首次实现无维度依赖的下推朗之万采样收敛分析
Dimension-Independent Convergence of Underdamped Langevin Monte Carlo in KL Divergence
- 改进KL局部误差框架，摆脱维度束缚
- 收敛速度依赖梯度矩阵迹，非维度数
- 在高维稀疏场景下优于传统方法
下推朗之万动力学（ULD）是采样吉布斯分布π∝e⁻ᵛ的常用方法，常在高维场景中表现良好。然而，现有离散化ULD的非渐近收敛保证通常随环境维度d多项式增长，当d较大时趋于无效。目前唯一已知的无维度结果针对随机中点离散化在Wasserstein-2距离下的情形（Liu et al., 2023），而离散化ULD在KL散度下的无维度保证长期悬而未决。本文首次证明了离散化ULD在KL散度下的无维度边界。分析将KL局部误差框架（Altschuler et al., 2025）推广至无维度设定，所得界依赖于∇²V的上界矩阵H的迹tr(H)，而非维度d。因此，在tr(H)≪d的场景下，下推朗之万蒙特卡洛的迭代复杂度优于过阻尼朗之万方法。
原文摘要 · Abstract (English)
Underdamped Langevin dynamics (ULD) is a widely-used sampler for Gibbs distributions $π\propto e^{-V}$, and is often empirically effective in high dimensions. However, existing non-asymptotic convergence guarantees for discretized ULD typically scale polynomially with the ambient dimension $d$, leading to vacuous bounds when $d$ is large. The main known dimension-free result concerns the randomized midpoint discretization in Wasserstein-2 distance (Liu et al.,2023), while dimension-independent guarantees for ULD discretizations in KL divergence have remained open. We close this gap by proving the first dimension-free KL divergence bounds for discretized ULD. Our analysis refines the KL local error framework (Altschuler et al., 2025) to a dimension-free setting and yields bounds that depend on $\mathrm{tr}(\mathbf{H})$, where $\mathbf{H}$ upper bounds the Hessian of $V$, rather than on $d$. As a consequence, we obtain improved iteration complexity for underdamped Langevin Monte Carlo relative to overdamped Langevin methods in regimes where $\mathrm{tr}(\mathbf{H})\ll d$.
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