Sampling from a target measure whose density is only known up to a normalization constant is a fundamental problem in computational statistics and machine learning. In this paper, we present a new optimization-based method for sampling called mollified interaction energy descent (MIED). MIED minimizes a new class of energies on probability measures called mollified interaction energies (MIEs). These energies rely on mollifier functions -- smooth approximations of the Dirac delta originated from PDE theory. We show that as the mollifier approaches the Dirac delta, the MIE converges to the chi-square divergence with respect to the target measure and the gradient flow of the MIE agrees with that of the chi-square divergence. Optimizing this energy with proper discretization yields a practical first-order particle-based algorithm for sampling in both unconstrained and constrained domains. We show experimentally that for unconstrained sampling problems our algorithm performs on par with existing particle-based algorithms like SVGD, while for constrained sampling problems our method readily incorporates constrained optimization techniques to handle more flexible constraints with strong performance compared to alternatives.

翻译：从仅已知密度形式的归一化常数的目标测度中采样，是计算统计与机器学习中的基本问题。本文提出一种新型基于优化的采样方法——平滑化相互作用能量下降（MIED）。MIED方法最小化概率测度上一类新型能量函数，即平滑化相互作用能量（MIE）。这类能量依赖平滑化函数——源于偏微分方程理论的狄拉克δ函数的光滑逼近。我们证明：当平滑化函数趋近狄拉克δ函数时，MIE收敛于目标测度的卡方散度，且MIE的梯度流与卡方散度的梯度流一致。通过适当离散化优化该能量，可得到适用于无约束与有约束域的实用一阶粒子算法。实验表明：在无约束采样问题中，本算法性能与SVGD等现有粒子算法相当；而在有约束采样问题中，本方法可自然融入约束优化技术处理更灵活的约束条件，且性能显著优于替代方案。