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Title: Coulomb GANs: Provably Optimal Nash Equilibria via Potential Fields

Paper content:
Generative adversarial networks (GANs) evolved into one of the most successful unsupervised techniques for generating realistic images.
Even though it has recently been shown that GAN training converges, GAN models often end up in local Nash equilibria that are associated with mode collapse or otherwise fail to model the target distribution.
We introduce Coulomb GANs, which pose the GAN learning problem as a potential field, where generated samples are attracted to training set samples but repel each other.
The discriminator learns a potential field while the generator decreases the energy by moving its samples along the vector (force) field determined by the gradient of the potential field.
Through decreasing the energy, the GAN model learns to generate samples according to the whole target distribution and does not only cover some of its modes.
We prove that Coulomb GANs possess only one Nash equilibrium which is optimal in the sense that the model distribution equals the target distribution.
We show the efficacy of Coulomb GANs on LSUN bedrooms, CelebA faces, CIFAR-10 and the Google Billion Word text generation.
Generative adversarial networks (GANs) excel at constructing realistic images BID28 BID24 BID3 and text BID18 .
In GAN learning, a discriminator network guides the learning of another, generative network.
This procedure can be considered as a game between the generator which constructs synthetic data and the discriminator which separates synthetic data from training set data BID16 .
The generator's goal is to construct data which the discriminator cannot tell apart from training set data.
GAN convergence points are local Nash equilibria.
At these local Nash equilibria neither the discriminator nor the generator can locally improve its objective.Despite their recent successes, GANs have several problems.
First (I), until recently it was not clear if in general gradient-based GAN learning could converge to one of the local Nash equilibria BID38 BID15 .
It is even possible to construct counterexamples BID16 .
Second (II), GANs suffer from "mode collapsing", where the model generates samples only in certain regions which are called modes.
While these modes contain realistic samples, the variety is low and only a few prototypes are generated.
Mode collapsing is less likely if the generator is trained with batch normalization, since the network is bound to create a certain variance among its generated samples within one batch .
However batch normalization introduces fluctuations of normalizing constants which can be harmful BID16 .
To avoid mode collapsing without batch normalization, several methods have been proposed BID5 BID38 .
Third (III), GANs cannot assure that the density of training samples is correctly modeled by the generator.
The discriminator only tells the generator whether a region is more likely to contain samples from the training set or synthetic samples.
Therefore the discriminator can only distinguish the support of the model distribution from the support of the target distribution.
Beyond matching the support of distributions, GANs with proper objectives may learn to locally align model and target densities via averaging over many training examples.
On a global scale, however, GANs fail to equalize model and target densities.
The discriminator does not inform the generator globally where probability mass is missing.
Consequently, standard GANs are not assured to capture the global sample density and are prone to neglect large parts of the target distribution.
The next paragraph gives an example of this.
Fourth (IV), the discriminator of GANs may forget previous modeling errors of the generator which then may reappear, a property that leads to oscillatory behavior instead of convergence BID16 .Recently
, problem (I) was solved by proving that GAN learning does indeed converge when discriminator and generator are learned using a two time-scale learning rule BID20 . Convergence
means that the expected SGD-gradient of both the discriminator objective and the generator objective are zero. Thus, neither
the generator nor the discriminator can locally improve, i.e., learning has reached a local Nash equilibrium. However, convergence
alone does not guarantee good generative performance. It is possible to converge
to sub-optimal solutions which are local Nash equilibria. Mode collapse is a special
case of a local Nash equilibrium associated with suboptimal generative performance. For example, assume a two
mode real world distribution where one mode contains too few and the other mode too many generator samples. If no real world samples
are between these two distinct modes, then the discriminator penalizes to move generated samples outside the modes. Therefore the generated
samples cannot be correctly distributed over the modes. Thus, standard GANs can
not capture the global sample density such that the resulting generators are prone to neglect large parts of the real world distribution. A more detailed example
is listed in the Appendix in Section A.1.In this paper, we introduce a novel GAN model, the Coulomb GAN, which has only one Nash equilibrium. We are later going to show
that this Nash equilibrium is optimal, i.e., the model distribution matches the target distribution. We propose Coulomb GANs to
avoid the GAN shortcoming (II) to (IV) by using a potential field created by point charges analogously to the electric field in physics. The next section will introduce
the idea of learning in a potential field and prove that its only solution is optimal. We will then show how learning
