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Title: A Variational Inequality Perspective on Generative Adversarial Networks

Paper content:
Generative adversarial networks (GANs) form a generative modeling approach known for producing appealing samples, but they are notably difficult to train.
One common way to tackle this issue has been to propose new formulations of the GAN objective.
Yet, surprisingly few studies have looked at optimization methods designed for this adversarial training.
In this work, we cast GAN optimization problems in the general variational inequality framework.
Tapping into the mathematical programming literature, we counter some common misconceptions about the difficulties of saddle point optimization and propose to extend methods designed for variational inequalities to the training of GANs.
We apply averaging, extrapolation and a computationally cheaper variant that we call extrapolation from the past to the stochastic gradient method (SGD) and Adam.
Generative adversarial networks (GANs) BID12 ) form a generative modeling approach known for producing realistic natural images (Karras et al., 2018) as well as high quality super-resolution (Ledig et al., 2017) and style transfer (Zhu et al., 2017) .
Nevertheless, GANs are also known to be difficult to train, often displaying an unstable behavior BID11 .
Much recent work has tried to tackle these training difficulties, usually by proposing new formulations of the GAN objective (Nowozin et al., 2016; .
Each of these formulations can be understood as a two-player game, in the sense of game theory (Von Neumann and Morgenstern, 1944) , and can be addressed as a variational inequality problem (VIP) BID15 , a framework that encompasses traditional saddle point optimization algorithms (Korpelevich, 1976) .Solving
such GAN games is traditionally approached by running variants of stochastic gradient descent (SGD) initially developed for optimizing supervised neural network objectives. Yet it
is known that for some games (Goodfellow, 2016, §8. 2) SGD exhibits oscillatory behavior and fails to converge. This oscillatory
behavior, which does not arise from stochasticity, highlights a fundamental problem: while a direct application of basic gradient descent is an appropriate method for regular minimization problems, it is not a sound optimization algorithm for the kind of two-player games of GANs. This constitutes
a fundamental issue for GAN training, and calls for the use of more principled methods with more reassuring convergence guarantees.Contributions. We point out that
multi-player games can be cast as variational inequality problems (VIPs) and consequently the same applies to any GAN formulation posed as a minimax or non-zerosum game. We present two techniques
from this literature, namely averaging and extrapolation, widely used to solve VIPs but which have not been explored in the context of GANs before. 1 We extend standard GAN
training methods such as SGD or Adam into variants that incorporate these techniques (Alg. 4 is new). We also explain that the
oscillations of basic SGD for GAN training previously noticed BID11 can be explained by standard variational inequality optimization results and we illustrate how averaging and extrapolation can fix this issue.We introduce a technique, called extrapolation from the past, that only requires one gradient computation per update compared to extrapolation which requires to compute the gradient twice, rediscovering, with a VIP perspective, a particular case of optimistic mirror descent (Rakhlin and Sridharan, 2013) . We prove its convergence
for strongly monotone operators and in the stochastic VIP setting.Finally, we test these techniques in the context of GAN training. We observe a 4-6% improvement
over Miyato et al. (2018) on the inception score and the Fréchet inception distance on the CIFAR-10 dataset using a WGAN-GP BID14 ) and a ResNet generator.
We newly addressed GAN objectives in the framework of variational inequality.
We tapped into the optimization literature to provide more principled techniques to optimize such games.
We leveraged these techniques to develop practical optimization algorithms suitable for a wide range of GAN training objectives (including non-zero sum games and projections onto constraints).
We experimentally verified that this could yield better trained models, improving the previous state of the art.
The presented techniques address a fundamental problem in GAN training in a principled way, and are orthogonal to the design of new GAN architectures and objectives.
They are thus likely to be widely applicable, and benefit future development of GANs.

assistant: text: We cast GANs in the variational inequality framework and import techniques from this literature to optimize GANs better; we give algorithmic extensions and empirically test their performance for training GANs.