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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?

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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.

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