### [Lecture 12.a] GAN equilibrum computation

Hi,

When studying the convergence of the loss function (in section "The GAN framework: Equilibrium at pg = pd"), I don't understand how we can be sure that the point we find is a maximum.

Here are my computations:
f(y) = a log(y) + b log(1-y)
f'(y) = a/y - b/(1-y)
f''(y) = -a/y^2 - b/(1-y)^2

f''(a/(a+b)) = -(a + b)^2 (1/a + 1/b)
which is not < 0 (except if a and b are both positive or if 1/a > 1/b which we don't know) as stated in the lecture.

Am I missing something in my computations ?

Top comment

Sorry, yes you are right. But then there is no problem, the second derivative is negative, what is your problem?

Just to make sure, a and b are positive (They are probabilities).

The second derivative is $$-a/y^2 - b/(1-y)^2$$.

In my computation (verified by Wolframalpha) it is a minus, as we already have a minus, then the minus from the exponent and then the minus from (1 - y).

Still it wouldn't show me why f''(a/(a+b)) is < 0.

Top comment

Sorry, yes you are right. But then there is no problem, the second derivative is negative, what is your problem?

Just to make sure, a and b are positive (They are probabilities).