Learning GANs: Game, Theory, Training, Variants

Definition and zero-sum game framing

In a GAN, the generator and discriminator are framed as two players in a zero-sum game: the generator tries to produce samples that look real, while the discriminator tries to distinguish real data from generated data. Formally, the original GAN defines a game between a generator strategy (a probability measure over the data space) and a discriminator strategy (a Markov kernel mapping inputs to probabilities in [0,1]). The objective function L(μG, μD) is constructed so that the generator minimizes L while the discriminator maximizes it, making one player’s gain the other’s loss. In practice, this game is implemented by alternating updates: the discriminator is trained on real samples and generator-produced samples to improve its ability to label inputs as real vs. fake, while the generator is trained based on how well it “fools” the discriminator. The generator is typically implemented via a latent variable z passed through a neural network G (a pushforward distribution), which turns the game into the common minG maxD formulation. The topic also emphasizes that, because the strategy spaces are not finite, equilibrium concepts can depend on move order (generator-first vs. discriminator-first), and the existence/uniqueness of equilibria becomes more subtle than in finite games. A key mathematical result in this framing is that for a fixed generator distribution μG, the optimal discriminator can be expressed using a Radon–Nikodym derivative, and the resulting value of the objective relates directly to the Jensen–Shannon divergence between the generator distribution and the reference (real) distribution. This connects the adversarial zero-sum game to a well-known statistical distance between distributions, clarifying what it means for the generator to match the data distribution.

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Generator and discriminator roles

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