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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2407.16936 |
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| _version_ | 1866915153470029824 |
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| author | Guo, Wei Tao, Molei Chen, Yongxin |
| author_facet | Guo, Wei Tao, Molei Chen, Yongxin |
| contents | We consider the outstanding problem of sampling from an unnormalized density that may be non-log-concave and multimodal. To enhance the performance of simple Markov chain Monte Carlo (MCMC) methods, techniques of annealing type have been widely used. However, quantitative theoretical guarantees of these techniques are under-explored. This study takes a first step toward providing a non-asymptotic analysis of annealed MCMC. Specifically, we establish, for the first time, an oracle complexity of $\widetilde{O}\left(\frac{dβ^2{\cal A}^2}{\varepsilon^6}\right)$ for the simple annealed Langevin Monte Carlo algorithm to achieve $\varepsilon^2$ accuracy in Kullback-Leibler divergence to the target distribution $π\propto{\rm e}^{-V}$ on $\mathbb{R}^d$ with $β$-smooth potential $V$. Here, ${\cal A}$ represents the action of a curve of probability measures interpolating the target distribution $π$ and a readily sampleable distribution. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_16936 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Provable Benefit of Annealed Langevin Monte Carlo for Non-log-concave Sampling Guo, Wei Tao, Molei Chen, Yongxin Machine Learning Statistics Theory Computation We consider the outstanding problem of sampling from an unnormalized density that may be non-log-concave and multimodal. To enhance the performance of simple Markov chain Monte Carlo (MCMC) methods, techniques of annealing type have been widely used. However, quantitative theoretical guarantees of these techniques are under-explored. This study takes a first step toward providing a non-asymptotic analysis of annealed MCMC. Specifically, we establish, for the first time, an oracle complexity of $\widetilde{O}\left(\frac{dβ^2{\cal A}^2}{\varepsilon^6}\right)$ for the simple annealed Langevin Monte Carlo algorithm to achieve $\varepsilon^2$ accuracy in Kullback-Leibler divergence to the target distribution $π\propto{\rm e}^{-V}$ on $\mathbb{R}^d$ with $β$-smooth potential $V$. Here, ${\cal A}$ represents the action of a curve of probability measures interpolating the target distribution $π$ and a readily sampleable distribution. |
| title | Provable Benefit of Annealed Langevin Monte Carlo for Non-log-concave Sampling |
| topic | Machine Learning Statistics Theory Computation |
| url | https://arxiv.org/abs/2407.16936 |