In generative adversarial models (GANs), why should we solve min-max problem and not max-min? I know that in GANs model, there is min-max game between generator and discriminator which discriminator tries to maximize the loss function and the goal of generator is to minimize it. But why we write the loss function as min-max problem and not max-min? As I know, the loss function is not convex, so there is difference between these approaches.
You can use either a min-max or a max-min game, it doesn't matter. Look, this is the &quot;usual&quot; loss function (i. e. if you use the original loss proposed by Goodfellow et al. in their paper on GANs, though I use a slightly different notation):
$$\min_{G}\max_{D}\mathcal L_{\text{GAN}}\left(D, G \right) = \mathbb E_{x \sim p_{\text{data}\left( x\right) }}\left[ \log \left\{ D\left( x \right) \right\} \right] + \mathbb E_{z\sim p_{z}\left( z\right)}\left[ \log\left\{ 1 - D\left( G\left( z\right) \right) \right\}\right].$$
Now, you could also write this as:
$$\max_{G}\min_{D}\mathcal L_{\text{GAN}}\left(D, G \right) = \mathbb E_{x \sim p_{\text{data}\left( x\right) }}\left[ \log \left\{ 1 - D\left( x \right) \right\} \right] + \mathbb E_{z\sim p_{z}\left( z\right)}\left[ \log\left\{ D\left( G\left( z\right) \right) \right\}\right].$$
I hope that clarifies your doubt.