We propose two novel samplers to produce high-quality samples from a given (un-normalized) probability density. The sampling is achieved by transforming a reference distribution to the target distribution with neural networks, which are trained separately by minimizing two kinds of Stein Discrepancies, and hence our method is named as Stein neural sampler. Theoretical and empirical results suggest that, compared with traditional sampling schemes, our samplers share the following three advantages: 1. Being asymptotically correct; 2. Experiencing less convergence issue in practice; 3. Generating samples instantaneously.
The fusion of deep learning and sampling is not new. @cite_0 proposed A-NICE-MC, where the proposal distribution in MCMC is, instead of domain-agnostic, adversarially trained using neural networks. Stein GAN also proposes to train a neural network to draw samples from given target distributions for probabilistic inference. Their method is by iteratively adjusting the weights according to the SVGD updates. From a GAN perspective, in each iteration of Stein GAN, the discriminator is performing a two sample test between the currently generated samples and the one-step updated samples by SVGD. This method generalized SVGD to training neural networks and it is minimizing the Kullback-Leibler divergence between the sampling distribution and the target inside a RKHS.
@cite_0
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Stein variational gradient descent (SVGD) is a deterministic sampling algorithm that iteratively transports a set of particles to approximate given distributions, based on a gradient-based update constructed to optimally decrease the KL divergence within a function space. This paper develops the first theoretical analysis on SVGD. We establish that the empirical measures of the SVGD samples weakly converge to the target distribution, and show that the asymptotic behavior of SVGD is characterized by a nonlinear Fokker-Planck equation known as Vlasov equation in physics. We develop a geometric perspective that views SVGD as a gradient flow of the KL divergence functional under a new metric structure on the space of distributions induced by Stein operator.