无维度多模态采样：基于预处理退火朗之万动力学
Dimension-Free Multimodal Sampling via Preconditioned Annealed Langevin Dynamics
AI总结：
针对函数空间问题有限维近似细化下保持稳定的多模态目标采样算法，提出预处理退火朗之万动力学，在谱条件下实现维度一致的时间复杂度内的高斯混合目标采样。
摘要：
设计对于多模态目标的采样算法，使其在底层函数空间问题的有限维近似细化下保持稳定，是一个核心挑战。退火朗之万动力学（ALD）是经典朗之万动力学在此背景下的自然替代方案，因为它通常被观察到能改善跨模态的探索。然而，其经验成功与现有理论之间仍存在差距：在何种条件下可以保证ALD在维度间保持稳定？在本文中，我们通过提供连续时间ALD对高斯混合目标的维度一致分析来弥合这一差距。沿着一条显式退火路径（通过逐渐从目标中移除高斯平滑获得），我们识别出谱条件，将平滑协方差与分量协方差联系起来，在该条件下ALD在维度一致的时间范围内达到KL散度的指定精度。然后，我们在具有不完美初始化和近似分数的微扰区域内建立稳定性。在错误指定的混合分数模型下，我们表明，用谱衰减足够快的算子对ALD进行预处理，可以防止误差项在坐标间累积，从而保持维度一致的控制。
英文摘要：
Designing sampling algorithms for multimodal targets that remain stable under refinement of the finite-dimensional approximation of an underlying function-space problem is a central challenge. Annealed Langevin dynamics (ALD) is a natural alternative to classical Langevin in this context, since it is often observed to improve exploration across modes. Yet a gap remains between its empirical success and existing theory: under which conditions can ALD be guaranteed to remain stable across dimensions? In this paper, we bridge this gap by providing a uniform-in-dimension analysis of continuous-time ALD for Gaussian-mixture targets. Along an explicit annealing path obtained by gradually removing Gaussian smoothing from the target, we identify spectral conditions linking the smoothing covariance to the component covariances under which ALD achieves a prescribed accuracy in Kullback-Leibler divergence within a dimension-uniform time horizon. We then establish stability in a perturbative regime with imperfect initialization and approximate scores. Under a misspecified-mixture score model, we show that preconditioning ALD with an operator whose spectrum decays sufficiently fast prevents error terms from accumulating across coordinates and thereby preserves dimension-uniform control.
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