GAN Adversarial Training

GANs pit a generator $G$ against a discriminator $D$ in a zero-sum min-max game: $\min_G\max_D V(D,G) = \mathbb{E}_{x \sim p_{data}}[\log D(x)] + \mathbb{E}_{z \sim p_z}[\log(1 - D(G(z)))]$. For fixed $G$, differentiating w.r.t. $D$ yields the optimal discriminator as a likelihood ratio $D^*(x) = \frac{p_{data}(x)}{p_{data}(x) + p_g(x)}$; plugging it back gives $V(D^*, G) = -\log 4 + 2 \cdot JSD(p_{data} \Vert p_g)$, i.e. GAN minimizes the Jensen-Shannon divergence between the data and generated distributions, reaching the global optimum $-\log 4$ iff $p_{data} = p_g$.

image generation (DCGAN/ProGAN/StyleGAN), super-resolution, image-to-image translation; interviews drill the min-max objective, the $D^*$ derivation, and instability/mode-collapse mechanisms.

unlike VAEs with explicit density approximation and pixel-wise reconstruction losses, GANs make no distributional assumption and produce sharper samples; the cost is that alternating D/G training approximates a Nash equilibrium rather than minimizing a loss — with a too-strong $D$ the JSD saturates and generator gradients vanish, with a too-weak $D$ the signal is noisy, leading to mode collapse and oscillation.

Write GAN's min-max objective and derive the optimal discriminator $D^*(x)$?

Why does substituting $D^*$ reduce GAN to minimizing $JSD(p_{data} \Vert p_g)$?

Why is GAN training unstable? Mechanisms behind vanishing gradients and mode collapse?

Why do GANs suffer mode collapse while VAEs do not? How do their sample qualities differ?

Common techniques to stabilize GAN training (label smoothing / TTUR / spectral norm)?