22. Global Optimum Exists
V (D, G) = Ex⇠pdata(x)[logD(x)] + Ez⇠pz(z)[log(1 D(G(z))]
x = G(z) ) z = G 1
(x) ) dz = (G 1
)0
(x)dx
) pg(x) = pz(G 1
(x))(G 1
)0
(x)
=
Z
x
pdata(x)log(D(x))dx +
Z
x
pz(G 1
(x))log(1 D(x))(G 1
)0
(x)dx
=
Z
x
pdata(x)log(D(x))dx +
Z
x
pg(x)log(1 D(x))dx
=
Z
x
pdata(x)log(D(x)) + pg(x)log(1 D(x))dx
=
Z
x
pdata(x)log(D(x))dx +
Z
z
pz(z)log(1 D(G(z)))dz
23. Global Optimum Exists
max
D
V (D, G) = max
D
Z
x
pdata(x)log(D(x)) + pg(x)log(1 D(x))dx
)
pdata(x)
D(x)
pg(x)
1 D(x)
= 0
@
@D(x)
(pdata(x)log(D(x)) + pg(x)log(1 D(x))) = 0
) D(x) =
pdata(x)
pdata(x) + pg(x)