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) $V$ 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 this problem 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 unbounded. 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 (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 seemingly moderate strengthening from (1) to (1*) yields an exponential gap in the query complexity. Furthermore, 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 $\mathcal{O}(λ)^{2(L+1)}$. We also establish a modified log-Sobolev inequality for $μ$ under these conditions. As an application of our technique, we obtain a new estimate of the modified log-Sobolev constant for a specific class of mixtures of strongly log-concave distributions.

翻译：我们研究从分布 $\mu$ 中采样的问题，其密度满足 $\propto e^{-V}$，其中势函数 $V:\mathbb R^d\to \mathbb R$ 及其梯度 $\nabla V$ 可通过查询访问。我们从以下标准假设出发：(1) $V$ 是 $L$-光滑的。(2) 二阶矩 $\mathbf{E}_{X\sim \mu}[\|X\|^2]\leq M$。最近，He 与 Zhang (COLT'25) 证明了该问题的查询复杂度至少为 $\left(\frac{LM}{dε}\right)^{Ω(d)}$，其中 $ε$ 是总变差距离上的期望精度，且泊松常数可能无界。与此同时，基于扩散的采样器研究中另一个常见假设（参见 Chen、Chewi、Li、Li、Salim 与 Zhang (ICLR'23) 的工作）将 (1) 强化为：(1*) 从 $\mu$ 出发的 Ornstein-Uhlenbeck 过程所对应的*每一个*分布的势函数均为 $L$-光滑的。我们证明，在假设 (1*) 与 (2) 下，从 $\mu$ 采样的查询复杂度可达 $\mathrm{poly}(L,d)\cdot \left(\frac{Ld+M}{ε^2}\right)^{\mathcal{O}(L+1)}$，当 $L=\mathcal{O}(1)$ 且 $M=\mathrm{poly}(d)$ 时，该复杂度关于 $d$ 与 $\frac{1}ε$ 为多项式阶。这改进了 Huang 等人 (COLT'24) 提出的具有拟多项式查询复杂度的算法。我们的结果表明，从 (1) 到 (1*) 看似温和的强化在查询复杂度上产生了指数级的差距。此外，我们证明在假设 (1*) 与更强的矩假设（即对于 $X\sim\mu$，$\|X\|$ 是 $λ$-次高斯分布）共同成立时，$\mu$ 的泊松常数至多为 $\mathcal{O}(λ)^{2(L+1)}$。我们还在这些条件下为 $\mu$ 建立了一个修正的对数索博列夫不等式。作为我们技术的应用，我们针对一类特定的强对数凹分布混合，得到了其修正对数索博列夫常数的新估计。