Editing Generative Adversarial Networks (section)

== <span style="color: #FFFFFF;">Understanding</span> ==
The GAN training objective is a minimax game:

min_G max_D  E[log D(x)] + E[log(1 - D(G(z)))]

* D wants to maximize this: output high probabilities for real data x and low for G(z)
* G wants to minimize this: produce G(z) that D assigns high probability to

Think of it as a forger (G) and an art expert (D). The forger gets better at creating convincing fakes; the expert gets better at detecting them. Both improve through competition. In theory, the game converges when the forger is so good that the expert can't tell real from fake — the Nash equilibrium.

'''Why is training hard?''' The minimax game is not convex — there's no guarantee of convergence. Several failure modes are common:
* If D is too strong early, G receives near-zero gradients and cannot learn (vanishing gradient)
* If G is stronger, D cannot discriminate and provides no useful training signal
* Mode collapse: G finds one or a few "safe" outputs that always fool D and gets stuck

'''Wasserstein distance''' addresses vanishing gradients. Instead of a probability (0–1), WGAN trains D (called the "critic") to output a real number representing how real the sample is, using the Wasserstein-1 distance as the objective. This provides a smooth, meaningful gradient even when the distributions are far apart — fixing the vanishing gradient problem.

'''Conditional generation''' lets you control what the GAN produces. By feeding both G and D a conditioning signal (e.g., a class label "cat" or a source image), the generator learns to produce outputs matching that condition, enabling text-to-image, image-to-image, and class-conditional generation.
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