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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 sets. 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 Formula: see text distance. The generalization to non-gradient drifts is also included.
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Li et al. (Thu,) studied this question.
www.synapsesocial.com/papers/68e5dfdcb6db64358757469f — DOI: https://doi.org/10.1142/s0219493724500357
Lei Li
Jian‐Guo Liu
Yuliang Wang
Stochastics and Dynamics
Duke University
Shanghai Jiao Tong University
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