Designing algorithms that can explore multimodal target distributions accurately across successive refinements of an underlying high-dimensional problem is a central challenge in sampling. Annealed Langevin dynamics (ALD) is a widely used alternative to classical Langevin since it often yields much faster mixing on multimodal targets, but there is still a gap between this empirical success and existing theory: when, and under which design choices, can ALD be guaranteed to remain stable as dimension increases? In this paper, we help bridge this gap by providing a uniform-in-dimension analysis of continuous-time ALD for multimodal targets that can be well-approximated by Gaussian mixture models. Along an explicit annealing path obtained by progressively removing Gaussian smoothing of the target, we identify sufficient spectral conditions - linking smoothing covariance and the covariances of the Gaussian components of the mixture - under which ALD achieves a prescribed accuracy within a single, dimension-uniform time horizon. We then establish dimension-robustness to imperfect initialization and score approximation: under a misspecified-mixture score model, we derive explicit conditions showing that preconditioning the ALD algorithm with a sufficiently decaying spectrum is necessary to prevent error terms from accumulating across coordinates and destroying dimension-uniform control. Finally, numerical experiments illustrate and validate the theory.

翻译：设计能够在底层高维问题连续细化过程中准确探索多模态目标分布的算法是采样领域的核心挑战。退火朗之万动力学（ALD）作为经典朗之万方法的常用替代方案，通常能在多模态目标上实现更快的混合速度，但其经验成功与现有理论之间仍存在差距：在何种条件下、通过何种设计选择，能保证ALD在维度增加时保持稳定性？本文通过建立对连续时间ALD的均匀维度分析，帮助弥合这一理论鸿沟，分析对象为可通过高斯混合模型良好逼近的多模态目标。沿着通过逐步移除目标高斯平滑而获得的显式退火路径，我们确定了充分的光谱条件——关联平滑协方差与混合模型中高斯分量的协方差——在此条件下，ALD能在单一且维度均匀的时间范围内达到预设精度。我们进一步建立了算法对非理想初始化和分数逼近的维度鲁棒性：在误设混合分数模型下，我们推导出显式条件，证明对ALD算法采用具有充分衰减谱的预处理是必要的，这能防止误差项在坐标间累积并破坏维度均匀控制。最后，数值实验验证并支撑了理论结果。