Generative neural samplers are probabilistic models that implement sampling using feedforward neural networks: they take a random input vector and produce a sample from a probability distribution defined by the network weights. These models are expressive and allow efficient computation of samples and derivatives, but cannot be used for computing likelihoods or for marginalization. The generative-adversarial training method allows to train such models through the use of an auxiliary discriminative neural network. We show that the generative-adversarial approach is a special case of an existing more general variational divergence estimation approach. We show that any f-divergence can be used for training generative neural samplers. We discuss the benefits of various choices of divergence functions on training complexity and the quality of the obtained generative models.
(NCE) @cite_19 is a method that estimates the parameters of unnormalized probabilistic models by performing non-linear logistic regression to discriminate the data from artificially generated noise. NCE can be viewed as a special case of GAN where the discriminator is constrained to a specific form that depends on the model (logistic regression classifier) and the generator (kept fixed) is providing the artificially generated noise (see supplementary material).
@cite_19
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Two recently introduced criteria for estimation of generative models are both based on a reduction to binary classification. Noise-contrastive estimation (NCE) is an estimation procedure in which a generative model is trained to be able to distinguish data samples from noise samples. Generative adversarial networks (GANs) are pairs of generator and discriminator networks, with the generator network learning to generate samples by attempting to fool the discriminator network into believing its samples are real data. Both estimation procedures use the same function to drive learning, which naturally raises questions about how they are related to each other, as well as whether this function is related to maximum likelihood estimation (MLE). NCE corresponds to training an internal data model belonging to the discriminator network but using a fixed generator network. We show that a variant of NCE, with a dynamic generator network, is equivalent to maximum likelihood estimation. Since pairing a learned discriminator with an appropriate dynamically selected generator recovers MLE, one might expect the reverse to hold for pairing a learned generator with a certain discriminator. However, we show that recovering MLE for a learned generator requires departing from the distinguishability game. Specifically: (i) The expected gradient of the NCE discriminator can be made to match the expected gradient of MLE, if one is allowed to use a non-stationary noise distribution for NCE, (ii) No choice of discriminator network can make the expected gradient for the GAN generator match that of MLE, and (iii) The existing theory does not guarantee that GANs will converge in the non-convex case. This suggests that the key next step in GAN research is to determine whether GANs converge, and if not, to modify their training algorithm to force convergence. We consider the task of estimating, from observed data, a probabilistic model that is parameterized by a finite number of parameters. In particular, we are considering the situation where the model probability density function is unnormalized. That is, the model is only specified up to the partition function. The partition function normalizes a model so that it integrates to one for any choice of the parameters. However, it is often impossible to obtain it in closed form. Gibbs distributions, Markov and multi-layer networks are examples of models where analytical normalization is often impossible. Maximum likelihood estimation can then not be used without resorting to numerical approximations which are often computationally expensive. We propose here a new objective function for the estimation of both normalized and unnormalized models. The basic idea is to perform nonlinear logistic regression to discriminate between the observed data and some artificially generated noise. With this approach, the normalizing partition function can be estimated like any other parameter. We prove that the new estimation method leads to a consistent (convergent) estimator of the parameters. For large noise sample sizes, the new estimator is furthermore shown to behave like the maximum likelihood estimator. In the estimation of unnormalized models, there is a trade-off between statistical and computational performance. We show that the new method strikes a competitive trade-off in comparison to other estimation methods for unnormalized models. As an application to real data, we estimate novel two-layer models of natural image statistics with spline nonlinearities. We present a new estimation principle for parameterized statistical models. The idea is to perform nonlinear logistic regression to discriminate between the observed data and some artificially generated noise, using the model log-density function in the regression nonlinearity. We show that this leads to a consistent (convergent) estimator of the parameters, and analyze the asymptotic variance. In particular, the method is shown to directly work for unnormalized models, i.e. models where the density function does not integrate to one. The normalization constant can be estimated just like any other parameter. For a tractable ICA model, we compare the method with other estimation methods that can be used to learn unnormalized models, including score matching, contrastive divergence, and maximum-likelihood where the normalization constant is estimated with importance sampling. Simulations show that noise-contrastive estimation offers the best trade-off between computational and statistical efficiency. The method is then applied to the modeling of natural images: We show that the method can successfully estimate a large-scale two-layer model and a Markov random field. We introduce the "Energy-based Generative Adversarial Network" model (EBGAN) which views the discriminator as an energy function that attributes low energies to the regions near the data manifold and higher energies to other regions. Similar to the probabilistic GANs, a generator is seen as being trained to produce contrastive samples with minimal energies, while the discriminator is trained to assign high energies to these generated samples. Viewing the discriminator as an energy function allows to use a wide variety of architectures and loss functionals in addition to the usual binary classifier with logistic output. Among them, we show one instantiation of EBGAN framework as using an auto-encoder architecture, with the energy being the reconstruction error, in place of the discriminator. We show that this form of EBGAN exhibits more stable behavior than regular GANs during training. We also show that a single-scale architecture can be trained to generate high-resolution images.