This work introduces a new approximate proximal sampler that operates solely with zeroth-order information of the potential function. Prior theoretical analyses have revealed that proximal sampling corresponds to alternating forward and backward iterations of the heat flow. The backward step was originally implemented by rejection sampling, whereas we directly simulate the dynamics. Unlike diffusion-based sampling methods that estimate scores via learned models or by invoking auxiliary samplers, our method treats the intermediate particle distribution as a Gaussian mixture, thereby yielding a Monte Carlo score estimator from directly samplable distributions. Theoretically, when the score estimation error is sufficiently controlled, our method inherits the exponential convergence of proximal sampling under isoperimetric conditions on the target distribution. In practice, the algorithm avoids rejection sampling, permits flexible step sizes, and runs with a deterministic runtime budget. Numerical experiments demonstrate that our approach converges rapidly to the target distribution, driven by interactions among multiple particles and by exploiting parallel computation.

翻译：本文提出了一种新型近似近端采样器，仅需利用势函数的零阶信息即可运行。先前的理论分析表明，近端采样对应热流的交替正向与反向迭代。原始方法通过拒绝采样实现反向步骤，而本工作直接模拟动力学过程。不同于依赖学习模型或辅助采样器估计分数的扩散采样方法，本方法将中间粒子分布视为高斯混合模型，从而通过可直接采样的分布获得蒙特卡洛分数估计。理论上，当分数估计误差得到充分控制时，该方法在目标分布满足等周条件的情况下继承了近端采样的指数收敛性。在实际应用中，该算法避免了拒绝采样，允许灵活的步长，且具有确定性的运行时预算。数值实验表明，在多粒子交互与并行计算驱动下，本方法能快速收敛至目标分布。