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A GAN consists of two networks that are competing against each other.

They are linked together in a GAN:

The generative network is trying to fool the discriminative network into misclassifying “fake” (generated inputs). Meanwhile, the discriminative network is also learning how to classify inputs as real or fake: i.e., does it come from the dataset, or from the generative network?

$D(x;\theta )$ outputs the probability that $x$ is from the real data:

$$x\in R\therefore t=1$$

$G(z;\varphi )$ creates an input sample from random noise $z$ with distribution $p_z$:

$$x^{\prime }=G(z;\varphi )\in F\therefore t=0$$

The loss for the GAN has two parts:

$$E(\theta ,\varphi )={\mathbb{E}}_{R,F}[L(D(x;\theta ),t)]+{\mathbb{E}}_z[L(D(G(z;\varphi ),\theta ),1)]$$

• The first term minimizes with respect to $\theta$ to make the discriminator better at detecting real vs. fake. $x$ are real inputs, and the target $t=1$ for $x\in R$ and $0$ for $x\in F$.
• The second term minimizes with respect to $\varphi$ to make the generator better at producing fake inputs that look real to the discriminator.

Training the generator:

• generate a fake sample
• pass it through the discriminator
• measure loss against the label 1
• update the generator so that the discriminator becomes more likely to output “real”.