# Why GAN use adversarial MinMax formulation rather than Min formulation?

For generative adversarial neural network, originally Goodfellow used a MinMax formulation as $$\text{Min}_D\text{Max}_G \mathbb{E}_{real}logD(x) dx+ \mathbb{E}_{fake}(1-D(G(z)))dz$$. As long as the generator $$G$$ is fixed, the optimal discriminator $$D$$ is explicit. My question is, as we have a clear understanding of $$D$$, why not just minimize the Jenson-Shannon divergence as a Min formulation, which is equivalent to the MinMax formulation?

BTW, other researchers raised a similar framework called f-GAN, which replaces the Jenson-Shannon divergence by other $$f$$-divergence. They also adopted the MinMax formulation, rather than minimizing the divergence directly. Why MinMax formulation is more popular here than Min formulation in these works?

## 1 Answer

Why not directly minimize the Jensen-Shannon divergence between the generator and empirical distribution? Because it's intractable to compute. The marginal distribution $$p(x) = \int p(x|z)p(z) dz$$ is very hard to work with, computationally.