Faster Sampling without Isoperimetry via Diffusion-based Monte Carlo
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866917565453828096 |
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| author | Huang, Xunpeng Zou, Difan Dong, Hanze Ma, Yian Zhang, Tong |
| author_facet | Huang, Xunpeng Zou, Difan Dong, Hanze Ma, Yian Zhang, Tong |
| contents | To sample from a general target distribution $p_*\propto e^{-f_*}$ beyond the isoperimetric condition, Huang et al. (2023) proposed to perform sampling through reverse diffusion, giving rise to Diffusion-based Monte Carlo (DMC). Specifically, DMC follows the reverse SDE of a diffusion process that transforms the target distribution to the standard Gaussian, utilizing a non-parametric score estimation. However, the original DMC algorithm encountered high gradient complexity, resulting in an exponential dependency on the error tolerance $ε$ of the obtained samples. In this paper, we demonstrate that the high complexity of DMC originates from its redundant design of score estimation, and proposed a more efficient algorithm, called RS-DMC, based on a novel recursive score estimation method. In particular, we first divide the entire diffusion process into multiple segments and then formulate the score estimation step (at any time step) as a series of interconnected mean estimation and sampling subproblems accordingly, which are correlated in a recursive manner. Importantly, we show that with a proper design of the segment decomposition, all sampling subproblems will only need to tackle a strongly log-concave distribution, which can be very efficient to solve using the Langevin-based samplers with a provably rapid convergence rate. As a result, we prove that the gradient complexity of RS-DMC only has a quasi-polynomial dependency on $ε$, which significantly improves exponential gradient complexity in Huang et al. (2023). Furthermore, under commonly used dissipative conditions, our algorithm is provably much faster than the popular Langevin-based algorithms. Our algorithm design and theoretical framework illuminate a novel direction for addressing sampling problems, which could be of broader applicability in the community. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_06325 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Faster Sampling without Isoperimetry via Diffusion-based Monte Carlo Huang, Xunpeng Zou, Difan Dong, Hanze Ma, Yian Zhang, Tong Machine Learning Optimization and Control Computation To sample from a general target distribution $p_*\propto e^{-f_*}$ beyond the isoperimetric condition, Huang et al. (2023) proposed to perform sampling through reverse diffusion, giving rise to Diffusion-based Monte Carlo (DMC). Specifically, DMC follows the reverse SDE of a diffusion process that transforms the target distribution to the standard Gaussian, utilizing a non-parametric score estimation. However, the original DMC algorithm encountered high gradient complexity, resulting in an exponential dependency on the error tolerance $ε$ of the obtained samples. In this paper, we demonstrate that the high complexity of DMC originates from its redundant design of score estimation, and proposed a more efficient algorithm, called RS-DMC, based on a novel recursive score estimation method. In particular, we first divide the entire diffusion process into multiple segments and then formulate the score estimation step (at any time step) as a series of interconnected mean estimation and sampling subproblems accordingly, which are correlated in a recursive manner. Importantly, we show that with a proper design of the segment decomposition, all sampling subproblems will only need to tackle a strongly log-concave distribution, which can be very efficient to solve using the Langevin-based samplers with a provably rapid convergence rate. As a result, we prove that the gradient complexity of RS-DMC only has a quasi-polynomial dependency on $ε$, which significantly improves exponential gradient complexity in Huang et al. (2023). Furthermore, under commonly used dissipative conditions, our algorithm is provably much faster than the popular Langevin-based algorithms. Our algorithm design and theoretical framework illuminate a novel direction for addressing sampling problems, which could be of broader applicability in the community. |
| title | Faster Sampling without Isoperimetry via Diffusion-based Monte Carlo |
| topic | Machine Learning Optimization and Control Computation |
| url | https://arxiv.org/abs/2401.06325 |