gan-optimal-discriminator-formula

IN premise — entries/2026/06/21/wiki-Generative_adversarial_network-chunk-2.md

Created 2026-06-21T09:55:49+00:00

The optimal discriminator for a fixed generator is D*(x) = μ_ref(x) / (μ_ref(x) + μ_G(x)), which is the Radon-Nikodym derivative dμ_ref/d(μ_ref + μ_G)

Dependents

These beliefs depend on this one:

• IN gan-architectural-asymmetry-mirrors-game-theory — GAN architectural asymmetry (deconvolutional generator vs convolutional discriminator in image tasks) parallels its game-theoretic asymmetry — the discriminator has a closed-form optimal solution while the generator does not, and these two asymmetries coexist in the minimax game structure, though the antecedents do not establish that the architectural choices were designed to instantiate the mathematical asymmetry.
• IN gan-complete-theoretical-characterization — GAN theory provides a complete analytical characterization of optimal behavior: the optimal discriminator has a closed-form solution (Radon-Nikodym derivative), the objective implicitly minimizes Jensen-Shannon divergence, and at the unique equilibrium the generator exactly recovers the data distribution.
• IN gan-training-stability-requires-multiple-interventions — GAN training stability benefits from several complementary design choices — non-saturating loss to address vanishing gradients when the generator is poor, two-timescale update rule for provable convergence to a stationary local Nash equilibrium (though not mode collapse prevention), and deterministic discriminators for optimality of the discriminator — each addressing a distinct failure mode, but none individually sufficient for overall stability.