用更优的梯度引导采样，让复杂分布采样更快更准。
Diffusion at Absolute Zero: Langevin Sampling Using Successive Moreau Envelopes [conference paper]
- 通过构造序列近似目标分布，逐步优化采样路径。
- 在多峰分布上收敛速度显著提升，有效避免局部最优。
- 适合高维多模态分布采样，对扩散模型有启发意义。
本文提出一种从形如 $π(x)/propto\ ext{exp}(-U(x))$ 的吉布斯分布中采样的新方法，其中 $U(x)$ 为势能函数。受扩散模型启发，我们构造一个近似序列 $(π^{t_k})_k$，当 $k$ 较小时 $π^{t_k}$ 接近目标分布 $π$，而 $k$ 较大时则具有利于采样的良好性质。该序列通过将 $U(x)$ 部分替换为其 Moreau 包络获得。采样采用类似退火朗之万过程：依次从 $π^{t_k}$ 中采样，随着 $k$ 递减，将样本从简单起始分布引导至复杂目标分布。除了理论分析外，实验结果表明该方法在多模态分布上显著加快收敛速度，展现出更强的适用性。
原文摘要 · Abstract (English)
In this article we propose a novel method for sampling from Gibbs distributions of the form $π(x)\propto\exp(-U(x))$ with a potential $U(x)$. In particular, inspired by diffusion models we propose to consider a sequence $(π^{t_k})_k$ of approximations of the target density, for which $π^{t_k}\approx π$ for $k$ small and, on the other hand, $π^{t_k}$ exhibits favorable properties for sampling for $k$ large. This sequence is obtained by replacing parts of the potential $U$ by its Moreau envelopes. Sampling is performed in an Annealed Langevin type procedure, that is, sequentially sampling from $π^{t_k}$ for decreasing $k$, effectively guiding the samples from a simple starting density to the more complex target. In addition to a theoretical analysis we show experimental results supporting the efficacy of the method in terms of increased convergence speed and applicability to multi-modal densities $π$.
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