### The two networks ```mermaid flowchart LR Z["z ~ N(0, I)
noise"] --> G["Generator
transposed convs"] G --> FAKE["Fake image"] REAL["Real image"] --> D["Discriminator
conv classifier"] FAKE --> D D --> OUT["P(real)"] style G fill:#dbeafe,stroke:#2563eb style D fill:#fef3c7,stroke:#d97706 style OUT fill:#dcfce7,stroke:#16a34a ``` The **generator** G takes a vector of noise `z` and outputs an image. The **discriminator** D takes an image and outputs a single scalar: the probability that the image is real. ### The game G wants D to be wrong. D wants to be right. Formally: ``` min_G max_D E_x[log D(x)] + E_z[log(1 - D(G(z)))] ``` Read right to left: D is maximising accuracy on real (`log D(real)`) and fake (`log (1 - D(fake))`) images. G is minimising D's accuracy on fakes — it wants `D(G(z))` to be high. Goodfellow proved that this minimax has a global equilibrium where `p_G = p_data`, D outputs 0.5 everywhere, and the Jensen-Shannon divergence between generated and real distributions is zero. The hard part is getting there. ### Non-saturating loss The form above is numerically unstable. Early in training, `D(G(z))` is near zero for every fake, so `log(1 - D(G(z)))` has vanishing gradients with respect to G. The fix: flip G's loss. ``` L_D = -E_x[log D(x)] - E_z[log(1 - D(G(z)))] L_G = -E_z[log D(G(z))] # non-saturating ``` Now when `D(G(z))` is near zero, G's loss is large and its gradient is informative. Every modern GAN trains with this variant. ### DCGAN architecture rules Radford, Metz, Chintala (2015) distilled years of failed experiments into five rules that make GAN training stable: 1. Replace pooling with strided convs (both nets). 2. Use batch norm in both generator and discriminator, except output of G and input of D. 3. Remove fully connected layers on deeper architectures. 4. G uses ReLU on all layers except output (tanh for output in [-1, 1]). 5. D uses LeakyReLU (negative_slope=0.2) on all layers. Every modern conv-based GAN (StyleGAN, BigGAN, GigaGAN) still starts from these rules and replaces pieces one at a time. ### Failure modes and their signatures ```mermaid flowchart LR M1["Mode collapse
G produces a narrow
set of outputs"] --> S1["D loss low,
G loss oscillating,
sample variety drops"] M2["Vanishing gradients
D wins completely"] --> S2["D accuracy ~100%,
G loss huge and static"] M3["Oscillation
G and D keep trading
wins forever"] --> S3["Both losses swing
wildly with no downward trend"] style M1 fill:#fecaca,stroke:#dc2626 style M2 fill:#fecaca,stroke:#dc2626 style M3 fill:#fecaca,stroke:#dc2626 ``` - **Mode collapse**: G finds one image that fools D and produces only that. Fix: add minibatch discrimination, spectral norm, or label-conditioning. - **Discriminator wins**: D becomes too strong too fast, G's gradients vanish. Fix: smaller D, lower D learning rate, or apply label smoothing on the real labels. - **Oscillation**: the two nets trade wins without ever approaching equilibrium. Fix: TTUR (D learns faster than G by a factor of 2-4), or switch to Wasserstein loss. ### Evaluation GANs have no ground truth, so how do you know they are working? - **Sample inspection** — just look at 64 samples at the end of every epoch. Non-negotiable. - **FID (Fréchet Inception Distance)** — distance between Inception-v3 feature distributions of real and generated sets. Lower is better. Community standard. - **Inception Score** — older, more brittle; prefer FID. - **Precision/Recall for generative models** — measures quality (precision) and coverage (recall) separately. More informative than FID alone. For a small synthetic-data run, sample inspection is enough.