# Proof (**GAN optimal point)**

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,* equals -2 log2.*V**

Proof:

If ** G** is fixed, the optimal Discriminator

**is**

*D**Find the optimal value for ** V**:

i.e.

Hence, from the equation above, the minimum (optimal point) for ** V** is when

**(when D_JS = 0)**

*p = q.***Optimal discriminator**

The optimal value ** y*** for

is

Proof: