2024-06-30-huang24a.md

title	section	abstract	layout	series	publisher	issn	id	month	tex_title	firstpage	lastpage	page	order	cycles	bibtex_author	author	date	address	container-title	volume	genre	issued	pdf	extras
Faster Sampling without Isoperimetry via Diffusion-based Monte Carlo	Original Papers	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 the original DMC algorithm originates from its redundant design of score estimation, and proposed a more efficient DMC 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 standard sampler (e.g., Langevin Monte Carlo) with a provably rapid convergence rate. As a result, we prove that the gradient complexity of RS-DMC exhibits merely a quasi-polynomial dependency on $\epsilon$. This finding is highly unexpected as it substantially enhances the prevailing belief of the necessity for exponential gradient complexity in all prior works such as Huang et al. (2023). 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.	inproceedings	Proceedings of Machine Learning Research	PMLR	2640-3498	huang24a	0	Faster Sampling without Isoperimetry via Diffusion-based Monte Carlo	2438	2493	2438-2493	2438	false	Huang, Xunpeng and Zou, Difan and Dong, Hanze and Ma, Yi-An and Zhang, Tong	given family Xunpeng Huang given family Difan Zou given family Hanze Dong given family Yi-An Ma given family Tong Zhang	2024-06-30		Proceedings of Thirty Seventh Conference on Learning Theory	247	inproceedings	date-parts 2024 6 30	https://proceedings.mlr.press/v247/huang24a/huang24a.pdf