• $x$: Data we already have
• $z$: Data we want to generate
• $\theta_g$: parameter for Generator
• $\theta_d$: parameter for D
• $G$: Generator
• $D$: Discriminator

target is training $G$ to minimize

$$\mathop{min}_G \mathop{max}_D V(D,G) = E_{x\sim p_{data(x)}}[\mathop{log}D(x)] + E_{z\sim p_z(z)}[\mathop{log}(1-D(G(z)))]$$

For player $D$, want V bigger by

• making more accurate estimate on real date x as $D(x)\rightarrow 1$
• discriminate the fake data by $D(G(z)) \rightarrow 0$ equals to $(1 - D(G(z))) \rightarrow 1$
• more accurate the $D$ is ,the larger value of $V$ can be

For player $G$, want V smaller by

• enlarge $D(G(z))\rightarrow 1$ for $(1 - D(G(z))) \rightarrow 0$
• better fake of $G$, $G(z)\rightarrow 1$, and less value of $V$

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Algorithm

for number of iterations do

for $k$ steps do

• Sample $m$ noise samples $\{ z^{(1)},…,z^{(m)} \}$ from noise prior $p_g(z)$
• Sample $m$ examples $\{ x^{(1)},…,x^{(m)} \}$ from data generating distribution $p_{data}(x)$
• Update the discriminator by ascending its stochastic gradient : $$\nabla_{\theta_g} \frac{1}{m} \sum_{i=1}^m \left[ \mathop{log}D(x^{(i)})+log\left(1-D(G(z^{(i)}))\right) \right]$$

end for

• Sample $m$ noise samples $\{ z^{(1)},…,z^{(m)} \}$ from noise prior $p_g(z)$
• Update the generator by descending its stochastic gradient : $$\nabla_{\theta_g} \frac{1}{m}\sum_{i=1}^m\mathop{log}\left( 1-D(G(z^{(i)})) \right)$$

end for

  ew
  - q
  - q