Title: Annealed Fisher Implicit Sampler. Abstract: Sampling from an un-normalized target distribution is an important problem in many scientific fields. An implicit sampler uses a parametric transform $x=G_\theta(z)$ to push forward an easy-to-sample latent code $z$ to obtain a sample $x$. Such samplers are favored for fast inference speed and flexible architecture. Thus it is appealing to train an implicit sampler for sampling from the un-normalized target. In this paper, we propose a novel approach to training an implicit sampler by minimizing the Fisher Divergence between sampler and target distribution. We find that the trained sampler works well for relatively simple targets but may fail for more complicated multi-modal targets. To improve the training for multi-modal targets, we propose another adaptive training approach that trains the sampler to gradually learn a sequence of annealed distributions. We construct the annealed distribution path to bridge a simple distribution and the complicated target. With the annealed approach, the sampler is capable of handling challenging multi-modal targets. In addition, we also introduce a few MCMC correction steps after the sampler to better spread the samples. We call our proposed sampler \emph{the Annealed Fisher Implicit Sampler} (AFIS). We test AFIS on several sampling benchmarks. The experiments show that our AFIS outperforms baseline methods in many aspects. We also show in theory that the added MC correction steps get faster mixing by using the learned sampler as MCMC's initialization. 
