We show how to sample in parallel from a distribution $\pi$ over $\mathbb R^d$ that satisfies a log-Sobolev inequality and has a smooth log-density, by parallelizing the Langevin (resp. underdamped Langevin) algorithms. We show that our algorithm outputs samples from a distribution $\hat\pi$ that is close to $\pi$ in Kullback--Leibler (KL) divergence (resp. total variation (TV) distance), while using only $\log(d)^{O(1)}$ parallel rounds and $\widetilde{O}(d)$ (resp. $\widetilde O(\sqrt d)$) gradient evaluations in total. This constitutes the first parallel sampling algorithms with TV distance guarantees. For our main application, we show how to combine the TV distance guarantees of our algorithms with prior works and obtain RNC sampling-to-counting reductions for families of discrete distribution on the hypercube $\{\pm 1\}^n$ that are closed under exponential tilts and have bounded covariance. Consequently, we obtain an RNC sampler for directed Eulerian tours and asymmetric determinantal point processes, resolving open questions raised in prior works.

翻译：我们展示了如何通过并行化Langevin（或欠阻尼Langevin）算法，从满足对数Sobolev不等式且具有光滑对数密度的 $\mathbb R^d$ 上的分布 $\pi$ 中进行并行采样。我们证明，该算法输出的样本来自一个分布 $\hat\pi$，该分布与 $\pi$ 在Kullback-Leibler（KL）散度（或总变差（TV）距离）上接近，同时仅使用 $\log(d)^{O(1)}$ 个并行轮次和总共 $\widetilde{O}(d)$（或 $\widetilde O(\sqrt d)$）次梯度评估。这构成了首个具有总变差距离保证的并行采样算法。对于我们的主要应用，我们展示了如何将算法的总变差距离保证与先前工作相结合，为超立方体 $\{\pm 1\}^n$ 上在指数倾斜下封闭且具有有界协方差的离散分布族，获得RNC（随机NC）采样到计数的归约。因此，我们得到了有向欧拉环游和非对称行列式点过程的RNC采样器，解决了先前工作中提出的开放性问题。