Computable Bounds on Convergence of Markov Chains in Wasserstein Distance via Contractive Drift
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916781404192768 |
|---|---|
| author | Qu, Yanlin Blanchet, Jose Glynn, Peter |
| author_facet | Qu, Yanlin Blanchet, Jose Glynn, Peter |
| contents | We introduce a unified framework to estimate the convergence of Markov chains to equilibrium in Wasserstein distance. The framework can provide convergence bounds with rates ranging from polynomial to exponential, all derived from a contractive drift condition that integrates not only contraction and drift but also coupling and metric design. The resulting bounds are computable, as they contain simple constants, one-step transition expectations, but no equilibrium-related quantities. We introduce the large M technique and the boundary removal technique to enhance the applicability of the framework, which is further enhanced by deep learning in Qu, Blanchet and Glynn (2024). We apply the framework to non-contractive or even expansive Markov chains arising from queueing theory, stochastic optimization, and Markov chain Monte Carlo. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_10341 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Computable Bounds on Convergence of Markov Chains in Wasserstein Distance via Contractive Drift Qu, Yanlin Blanchet, Jose Glynn, Peter Probability Optimization and Control 60J05 We introduce a unified framework to estimate the convergence of Markov chains to equilibrium in Wasserstein distance. The framework can provide convergence bounds with rates ranging from polynomial to exponential, all derived from a contractive drift condition that integrates not only contraction and drift but also coupling and metric design. The resulting bounds are computable, as they contain simple constants, one-step transition expectations, but no equilibrium-related quantities. We introduce the large M technique and the boundary removal technique to enhance the applicability of the framework, which is further enhanced by deep learning in Qu, Blanchet and Glynn (2024). We apply the framework to non-contractive or even expansive Markov chains arising from queueing theory, stochastic optimization, and Markov chain Monte Carlo. |
| title | Computable Bounds on Convergence of Markov Chains in Wasserstein Distance via Contractive Drift |
| topic | Probability Optimization and Control 60J05 |
| url | https://arxiv.org/abs/2308.10341 |