The GAN min–max game

The discriminator tries to maximise its reward $\mathcal{E}(\mathbb{X},{\mathcal{G}}_{\theta }(\mathcal{K}))$: for any given generator it solves $\arg {\max }_D\mathcal{E}$, i.e. it maximises the divergence between $\mathbb{X}$ and $\mathcal{G}(\mathcal{K})$.

The generator minimises the power of this test — the accuracy of $D$. The generative model's goal is $\arg {\min }_{\theta }\mathcal{E}(\mathbb{X},{\mathcal{G}}_{\theta }(\mathcal{K}))$, so the divergence between the two distributions has to be computed; the model is kept non-parametric for its high capacity.

The classic vanilla GAN instance is ${\min }_G{\max }_D{\mathbb{E}}_{x\sim \mathbb{X}}[\log D(x)]+{\mathbb{E}}_z[\log (1-D(G(z)))]$.