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. Existing Markov Chain Monte Carlo (MCMC) methods are either based on general-purpose and domain-agnostic schemes which can lead to slow convergence, or hand-crafting of problem-specific proposals by an expert. We propose A-NICE-MC, a novel method to train flexible parametric Markov chain kernels to produce samples with desired properties. First, we propose an efficient likelihood-free adversarial training method to train a Markov chain and mimic a given data distribution. Then, we leverage flexible volume preserving flows to obtain parametric kernels for MCMC. Using a bootstrap approach, we show how to train efficient Markov chains to sample from a prescribed posterior distribution by iteratively improving the quality of both the model and the samples. A-NICE-MC provides the first framework to automatically design efficient domain-specific MCMC proposals. Empirical results demonstrate that A-NICE-MC combines the strong guarantees of MCMC with the expressiveness of deep neural networks, and is able to significantly outperform competing methods such as Hamiltonian Monte Carlo. We develop Riemannian Stein Variational Gradient Descent (RSVGD), a Bayesian inference method that generalizes Stein Variational Gradient Descent (SVGD) to Riemann manifold. The benefits are two-folds: (i) for inference tasks in Euclidean spaces, RSVGD has the advantage over SVGD of utilizing information geometry, and (ii) for inference tasks on Riemann manifolds, RSVGD brings the unique advantages of SVGD to the Riemannian world. To appropriately transfer to Riemann manifolds, we conceive novel and non-trivial techniques for RSVGD, which are required by the intrinsically different characteristics of general Riemann manifolds from Euclidean spaces. We also discover Riemannian Stein's Identity and Riemannian Kernelized Stein Discrepancy. Experimental results show the advantages over SVGD of exploring distribution geometry and the advantages of particle-efficiency, iteration-effectiveness and approximation flexibility over other inference methods on Riemann manifolds. Generative adversarial networks (GANs) are a learning framework that rely on training a discriminator to estimate a measure of difference between a target and generated distributions. GANs, as normally formulated, rely on the generated samples being completely differentiable w.r.t. the generative parameters, and thus do not work for discrete data. We introduce a method for training GANs with discrete data that uses the estimated difference measure from the discriminator to compute importance weights for generated samples, thus providing a policy gradient for training the generator. The importance weights have a strong connection to the decision boundary of the discriminator, and we call our method boundary-seeking GANs (BGANs). We demonstrate the effectiveness of the proposed algorithm with discrete image and character-based natural language generation. In addition, the boundary-seeking objective extends to continuous data, which can be used to improve stability of training.