In GANs, you have two players with different goals:

• Discriminator: Wants to maximize its chance of correctly identifying real vs. fake data, this uses gradient ascent.
• Generator: Wants to minimize the discriminator’s ability to detect fakes this uses gradient descent.

This creates a minimax game, balanced by both updates during training.

$$ \min_{\theta_g} \max_{\theta_d} \frac{1}{2} \mathbb{E}{x \sim p{data}} [\log D(x)] + \frac{1}{2} \mathbb{E}_{z \sim p(z)} [\log (1 - D(G(z)))] $$

Discriminator : Using gradient ascent, it maximizes the loss with respect to discriminator parameters

$$ \theta_d \leftarrow \theta_d + \alpha \nabla_{\theta_d} V(\theta_d, \theta_g) $$

Generator : Using gradient descent, it minimizes the loss with respect to generator parameter

$$ \theta_g \leftarrow \theta_g - \alpha \nabla_{\theta_g} V(\theta_d, \theta_g) $$

Start of Training

• Discriminator D: Easily tells real vs fake
• Loss values:
  • First term (real):  small/close to zero loss
  • Second term (fake): small loss
• Both losses are low because the discriminator is strong; generator’s fake images are easily detected.

Mid Training

• Generator G improves to fool Discriminator D, so:
  • D(G(z)) increases from 0 → values closer to 0.5
• Discriminator tries to keep up: D(x)≈1 and D(G(z))≈0