Path Integral Sampler: a stochastic control approach for sampling
Path Integral Sampler: a stochastic control approach for sampling
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发表时间：
2021-11
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通讯作者：
Qinsheng Zhang;Yongxin Chen
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作者：
Qinsheng Zhang;Yongxin Chen
本文提出了一种从非归一化概率密度函数中抽取样本的新算法--路径积分采样器PIS。PIS是建立在Schr“odinger桥问题上的，该问题的目的是在给定扩散过程的初始分布和终端分布的情况下恢复扩散过程的最可能演化。PIS从初始分布中抽取样本，然后将样本通过薛定谔桥传播到终端分布。应用Girsanov定理，通过一个简单的先验扩散，将PIS表示为一个随机最优控制问题，其运行成本为控制能量，终端成本根据目标分布选择。通过将控制建模为神经网络，我们建立了一个可以端到端训练的采样算法。我们提供了理论上的理由的抽样质量的PIS在Wasserstein距离时，使用次优控制。此外，路径积分理论被用来计算样本的重要性权重，以补偿由控制器的次优性和时间离散化引起的偏差。我们通过实验证明了PIS与其他最先进的采样方法相比在各种任务上的优势。
We present Path Integral Sampler~(PIS), a novel algorithm to draw samples from unnormalized probability density functions. The PIS is built on the Schr\"odinger bridge problem which aims to recover the most likely evolution of a diffusion process given its initial distribution and terminal distribution. The PIS draws samples from the initial distribution and then propagates the samples through the Schr\"odinger bridge to reach the terminal distribution. Applying the Girsanov theorem, with a simple prior diffusion, we formulate the PIS as a stochastic optimal control problem whose running cost is the control energy and terminal cost is chosen according to the target distribution. By modeling the control as a neural network, we establish a sampling algorithm that can be trained end-to-end. We provide theoretical justification of the sampling quality of PIS in terms of Wasserstein distance when sub-optimal control is used. Moreover, the path integrals theory is used to compute importance weights of the samples to compensate for the bias induced by the sub-optimality of the controller and time-discretization. We experimentally demonstrate the advantages of PIS compared with other start-of-the-art sampling methods on a variety of tasks.