Explicit convergence bounds for Metropolis Markov chains: isoperimetry, spectral gaps and profiles
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2022
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| _version_ | 1866916409092603904 |
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| author | Andrieu, Christophe Lee, Anthony Power, Sam Wang, Andi Q. |
| author_facet | Andrieu, Christophe Lee, Anthony Power, Sam Wang, Andi Q. |
| contents | We derive the first explicit bounds for the spectral gap of a random walk Metropolis algorithm on $R^d$ for any value of the proposal variance, which when scaled appropriately recovers the correct $d^{-1}$ dependence on dimension for suitably regular invariant distributions. We also obtain explicit bounds on the ${\rm L}^2$-mixing time for a broad class of models. In obtaining these results, we refine the use of isoperimetric profile inequalities to obtain conductance profile bounds, which also enable the derivation of explicit bounds in a much broader class of models. We also obtain similar results for the preconditioned Crank--Nicolson Markov chain, obtaining dimension-independent bounds under suitable assumptions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_08959 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Explicit convergence bounds for Metropolis Markov chains: isoperimetry, spectral gaps and profiles Andrieu, Christophe Lee, Anthony Power, Sam Wang, Andi Q. Probability Computation We derive the first explicit bounds for the spectral gap of a random walk Metropolis algorithm on $R^d$ for any value of the proposal variance, which when scaled appropriately recovers the correct $d^{-1}$ dependence on dimension for suitably regular invariant distributions. We also obtain explicit bounds on the ${\rm L}^2$-mixing time for a broad class of models. In obtaining these results, we refine the use of isoperimetric profile inequalities to obtain conductance profile bounds, which also enable the derivation of explicit bounds in a much broader class of models. We also obtain similar results for the preconditioned Crank--Nicolson Markov chain, obtaining dimension-independent bounds under suitable assumptions. |
| title | Explicit convergence bounds for Metropolis Markov chains: isoperimetry, spectral gaps and profiles |
| topic | Probability Computation |
| url | https://arxiv.org/abs/2211.08959 |