# Generative Adversarial Networks and Game Theory

GAN(Generative Adversarial Network) is a Neural Network model in which there exist two neural networks, one commonly referred to be the Generator and the other as Discriminator. Adversarial Learning is a study of attacking neural networks, but it is being used as a tool to build the GAN model. In each iteration, the Generator will synthesize a product–commonly to be images in modern applications, and the Discriminator will take this product as input and judge if this product is real or fake(produced by neural networks); if it is the second case, the parameters of the Generator will be tuned, the goal is making the product as realistic as possible.

Such an idea could be modeled as a two-player zero-sum minimax game played by the Generator and the Discriminator. Zero-sum refers to the fact that the payoff for the Generator is the negative payoff but with the same magnitude for the Discriminator. And the minimax refers to each player trying to minimize the payoff of the opponent while maximizing the payoff of itself. Combining the two facts above, we are able to find out that GANs are designed to reach aNash equilibrium at which each player cannot increase their gain without reducing the other player’s gain[1]

However, the major problem with the initial GAN model facing was such a game-theoric model could be very difficult to converge[1]. That is, at some point, the “quality” of the output will not be able to increase. In this paper[3], Barbara Franci and Sergio Grammatico propose a stochastic relaxed forward-backward (SRFB) algorithm to alter the training process so that we are able to use many existing training techniques. At the very least, we are able to use gradient descent or even more advanced methods such as Adams and its variance.

GAN is an interesting model, and it seems promising to combine it with nowadays hyped text-image generation models like Dall-E and Stable-Diffusion.

[1] I. J. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley,

S. Ozair, A. Courville, and Y. Bengio, “Generative adversarial networks,”

arXiv preprint arXiv:1406.2661, 2014 https://arxiv.org/abs/1406.2661