Foundations — Generator and discriminator dynamics

Before you can read a single line of the parent note, you must be able to look at every symbol it uses and immediately know what it means and why it is there. This page builds them one at a time, in an order where each new symbol only uses symbols already defined. Nothing is assumed.

1. The two players: what actually moves

Everything starts with two machines. Let us draw them before we name them.

Why two machines and not one? Because we have no direct rule for "what makes a face look real". So we let one machine learn that rule (the detective $D$) and use it as the teacher for the painter $G$. That is the whole trick.

2. The input symbols: $z$, $x$, and the wavy arrow

We keep saying "noise" and "picture". Let us pin down the exact symbols.

Why do we need distributions at all? Because $G$ is never judged on one fake — it is judged on the whole population of fakes it can make. Distributions are the language for "the whole population at once".

3. The averaging symbol: $\mathbb{E}$ (expectation)

The parent note is full of $\mathbb{E}$. It looks scary; it is just an average.

Why average instead of one sample? Because a single lucky (or unlucky) picture tells you nothing about whether $G$ is generally good. The game is decided on averages, so the objective is written with averages.

4. The logarithm: $\log$ — why it appears everywhere

The parent note wraps every probability in $\log$. Here is the honest reason.

Two building blocks you will now recognise on sight:

• $\log D(x)$ — big (near 0) when $D$ correctly calls a real picture real.
• $\log (1-D(G(z)))$ — big (near 0) when $D$ correctly calls a fake picture fake. Here $1-D(\cdot )$ flips "probability real" into "probability fake".

5. Putting it together: the value function $V(D,G)$

Now every piece of the headline formula is a symbol you have already met.

6. The learning symbols: $\theta$, $\nabla$, gradient steps

How does a machine "tune itself"? Two more symbols.

Why gradients and not a formula for the best knobs? Because $D$ and $G$ are giant tangled networks with no closed-form best setting. Feeling your way uphill/downhill one small step at a time is the only tractable option — this is the machinery of the GAN training algorithm.

7. The prerequisite map

Every arrow says "you need the tail before you can understand the head." Read the parent note only once every box here feels obvious.

8. Where these lead next

Once the tug-of-war stalls perfectly you reach $D^{\ast }(x)=\frac{1}{2}$ everywhere — but the path there can break in two famous ways: mode collapse (the painter cheats with one trick) and vanishing gradients, which motivates the Wasserstein GAN. This page only equips you to read the dynamics; those pages explain when they go wrong.

Equipment checklist

Cover the right side. If you can answer each, you are ready for the parent note.