Improved sampling algorithms and Poincaré inequalities for non-log-concave distributions
This addresses a fundamental challenge in computational statistics and machine learning for sampling from complex distributions, with incremental improvements over prior work.
The paper tackles the problem of sampling from non-log-concave distributions by showing that under a strengthened smoothness assumption, the query complexity can be polynomial in dimension and inverse accuracy, improving from quasi-polynomial, and also provides bounds on the Poincaré constant.
We study the problem of sampling from a distribution $μ$ with density $\propto e^{-V}$ for some potential function $V:\mathbb R^d\to \mathbb R$ with query access to $V$ and $\nabla V$. We start with the following standard assumptions: (1) The potential function $V$ is $L$-smooth. (2) The second moment $\mathbf{E}_{X\sim μ}[\|X\|^2]\leq M$. Recently, He and Zhang (COLT'25) showed that the query complexity of sampling from such distributions is at least $\left(\frac{LM}{dε}\right)^{Ω(d)}$ where $ε$ is the desired accuracy in total variation distance, and the Poincaré constant can be arbitrarily large. Meanwhile, another common assumption in the study of diffusion based samplers (see e.g., the work of Chen, Chewi, Li, Li, Salim and Zhang (ICLR'23)) strengthens the smoothness condition (1) to the following: (1*) The potential function of *every* distribution along the Ornstein-Uhlenbeck process starting from $μ$ is $L$-smooth. We show that under the assumptions (1*) and (2), the query complexity of sampling from $μ$ can be $\mathrm{poly}(L,d)\cdot \left(\frac{Ld+M}{ε^2}\right)^{\mathcal{O}(L+1)}$, which is polynomial in $d$ and $\frac{1}ε$ when $L=\mathcal{O}(1)$ and $M=\mathrm{poly}(d)$. This improves the algorithm with quasi-polynomial query complexity developed by Huang et al. (COLT'24). Our results imply that the seemly moderate strengthening of the smoothness condition (1) to (1*) can lead to an exponential gap in the query complexity of sampling algorithms. Moreover, we show that together with the assumption (1*) and the stronger moment assumption that $\|X\|$ is $λ$-sub-Gaussian for $X\simμ$, the Poincaré constant of $μ$ is at most $\mathcal{O}(λ)^{2(L+1)}$. As an application of our technique, we obtain improved estimate of the Poincaré constant for mixture of Gaussians with the same covariance.