用漂移方法实现玻尔兹曼分布的单步采样，训练简单且高效。
One-Step Sampler for Boltzmann Distributions via Drifting
- 基于梯度漂移，通过神经网络一步生成目标分布样本。
- 四模混合分布下误差低至0.0754，相关性误差仅0.0425。
- 适用于非凸、弯曲等复杂几何结构，适合快速采样场景。
我们提出一种基于漂移的框架，用于对由能量函数定义的玻尔兹曼分布进行近似采样。该方法通过将当前模型分布的样本沿高斯平滑后的得分场投影，引导至目标玻尔兹曼分布，从而训练一个单步神经生成器。对于仅知未归一化常数的目标，我们推导出一种基于平滑能量的目标侧漂移，并使用两种估计器：局部重要性采样均值漂移估计和二阶曲率修正近似。结合小批量高斯均值漂移估计的采样侧平滑得分，得到一个简单的停梯度目标，实现稳定的一步训练。在四模高斯混合玻尔兹曼目标上，采样器达到均值误差0.0754、协方差误差0.0425和RBF MMD 0.0020。额外的双阱和香蕉形目标表明，该方法同样能处理非凸和弯曲的低能几何结构。总体结果支持漂移法可有效将迭代采样压缩为测试时的单次前向传播。
原文摘要 · Abstract (English)
We present a drifting-based framework for amortized sampling of Boltzmann distributions defined by energy functions. The method trains a one-step neural generator by projecting samples along a Gaussian-smoothed score field from the current model distribution toward the target Boltzmann distribution. For targets specified only up to an unknown normalization constant, we derive a practical target-side drift from a smoothed energy and use two estimators: a local importance-sampling mean-shift estimator and a second-order curvature-corrected approximation. Combined with a mini-batch Gaussian mean-shift estimate of the sampler-side smoothed score, this yields a simple stop-gradient objective for stable one-step training. On a four-mode Gaussian-mixture Boltzmann target, our sampler achieves mean error $0.0754$, covariance error $0.0425$, and RBF MMD $0.0020$. Additional double-well and banana targets show that the same formulation also handles nonconvex and curved low-energy geometries. Overall, the results support drifting as an effective way to amortize iterative sampling from Boltzmann distributions into a single forward pass at test time.
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