  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.
