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.
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.
[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 ?
Thanks in advance for your attention !
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).
1
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.
1
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).
1
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