This paper studies a curious phenomenon in learning energy-based model (EBM) using MCMC. In each learning iteration, we generate synthesized examples by running a non-convergent, non-mixing, and non-persistent short-run MCMC toward the current model, always starting from the same initial distribution such as uniform noise distribution, and always running a fixed number of MCMC steps. After generating synthesized examples, we then update the model parameters according to the maximum likelihood learning gradient, as if the synthesized examples are fair samples from the current model. We treat this non-convergent short-run MCMC as a learned generator model or a flow model, with the initial image serving as the latent variables, and discard the learned EBM. We provide arguments for treating the learned non-convergent short-run MCMC as a valid model. We show that the learned short-run MCMC is capable of generating realistic images. Moreover, unlike traditional EBM or MCMC, the learned short-run MCMC is also capable of reconstructing observed images and interpolating different images, like generator model or flow model. The code can be found in the Appendix.
@cite_30: It is possible to combine multiple latent-variable models of the same data by multiplying their probability distributions together and then renormalizing. This way of combining individual "expert" models makes it hard to generate samples from the combined model but easy to infer the values of the latent variables of each expert, because the combination rule ensures that the latent variables of different experts are conditionally independent when given the data. A product of experts (PoE) is therefore an interesting candidate for a perceptual system in which rapid inference is vital and generation is unnecessary. Training a PoE by maximizing the likelihood of the data is difficult because it is hard even to approximate the derivatives of the renormalization term in the combination rule. Fortunately, a PoE can be trained using a different objective function called "contrastive divergence" whose derivatives with regard to the parameters can be approximated accurately and efficiently. Examples are presented of contrastive divergence learning using several types of expert on several types of data.
@cite_38: 
@cite_14: 
@cite_28: A new algorithm for training Restricted Boltzmann Machines is introduced. The algorithm, named Persistent Contrastive Divergence, is different from the standard Contrastive Divergence algorithms in that it aims to draw samples from almost exactly the model distribution. It is compared to some standard Contrastive Divergence and Pseudo-Likelihood algorithms on the tasks of modeling and classifying various types of data. The Persistent Contrastive Divergence algorithm outperforms the other algorithms, and is equally fast and simple.
@cite_0: Training energy-based probabilistic models is confronted with apparently intractable sums, whose Monte Carlo estimation requires sampling from the estimated probability distribution in the inner loop of training. This can be approximately achieved by Markov chain Monte Carlo methods, but may still face a formidable obstacle that is the difficulty of mixing between modes with sharp concentrations of probability. Whereas an MCMC process is usually derived from a given energy function based on mathematical considerations and requires an arbitrarily long time to obtain good and varied samples, we propose to train a deep directed generative model (not a Markov chain) so that its sampling distribution approximately matches the energy function that is being trained. Inspired by generative adversarial networks, the proposed framework involves training of two models that represent dual views of the estimated probability distribution: the energy function (mapping an input configuration to a scalar energy value) and the generator (mapping a noise vector to a generated configuration), both represented by deep neural networks.
@cite_46: The convolutional neural networks (CNNs) have proven to be a powerful tool for discriminative learning. Recently researchers have also started to show interest in the generative aspects of CNNs in order to gain a deeper understanding of what they have learned and how to further improve them. This paper investigates generative modeling of CNNs. The main contributions include: (1) We construct a generative model for the CNN in the form of exponential tilting of a reference distribution. (2) We propose a generative gradient for pre-training CNNs by a non-parametric importance sampling scheme, which is fundamentally different from the commonly used discriminative gradient, and yet has the same computational architecture and cost as the latter. (3) We propose a generative visualization method for the CNNs by sampling from an explicit parametric image distribution. The proposed visualization method can directly draw synthetic samples for any given node in a trained CNN by the Hamiltonian Monte Carlo (HMC) algorithm, without resorting to any extra hold-out images. Experiments on the challenging ImageNet benchmark show that the proposed generative gradient pre-training consistently helps improve the performances of CNNs, and the proposed generative visualization method generates meaningful and varied samples of synthetic images from a large-scale deep CNN.
For short-run MCMC, contrastive divergence (CD) @cite_30 is the most prominent framework for theoretical underpinning. The difference between CD and our study is that in our study, the short-run MCMC is initialized from noise, while CD initializes from observed images. CD has been generalized to persistent CD @cite_28 , and has more recently been generalized to modified CD @cite_46 and adversarial CD @cite_0 @cite_38 @cite_14 . However, in all those CD-based framework, the goal is still to learn the EBM, whereas in our framework, we discard the learned EBM, and only keep the learned short-run MCMC.