Geometric ergodicity of SGLD via reflection coupling
This addresses convergence guarantees for SGLD, a key algorithm in Bayesian deep learning, but is incremental as it builds on existing coupling techniques for nonconvex optimization.
The paper tackled the geometric ergodicity of Stochastic Gradient Langevin Dynamics (SGLD) in nonconvex settings, proving Wasserstein contraction and establishing an invariant distribution with geometric ergodicity in W1 distance using reflection coupling.
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 $W_1$ distance. The generalization to non-gradient drifts is also included.