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Proof (GAN optimal point)

Jonathan Hui
2 min readJun 23, 2018


We have mentioned optimizing GAN is optimizing JS-divergence. This is not obvious from the cost function:

We will prove it here and also prove the corresponding optimal point is:

  • p = q,
  • the discriminator cannot distinguish the real from the fake D*(x) = 1/2,
  • V* equals -2 log2.


If G is fixed, the optimal Discriminator D* is

Find the optimal value for V:


Hence, from the equation above, the minimum (optimal point) for V is when p = q. (when D_JS = 0)

Optimal discriminator

The optimal value y* for