Convergence Bounds for Sequential Monte Carlo on Multimodal Distributions using Soft Decomposition
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914400445661184 |
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| author | Lee, Holden Santana-Gijzen, Matheau |
| author_facet | Lee, Holden Santana-Gijzen, Matheau |
| contents | We prove bounds on the variance of a function $f$ under the empirical measure of the samples obtained by the Sequential Monte Carlo (SMC) algorithm, with time complexity depending on local rather than global Markov chain mixing dynamics. SMC is a Markov Chain Monte Carlo (MCMC) method, which starts by drawing $N$ particles from a known distribution, and then, through a sequence of distributions, re-weights and re-samples the particles, at each instance applying a Markov chain for smoothing. In principle, SMC tries to alleviate problems from multi-modality. However, most theoretical guarantees for SMC are obtained by assuming global mixing time bounds, which are only efficient in the uni-modal setting. We show that bounds can be obtained in the truly multi-modal setting, with mixing times that depend only on local MCMC dynamics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_19553 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Convergence Bounds for Sequential Monte Carlo on Multimodal Distributions using Soft Decomposition Lee, Holden Santana-Gijzen, Matheau Statistics Theory Machine Learning Probability We prove bounds on the variance of a function $f$ under the empirical measure of the samples obtained by the Sequential Monte Carlo (SMC) algorithm, with time complexity depending on local rather than global Markov chain mixing dynamics. SMC is a Markov Chain Monte Carlo (MCMC) method, which starts by drawing $N$ particles from a known distribution, and then, through a sequence of distributions, re-weights and re-samples the particles, at each instance applying a Markov chain for smoothing. In principle, SMC tries to alleviate problems from multi-modality. However, most theoretical guarantees for SMC are obtained by assuming global mixing time bounds, which are only efficient in the uni-modal setting. We show that bounds can be obtained in the truly multi-modal setting, with mixing times that depend only on local MCMC dynamics. |
| title | Convergence Bounds for Sequential Monte Carlo on Multimodal Distributions using Soft Decomposition |
| topic | Statistics Theory Machine Learning Probability |
| url | https://arxiv.org/abs/2405.19553 |