The GAN formulation
A simple way to formulate the GAN is a zero-sum game.
Any loss for the discriminator is a gain for the generator and vice versa.
Therefore, the payoffs in GANs are exactly balanced to model a zero-sum game.
The discriminator receives the payoff \(v\left(\mTheta^{(g)}, \mTheta^{(d)}\right) \).
The discriminator payoff is defined as
\begin{equation}
v\left(\mTheta^{(g)}, \mTheta^{(d)}\right) = \expect{\vx \sim p_{\text{data}}}{ \log d(\vx) } + \expect{\vx \sim p_{\text{model}}}{\log \left(1 - d(\vx)\right)}
\label{eqn:gan-payoff}
\end{equation}
The first term, \( \expect{\vx \sim p_{\text{data}}}{ \log d(\vx) } \) is an expectation over available actual data. Therefore, \( \expect{\vx \sim p_{\text{data}}}{\cdot} \).
We wish \( d(\vx) \) for all examples drawn from the training distribution \( \vx \sim p_{\text{data}} \) to be close to 1, implying an accurate detection of true examples.
This will achieve a high value for the payoff.
The second term, \( \expect{\vx \sim p_{\text{model}}}{\log \left(1 - d(\vx)\right)} \) is an expectation over the fake data created by the generator. Therefore, \( \expect{\vx \sim p_{\text{model}}}{\cdot} \).
For fake examples, we wish the discriminator to emit a probability of 0, implying an accurate detection of fake examples.
This means, we want \( d(\vx) \) for all examples drawn from the generator distribution \( \vx \sim p_{\text{model}} \) to be close to 0.
This will achieve a high value for the payoff since we are taking a logarithm of \( \left(1 - d(\vx) \right) \).
To counterbalance the discriminator payoff, the generator payoff is set to the negative of the discriminator payoff. This means, the generator payoff is \( -v\left(\mTheta^{(g)}, \mTheta^{(d)}\right) \).
With the payoffs set up this way, the training algorithm attempts to maximize the gain of the discriminator and minimize the loss of the generator.
Therefore, at convergence, the final generator \( \star{g} \) is
\begin{equation}
\star{g} = \argmin_{g} \left[ \maxunder{d} v(g,d) \right]
\label{eqn:gan-convergence}
\end{equation}