the discriminator and generator works in a Coulomb GAN and discuss the assumptions needed for our optimality proof. In Section 3 we will then see
that the Coulomb GAN does indeed work well in practice and that the samples it produces have very large variability and appear to capture the original distribution very well.Related Work. Several GAN approaches have been
suggested for bringing the target and model distributions in alignment using not just local discriminator information: Geometric GANs combine samples via a linear support vector machine which uses the discriminator outputs as samples, therefore they are much more robust to mode collapsing BID31 . Energy-Based GANs BID41 and their
later improvement BEGANs BID3 ) optimize an energy landscape based on auto-encoders. McGANs match mean and covariance
of synthetic and target data, therefore are more suited than standard GANs to approximate the target distribution BID34 . In a similar fashion, Generative
Moment Matching Networks BID30 and MMD nets BID12 directly optimize a generator network to match a training distribution by using a loss function based on the maximum mean discrepancy (MMD) criterion BID17 . These approaches were later expanded
to include an MMD criterion with learnable kernels and discriminators . The MMD criterion that these later approaches
optimize has a form similar to the energy function that Coulomb GANs optimize (cf. Eq. (33)). However, all MMD approaches end up
using
either
Gaussian or Laplace kernels, which are not guaranteed to find the optimal solution where the model distribution matches the target distribution. In contrast, the Plummer kernel which is employed
in this work has been shown to lead to the optimal solution BID22 . We show that even a simplified version of the Plummer
kernel, the low-dimensional Plummer kernel, ensures that gradient descent convergences to the optimal solution as stated by Theorem 1. Furthermore, most MMD GAN approaches use the MMD directly
as loss function though the number of possible samples in a mini-batch is limited. Therefore MMD approaches face a sampling problem in high-dimensional
spaces. The Coulomb GAN instead learns a discriminator network that gradually
improves its approximation of the potential field via learning Figure 1 : The vector field of a Coulomb GAN. The basic idea behind the Coulomb GAN: true samples (blue) and generated
samples (red) create a potential field (scalar field). Blue samples act as sinks that attract the red samples, which repel each
other. The superimposed vector field shows the forces acting on the generator samples
to equalize potential differences, and the background color shows the potential at each position. Best viewed in color.on many mini-batches. The discriminator network also tracks
the slowly changing generator distribution
during learning. Most importantly however, our approach is, to the best of our knowledge, the first
one for which optimality, i.e., ability to perfectly learn a target distribution, can be proved.The use of the Coulomb potential for learning is not new. Coulomb Potential Learning was proposed to store arbitrary many patterns in a potential
field with perfect recall and without spurious patterns BID35 . Another related work is the Potential Support Vector Machine (PSVM), which minimizes Coulomb
potential differences BID21 BID23 . BID22 also used a potential function based on Plummer kernels for optimal unsupervised learning
, on which we base our work on Coulomb GANs.
The Coulomb GAN is a generative adversarial network with strong theoretical guarantees.
Our theoretical results show that the Coulomb GAN will be able to approximate the real distribution perfectly if the networks have sufficient capacity and training does not get stuck in local minima.
Our results show that the potential field used by the Coulomb GAN far outperforms MMD based approaches due to its low-dimensional Plummer kernel, which is better suited for modeling probability density functions, and is very effective at eliminating the mode collapse problem in GANs.
This is because our loss function forces the generated samples to occupy different regions of the learned distribution.
In practice, we have found that Coulomb GANs are able to produce a wide range of different samples.
However, in our experience, this sometimes leads to a small number of generated samples that are non-sensical interpolations of existing data modes.
While these are sometimes also present in other GAN models , we found that our model produces such images at a slightly higher rate.
This issue might be solved by finding better ways of learning the discriminator, as learning the correct potential field is crucial for the Coulomb GAN's performance.
We also observed that increasing the capacity of the discriminator seems to always increase the generative performance.
We thus hypothesize that the largest issue in learning Coulomb GANs is that the discriminator needs to approximate the potential field Φ very well in a high-dimensional space.
In summary, instead of directly optimizing a criterion based on local differences of densities which can exhibit many local minima, Coulomb GANs are based on a potential field that has no local minima.
The potential field is created by point charges in an analogy to electric field in physics.
We have proved that if learning converges then it converges to the optimal solution if the samples can be moved freely.
We showed that Coulomb GANs avoid mode collapsing, model the target distribution more truthfully than standard GANs, and do not overlook high probability regions of the target distribution.A APPENDIX

assistant: text: Coulomb GANs can optimally learn a distribution by posing the distribution learning problem as optimizing a potential field