Faster Sampling without Isoperimetry via Diffusion-based Monte Carlo

• URL: http://arxiv.org/abs/2401.06325v1
• Date: Fri, 12 Jan 2024 02:33:57 GMT
• Title: Faster Sampling without Isoperimetry via Diffusion-based Monte Carlo
• Authors: Xunpeng Huang and Difan Zou and Hanze Dong and Yian Ma and Tong Zhang
• Abstract summary: Diffusion-based Monte Carlo (DMC) is a method to sample from a general target distribution beyond the isoperimetric condition. DMC encountered high gradient complexity, resulting in an exponential dependency on the error tolerance $epsilon$ of the obtained samples. We propose RS-DMC, based on a novel recursion-based score estimation method. Our algorithm is provably much faster than the popular Langevin-based algorithms.
• Score: 30.4930148381328
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 $\epsilon$ 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 $\epsilon$, 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.

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