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Title: Training GANs with Optimism

Paper content:
We address the issue of limit cycling behavior in training Generative Adversarial Networks and propose the use of Optimistic Mirror Decent (OMD) for training Wasserstein GANs.
Recent theoretical results have shown that optimistic mirror decent (OMD) can enjoy faster regret rates in the context of zero-sum games.
WGANs is exactly a context of solving a zero-sum game with simultaneous no-regret dynamics.
Moreover, we show that optimistic mirror decent addresses the limit cycling problem in training WGANs.
We formally show that in the case of bi-linear zero-sum games the last iterate of OMD dynamics converges to an equilibrium, in contrast to GD dynamics which are bound to cycle.
We also portray the huge qualitative difference between GD and OMD dynamics with toy examples, even when GD is modified with many adaptations proposed in the recent literature, such as gradient penalty or momentum.
We apply OMD WGAN training to a bioinformatics problem of generating DNA sequences.
We observe that models trained with OMD achieve consistently smaller KL divergence with respect to the true underlying distribution, than models trained with GD variants.
Finally, we introduce a new algorithm, Optimistic Adam, which is an optimistic variant of Adam.
We apply it to WGAN training on CIFAR10 and observe improved performance in terms of inception score as compared to Adam.
Generative Adversarial Networks (GANs) BID4 have proven a very successful approach for fitting generative models in complex structured spaces, such as distributions over images.
GANs frame the question of fitting a generative model from a data set of samples from some distribution as a zero-sum game between a Generator (G) and a discriminator (D).
The Generator is represented as a deep neural network which takes as input random noise and outputs a sample in the same space of the sampled data set, trying to approximate a sample from the underlying distribution of data.
The discriminator, also modeled as a deep neural network is attempting to discriminate between a true sample and a sample generated by the generator.
The hope is that at the equilibrium of this zero-sum game the generator will learn to generate samples in a manner that is indistinguishable from the true samples and hence has essentially learned the underlying data distribution.Despite their success at generating visually appealing samples when applied to image generation tasks, GANs are very finicky to train.
One particular problem, raised for instance in a recent survey as a major issue BID5 is the instability of the training process.
Typically training of GANs is achieved by solving the zero-sum game via running simultaneously a variant of a Stochastic Gradient Descent algorithm for both players (potentially training the discriminator more frequently than the generator).The
latter amounts essentially to solving the zero-sum game via running no-regret dynamics for each player. However
, it is known from results in game theory, that no-regret dynamics in zerosum games can very often lead to limit oscillatory behavior, rather than converge to an equilibrium. Even in
convex-concave zero-sum games it is only the average of the weights of the two players that constitutes an equilibrium and not the last-iterate. In fact
recent theoretical results of Mertikopoulos et al. (2017) show the strong result that no variant of GD that falls in the large class of Follow-theRegularized-Leader (FTRL) algorithms can converge to an equilibrium in terms of the last-iterate and are bound to converge to limit cycles around the equilibrium.Averaging the weights of neural nets is a prohibitive approach in particular because the zero-sum game that is defined by training one deep net against another is not a convex-concave zero-sum game. Thus it
seems essential to identify training algorithms that make the last iterate of the training be very close to the equilibrium, rather than only the average.Contributions. In this
paper we propose training GANs, and in particular Wasserstein GANs BID1 , via a variant of gradient descent known as Optimistic Mirror Descent. Optimistic
Mirror Descent (OMD) takes advantage of the fact that the opponent in a zero-sum game is also training via a similar algorithm and uses the predictability of the strategy of the opponent to achieve faster regret rates. It has been
shown in the recent literature that Optimistic Mirror Descent and its generalization of Optimistic Follow-the-Regularized-Leader (OFTRL), achieve faster convergence rates than gradient descent in convex-concave zero-sum games (Rakhlin & Sridharan, 2013a; b) and even in general normal form games (Syrgkanis et al., 2015) . Hence, even
from the perspective of faster training, OMD should be preferred over GD due to its better worst-case guarantees and since it is a very small change over GD.Moreover, we prove the surprising theoretical result that for a large class of zero-sum games (namely bi-linear games), OMD actually converges to an equilibrium in terms of the last iterate. Hence, we give
strong theoretical evidence that OMD can help in achieving the long sought-after stability and last-iterate convergence required for GAN training. The latter theoretical
result is of independent interest, since solving zero-sum games via no-regret dynamics has found applications in many areas of machine learning, such as boosting BID2 . Avoiding limit cycles
in such approaches could help improve the performance of the resulting solutions.We complement our theoretical result with toy simulations that portray exactly the large qualitative difference between OMD as opposed to GD (and its many variants, including gradient penalty, momentum, adaptive step size etc.). We show that even in
a simple distribution learning setting where the generator simply needs to learn the mean of a multi-variate distribution, GD leads to limit cycles, while OMD converges pointwise.Moreover, we give a more complex application to the problem of learning to generate distributions of DNA sequences of the same cellular function. DNA sequences that carry
out the same function in the genome, such as binding to a specific transcription factor, follow the same nucleotide distribution. Characterizing the DNA distribution
of different cellular functions is essential for understanding the functional landscape of the human genome and predicting the clinical consequence of DNA mutations (Zeng et al., 2015; 2016; Zeng & Gifford, 2017) . We perform a simulation study where
we generate samples of DNA sequences from a known distribution. Subsequently we train a GAN to attempt
to learn this underlying distribution. We show that OMD achieves consistently
better performance than GD variants in terms of the Kullback-Leibler (KL) divergence between the distribution learned by the Generator and the true distribution.Finally, we apply optimism to training GANs for images and introduce the Optimistic Adam algorithm. We show that it achieves better performance
than Adam, in terms of inception score, when trained on CIFAR10.

assistant: text: The paper proposes to use optimistic gradient descent for GAN training that avoids the cycling behavior observed with SGD and its variants and provides promising results in GAN training.