Adversarial Nets

$D$ and $G$ play the following two-player minimax game with value function $V(G,D)$

$$ \min_{G} \max_{D}V(D,G) = \mathbb{E}{x \sim p{data}(x)}[\log D(x)] + \mathbb{E}_{z\sim p_z(z)}[\log (1-D(G(z)))] $$

• $p_g$ : Generator’s distribution
• $p_z(z)$ : prior on input noise variables
• $G(z;\theta_g)$ : Generator mapping noise to data space with parameters $\theta_g$
• $D(x;\theta_d)$ : Discriminator outputing probability that $x$ came from data with parameters $\theta_d$

Pedagogical Explanation

$\cdots$ : Discriminative distribution $D$

$\cdots$ : Data generating distribution $p_x$

—— : Generative distribution $p_g(G)$

$\uparrow$ : how mapping $x = G(z)$ imposes the non-uniform distribution $p_g$ on transformed samples

• $p_g$ is similar to $p_{data}$ and $D$ is partially an accurate classifier

• Inner loop of algorithm $D$ is trained to discriminate samples from data, converging to $D^{*}(x) = \frac{p_{data}(x)}{p_{data}(x)+p_g(x)}$

• After an update to $G$, gradient of $D$ has guided $G(z)$ to dlow to regions that are more likely to be classified as data

• After several steps of training, $p_g = p_{data}$.
• Discriminator is unable to differentiate between two distributions
• $D(x) = \frac{1}{2}$