#optimal-discriminator

Question 1: Single choice

hardAI-checkedGenerative Adversarial Networks

A least-squares GAN trains its discriminator by minimising $\tfrac{1}{2}\,\mathbb{E}_{x \sim p_{\text{data}}}\big[(D(x) - b)^2\big] + \tfrac{1}{2}\,\mathbb{E}_{z \sim p_z}\big[(D(G(z)) - a)^2\big],$ where $b$ is the target for real samples and $a$ the target for generated ones. For a fixed generator whose samples have density $p_g$ , what is the optimal discriminator $D^*(x)$ ?

$D^*(x) = \dfrac{b\,p_{\text{data}}(x) + a\,p_g(x)}{p_{\text{data}}(x) + p_g(x)}$ 1
$D^*(x) = \dfrac{a\,p_{\text{data}}(x) + b\,p_g(x)}{p_{\text{data}}(x) + p_g(x)}$ 2
$D^*(x) = \dfrac{p_{\text{data}}(x)}{p_{\text{data}}(x) + p_g(x)}$ 3
$D^*(x) = \dfrac{p_{\text{data}}(x) + p_g(x)}{b\,p_{\text{data}}(x) + a\,p_g(x)}$ 4

#optimal-discriminator

Timed test