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Advanced•Modern AI
Generative Models
GANs, VAEs, and Diffusion Probabilistic Models (DDPM/DDIM).
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Research-Level Deep Dive & Equations
Generative Adversarial Networks (Goodfellow et al., 2014) frame generative modeling as a game-theoretic competition between two networks: a **Generator** that maps a low-dimensional noise vector to the data space, and a **Discriminator** that outputs the probability that a sample came from the empirical data distribution rather than the generated distribution .
The system is optimized by solving a zero-sum, two-player minimax objective:
**Mathematical Proof of the Optimal Discriminator **:
For any fixed generator , we want to find the discriminator that maximizes the objective . We can express the expectation terms as continuous integrals:
Using the change of variables under the push-forward measure , this becomes:
To maximize this integral with respect to at each point , we differentiate the integrand with respect to :
Substituting and , we obtain the unique optimal discriminator:
**Jensen-Shannon Divergence Convergence Bound**:
Substituting back into the minimax objective:
where is the Jensen-Shannon divergence. This proves that the global minimum of the minimax game is achieved if and only if , where .
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