynamics under local conditions in nonconvex optimization](https://mlsanity.com/paper/1910.02008/) (2019)
- [Coupling and Convergence for Hamiltonian Monte Carlo](https://mlsanity.com/paper/1805.00452/) (2018)
- [Non-asymptotic bounds for sampling algorithms without log-concavity](https://mlsanity.com/paper/1808.07105/) (2018)
- [Faster Convergence of Stochastic Gradient Langevin Dynamics for Non-Log-Concave Sampling](https://mlsanity.com/paper/2010.09597/) (2020)
- [Time-independent Generalization Bounds for SGLD in Non-convex Settings](https://mlsanity.com/paper/2111.12876/) (2021)
- [A sharp uniform-in-time error estimate for Stochastic Gradient Langevin Dynamics](https://mlsanity.com/paper/2207.09304/) (2022)

## Built on by (most cited first, 1 of 1)

- [Subsampling Error in Stochastic Gradient Langevin Diffusions](https://mlsanity.com/paper/2305.13882/) (2023)

## Abstract (arXiv)

We consider the geometric ergodicity of the Stochastic Gradient Langevin Dynamics (SGLD) algorithm under nonconvexity settings. Via the technique of reflection coupling, we prove the Wasserstein contraction of SGLD when the target distribution is log-concave only outside some compact set. The time discretization and the minibatch in SGLD introduce several difficulties when applying the reflection coupling, which are addressed by a series of careful estimates of conditional expectations. As a direct corollary, the SGLD with constant step size has an invariant distribution and we are able to obtain its geometric ergodicity in terms of W1 distance. The generalization to non-gradient drifts is also included.