Faster Sampling without Isoperimetry via Diffusion-based Monte Carlo

Faster Sampling without Isoperimetry via Diffusion-based Monte Carlo

12 Jan 2024 | Xunpeng Huang*, Difan Zou§, Hanze Dong†, Yian Ma‡, and Tong Zhang‡
This paper introduces a new diffusion-based Monte Carlo (DMC) algorithm, RS-DMC, for sampling from general non-log-concave distributions. The original DMC algorithm suffers from high gradient complexity due to its inefficient score estimation. RS-DMC improves this by using a recursive score estimation method, dividing the diffusion process into segments and solving related mean estimation and sampling subproblems recursively. This approach ensures that all subproblems only involve strongly log-concave distributions, which can be efficiently sampled using standard methods like Langevin Monte Carlo. The gradient complexity of RS-DMC is quasi-polynomial in the target error ε and dimension d, significantly improving upon the exponential complexity of the original DMC. The algorithm is also proven to be faster than popular Langevin-based methods under common dissipative conditions. The key contributions include the development of RS-DMC, establishing convergence guarantees under mild assumptions, and demonstrating the algorithm's efficiency in terms of gradient complexity. The paper shows that RS-DMC can be applied to a broader class of distributions with rigorous theoretical guarantees. The algorithm's design and theoretical framework open a new direction for addressing sampling problems, with potential broader applicability in the community.This paper introduces a new diffusion-based Monte Carlo (DMC) algorithm, RS-DMC, for sampling from general non-log-concave distributions. The original DMC algorithm suffers from high gradient complexity due to its inefficient score estimation. RS-DMC improves this by using a recursive score estimation method, dividing the diffusion process into segments and solving related mean estimation and sampling subproblems recursively. This approach ensures that all subproblems only involve strongly log-concave distributions, which can be efficiently sampled using standard methods like Langevin Monte Carlo. The gradient complexity of RS-DMC is quasi-polynomial in the target error ε and dimension d, significantly improving upon the exponential complexity of the original DMC. The algorithm is also proven to be faster than popular Langevin-based methods under common dissipative conditions. The key contributions include the development of RS-DMC, establishing convergence guarantees under mild assumptions, and demonstrating the algorithm's efficiency in terms of gradient complexity. The paper shows that RS-DMC can be applied to a broader class of distributions with rigorous theoretical guarantees. The algorithm's design and theoretical framework open a new direction for addressing sampling problems, with potential broader applicability in the community.
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