I have 3 questions related to the GAN training objective (in lecture notes 12a, page 7), and the 2P-G formulation (in lecture notes 12a, page 2):
1) Why do we choose a min-max order for the GAN objective, rather than a max-min order? Wouldn't both be a solution to the zero sum 2P-G formulation? Is it because we are ultimately interested in the generator? Would we need to change the update order in the training loop if we chose a max-min order instead?
2) Couldn't there be parameters theta and phi that satisfy the 2P-G formulation, but are not a solution for either of the min-max or max-min problems? (I think both the min-max and max-min solutions do satisfy the 2P-G condition, but I can not see why the reverse argument holds true)
3) For the given GAN objective, what term is constricting the generator to model the entire probability distribution, rather than a subset of it? To give an example, if my data distribution contains cat and dog images, and if my generator only creates perfect fake dog images for every realization of the noise (and no cat images at all), wouldn't that be a local optimum (since the discriminator can't distinguish the fake dog images from a real dog image)? What part of the GAN objective forces such a generator to learn to generate cat images as well?
I apologize for the long question. Thanks in advance.
Hi,
I will answer only the third part of your question. Indeed such an inability to support a large part of the domain is one of the weaknesses of GAN's. In practice, GAN generator's output variability is often lower than other likelihood-based models, such as variational autoencoders (although often output quality of GAN's are better). However, you also need to realize that if the generator is only limited to a very small subset of the real data, then the discriminator can start rejecting every example coming from that region (if there are 1000 classes of objects as in ImageNet, the generator cannot just produce cats or dogs as it will be too easy for the discriminator.)
Best,
Semih
If your objective is convex-concave (and your domain in cvx and compact), min-max is equivalent to max-min [shown by von Neumann 1928]. In all other cases, these are not equivalent, and the order matters. These are out of the scope of the course, but if you're interested lookup Stackelberg games and some nice references that connect these to GANs are [1,2,3].
The reverse argument does not hold true, and 2P-G seeks joint point. See for example the 'almost bilinear' toy problem in Hsieh et al. 2020--where we don't want to converge to such point (and in fact such point is unstable).
see the above answer. such an issue is referred to as 'mode collapse' in the literature (although the choice of this term is not good)
Some bits of your questions are out of the scope for the course (as well as the references I listed), but if you're further curious, please ping me by email for pointers. Thanks for the nice questions!
[1] Implicit Learning Dynamics in Stackelberg Games: Equilibria Characterization, Convergence Analysis, and Empirical Study, ICML, 2020.
[2] What is Local Optimality in Nonconvex-Nonconcave Minimax Optimization?, ICML 2020.
[3] On Solving Minimax Optimization Locally: A Follow-the-Ridge Approach, ICLR 2020.
Questions about GAN Objective
Greetings,
I have 3 questions related to the GAN training objective (in lecture notes 12a, page 7), and the 2P-G formulation (in lecture notes 12a, page 2):
1) Why do we choose a min-max order for the GAN objective, rather than a max-min order? Wouldn't both be a solution to the zero sum 2P-G formulation? Is it because we are ultimately interested in the generator? Would we need to change the update order in the training loop if we chose a max-min order instead?
2) Couldn't there be parameters theta and phi that satisfy the 2P-G formulation, but are not a solution for either of the min-max or max-min problems? (I think both the min-max and max-min solutions do satisfy the 2P-G condition, but I can not see why the reverse argument holds true)
3) For the given GAN objective, what term is constricting the generator to model the entire probability distribution, rather than a subset of it? To give an example, if my data distribution contains cat and dog images, and if my generator only creates perfect fake dog images for every realization of the noise (and no cat images at all), wouldn't that be a local optimum (since the discriminator can't distinguish the fake dog images from a real dog image)? What part of the GAN objective forces such a generator to learn to generate cat images as well?
I apologize for the long question. Thanks in advance.
2
That's a nice question, any update on that ?
1
Hi,
I will answer only the third part of your question. Indeed such an inability to support a large part of the domain is one of the weaknesses of GAN's. In practice, GAN generator's output variability is often lower than other likelihood-based models, such as variational autoencoders (although often output quality of GAN's are better). However, you also need to realize that if the generator is only limited to a very small subset of the real data, then the discriminator can start rejecting every example coming from that region (if there are 1000 classes of objects as in ImageNet, the generator cannot just produce cats or dogs as it will be too easy for the discriminator.)
Best,
Semih
1
Hi,
If your objective is convex-concave (and your domain in cvx and compact), min-max is equivalent to max-min [shown by von Neumann 1928]. In all other cases, these are not equivalent, and the order matters. These are out of the scope of the course, but if you're interested lookup Stackelberg games and some nice references that connect these to GANs are [1,2,3].
The reverse argument does not hold true, and 2P-G seeks joint point. See for example the 'almost bilinear' toy problem in Hsieh et al. 2020--where we don't want to converge to such point (and in fact such point is unstable).
see the above answer. such an issue is referred to as 'mode collapse' in the literature (although the choice of this term is not good)
Some bits of your questions are out of the scope for the course (as well as the references I listed), but if you're further curious, please ping me by email for pointers. Thanks for the nice questions!
[1] Implicit Learning Dynamics in Stackelberg Games: Equilibria Characterization, Convergence Analysis, and Empirical Study, ICML, 2020.
[2] What is Local Optimality in Nonconvex-Nonconcave Minimax Optimization?, ICML 2020.
[3] On Solving Minimax Optimization Locally: A Follow-the-Ridge Approach, ICLR 2020.
1
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