16.8  Further Reading

The chapter taught a few ideas on problems small enough to check: what a critic estimates, what a ranking critic changes, why the game needs a penalty to converge, how to score images and condition them, and where a learned comparison is still needed. The field produced far more objectives, architectures, scores and systems than these ideas require. This page points to the ones a reader of the chapter is best placed to read next. Each entry states the problem it addresses, what it changes relative to the chapter, and why to read it; the topics follow the order of the chapter, and nothing in the book depends on them.

16.8.1 Objectives and Critic Constraints

Least squares as a divergence. Section 16.2 derived the divergence that the least-squares critic loss evaluates. Mao et al. (2017) started from the other end and asked which target codes make the generator’s loss a known divergence: with the real and fake codes two apart and the generator’s target one above the fake code, minimizing the generator’s loss minimizes the Pearson \(\chi^2\) divergence between \(p + q\) and \(2q\). In the paper’s convention this is \(\tfrac12 \int (p-q)^2/(p+q)\), symmetric and bounded by one, not the unbounded \(\chi^2(p \,\|\, q)\) of Section 28.3.2.1. The zero–one coding of the chapter’s code, with the generator’s target equal to the real code, falls outside the derivation; its critic still evaluates the square row of the chapter’s table. The derivation takes a few lines in the template of that section, and it shows how much the coding constants matter.

Output activations for \(f\)-GAN critics. The variational bound of Section 16.2 asks the critic’s output \(t\) to lie where the conjugate \(f^*(t)\) is finite. Nowozin et al. (2016) meet this with an output activation chosen per divergence. The forward-KL conjugate is finite on all of \(\mathbb{R}\) and needs none; the reverse-KL conjugate is finite only for \(t < 0\), enforced by \(-e^{-v}\) on the network’s raw output \(v\); the Jensen–Shannon conjugate is finite for \(t < \log 2\), enforced by \(\log 2 - \mathrm{softplus}(-v)\); and the original GAN’s conjugate is finite for \(t < 0\), enforced by \(-\mathrm{softplus}(-v)\), the log-sigmoid that Section 16.1 already uses. Their table is the place to look before training a divergence that the chapter did not. A trained critic certifies only a lower bound, and a minibatch estimate of that bound can flatter it: an overshooting critic is penalized only where \(q\) is small, which minibatches rarely sample (Exercise 7 of Section 16.2 measures both on the chapter’s three-mode mixture).

Three ways to a Lipschitz critic. The Wasserstein-1 game needs a 1-Lipschitz critic, and no network is 1-Lipschitz by construction. Weight clipping (Arjovsky et al. 2017) and spectral normalization (Miyato et al. 2018) bound the product of the layers’ Lipschitz constants, a bound that holds everywhere but is often loose. The one-centered gradient penalty (Gulrajani et al. 2017) constrains the gradient only at sampled points, tightly there and not at all elsewhere. These guarantees decide how close a trained critic can come to the Kantorovich potential; Section 16.4 explains why zero-centered penalties replaced the one-centered one.

The relativistic average critic. Jolicoeur-Martineau (2019) proposed, besides the pairing loss of Section 16.3, a cheaper variant that compares each sample’s score with the mean score of the other batch: a real sample against the average generated score, and a generated sample against the average real score. The comparison makes the loss shift-invariant, like the pairing loss, but each sample is still judged against one number, a threshold rather than a rank. The super-resolution network ESRGAN trains with it (Wang et al. 2018), so it is the variant a reader most often meets in image-restoration code.

The relativistic divergence for general payoffs. Section 16.3 computed the value of the pairing game for the logistic payoff, for which the best pair critic is a difference of single-sample scores. For other payoffs it is not, and the value has no closed form. Jolicoeur-Martineau (2020) shows that the pairing objective is still a divergence: its calibrated value is nonnegative and vanishes exactly at \(q = p\) for every concave payoff \(\ell\) with \(\ell(0) = 0\), \(\ell'(0) \neq 0\), and a positive supremum attained or approached at a positive argument. Her normalization is twice the chapter’s, so her divergence is \(2\, d_{\mathrm{Rp}}\). Read it for the general conditions and for the relativistic least-squares and hinge losses, whose payoffs, shifted to vanish at zero, satisfy them.

16.8.2 Architectures and Training Devices

Architectural devices, 2015 to 2021. Between the 2015 DCGAN network and the plain penalized backbone of Section 16.5, most progress on image GANs came from devices added to the architecture, each aimed at one failure. Progressive growing starts both networks at \(4 \times 4\) and adds layers during training, against instability at high resolution (Karras et al. 2018). Spectral normalization divides each weight matrix by its largest singular value, which bounds the critic’s steepness everywhere, where the penalties of Section 16.4 act only at the samples (Miyato et al. 2018). Self-attention gives both networks image-wide receptive fields for long-range structure (Zhang et al. 2019). BigGAN scales batches and widths and finds that collapse is delayed, not removed (Brock et al. 2019). StyleGAN adds a mapping network and per-layer style modulation to disentangle latent factors (Karras et al. 2019); StyleGAN2 replaces the normalization that caused droplet artifacts by weight demodulation (Karras, Laine, et al. 2020); StyleGAN3 removes texture that sticks to pixel coordinates by band-limited resampling (Karras et al. 2021). Some impose by architecture what the penalties impose through the loss; reading them against R3GAN shows which ones a well-behaved loss makes unnecessary.

