Understanding the dimension dependency of computational complexity in high-dimensional sampling problem is a fundamental problem, both from a practical and theoretical perspective. Compared with samplers with unbiased stationary distribution, e.g., Metropolis-adjusted Langevin algorithm (MALA), biased samplers, e.g., Underdamped Langevin Dynamics (ULD), perform better in low-accuracy cases just because a lower dimension dependency in their complexities. Along this line, Freund et al. (2022) suggest that the modified Langevin algorithm with prior diffusion is able to converge dimension independently for strongly log-concave target distributions. Nonetheless, it remains open whether such property establishes for more general cases. In this paper, we investigate the prior diffusion technique for the target distributions satisfying log-Sobolev inequality (LSI), which covers a much broader class of distributions compared to the strongly log-concave ones. In particular, we prove that the modified Langevin algorithm can also obtain the dimension-independent convergence of KL divergence with different step size schedules. The core of our proof technique is a novel construction of an interpolating SDE, which significantly helps to conduct a more accurate characterization of the discrete updates of the overdamped Langevin dynamics. Our theoretical analysis demonstrates the benefits of prior diffusion for a broader class of target distributions and provides new insights into developing faster sampling algorithms.

翻译：理解高维采样问题中计算复杂度的维度依赖性是一个基础性问题，兼具实践与理论意义。与具有无偏平稳分布的采样器（如Metropolis-adjusted Langevin算法，MALA）相比，有偏采样器（如欠阻尼Langevin动力学，ULD）在低精度情况下表现更优，其优势源于更低的维度复杂度依赖性。沿着这一方向，Freund等人（2022）指出，带有先验扩散的改进Langevin算法能够对强对数凹目标分布实现维度无关收敛。然而，该性质是否适用于更一般的情形仍是一个开放问题。本文研究满足对数Sobolev不等式（LSI）的目标分布的先验扩散技术，其覆盖的分布类别远广于强对数凹分布。特别地，我们证明了改进的Langevin算法能在不同步长调度下获得KL散度的维度无关收敛。本文证明技术的核心在于创新性地构造了一个插值随机微分方程，该方程显著提升了对过阻尼Langevin动力学离散更新的精确刻画能力。我们的理论分析揭示了先验扩散对更广泛目标分布的优势，并为开发更快速采样算法提供了新见解。