Pretrained features in the critic. Projected GAN discriminates in the feature pyramid of a frozen pretrained classifier and reports reaching previous best FID values up to forty times faster (Sauer et al. 2021). The frozen features change what the critic can estimate: it sees the log ratio only through features trained for another task. StyleGAN-T keeps the construction but uses self-supervised features, partly to reduce the overlap with the network that computes FID (Sauer et al. 2023); the second entry under Evaluation explains why that overlap matters.

Training at scale. Large runs keep R3GAN’s recipe but schedule its constants over a burn-in of 20 million images instead of fixing them (Huang et al. 2024). The learning rate decays from \(2 \cdot 10^{-4}\) to \(5 \cdot 10^{-5}\) and the penalty weight \(\gamma\) falls tenfold with it, from 0.05 to 0.005 on CIFAR-10 and from 150 to 15 on FFHQ at \(256 \times 256\). Adam’s \(\beta_2\) rises from 0.9 to 0.99, and the EMA half-life grows from zero to millions of images. The same paper reports that mixed-precision training fails with FP16 and works with BFloat16. It also rebuilds StyleGAN2 on FFHQ at \(256 \times 256\) at about the same parameter count and budget. Removing StyleGAN2’s ten devices, from the mapping network and weight demodulation to lazy regularization, raises the reported FID from 7.5 to 12.5. The pairing loss with \(R_1\) and \(R_2\) brings it to 11.7, a modern residual backbone to 10.0, and grouped convolutions with inverted bottlenecks to 7.0, below the starting point.

The cost of the penalty. An input-gradient penalty needs a second backward pass through the critic. StyleGAN2 computes \(R_1\) only once every 16 minibatches, with its weight scaled up to compensate, and calls this lazy regularization (Karras, Laine, et al. 2020). StyleGAN2 also trains well on faces with \(R_1\) alone, so the StackedMNIST ablation of Section 16.4, where \(R_1\) alone diverged, does not settle whether one penalty suffices. Adversarial post-training of a video diffusion transformer could not take the second backward pass at all, because its fused attention kernels and sharded training did not support it. It penalizes instead the squared change of the critic’s output when a real sample is perturbed by small Gaussian noise, which for small noise approximates the \(R_1\) integrand scaled by the noise variance; without this term its training collapsed (Lin et al. 2025). A reader who needs the penalty at a large scale will find the trade-off between its cost and its frequency in these papers.

Training on limited data. Section 16.5 reads overfitting from the critic’s train–holdout gap and argues that an augmentation must be applied to both inputs of the critic and must not map two distributions to one. Differentiable augmentation applies the same random color, translation and cutout transformations to real and generated images in both updates, inside the generator’s gradient path (Zhao et al. 2020). Adaptive discriminator augmentation adds two things (Karras, Aittala, et al. 2020). It characterizes which augmentations are non-leaking, applied with probability below one so that the clean distribution can be recovered, and it sets that probability during training from an overfitting statistic, the mean sign of the critic’s scores on training images, \(r_t = \mathbb{E}[\operatorname{sign} D(x_{\mathrm{train}})]\). The statistic presumes a meaningful zero level of the critic’s score, which a pairing critic does not have, since adding a constant to its scores leaves its loss unchanged; with the loss of Section 16.3 a pairwise statistic is needed instead. With adaptive augmentation StyleGAN2 trains on a few thousand images, and the two papers are the place to start for a dataset that small.

16.8.3 Evaluation

Fidelity and coverage separately. FID combines two failures in one number: generated samples that look wrong, and data modes that no generated sample covers. The precision and recall of Kynkäänniemi et al. (2019) separate them through neighborhoods in feature space. Precision is the fraction of generated samples that fall inside the estimated support of the real features, recall the fraction of real samples inside the support of the generated ones. A generator that collapses to one good image keeps a high precision while its recall falls to zero, which FID reports only as a worse score. R3GAN reports recall next to FID for this reason.

The feature network behind the score. FID and KID are only as good as the features they compare. Kynkäänniemi et al. (2023) show that Inception features respond strongly to ImageNet class information, so a generator can lower FID by matching the distribution of ImageNet classes in its samples without improving them visibly, and critics built on ImageNet-pretrained features can lower FID in the same way. Stein et al. (2023) compare seventeen scores with a large human study, find that Inception-based FID underrates the realism of diffusion models, and recommend DINOv2 features; recent image-generation papers report the Fréchet distance in DINOv2 features next to FID (Oquab et al. 2024; Karras et al. 2024). CMMD replaces both the Inception features and the Gaussian fit of FID: it computes the MMD of Section 16.2 with a Gaussian kernel on CLIP embeddings, which is unbiased and assumes nothing about the shape of the feature distribution (Jayasumana et al. 2024). Two further effects change the numbers without changing the images. Aliased resizing of the images before the feature network shifts FID by as much as the differences between published methods (Parmar et al. 2022). The finite-sample bias of FID depends on the generator as well as on \(n\), so at small \(n\) it alone can reverse a ranking; extrapolating FID linearly in \(1/n\) removes most of it (Chong and Forsyth 2020). A reader who trains a critic on pretrained features should evaluate with a different feature network.

Conditional scores. Section 16.6 measures fidelity, within-class diversity and alignment with three separate numbers. Two published scores combine them. The Fréchet joint distance fits a Gaussian to image and condition embeddings together, so a realistic image with the wrong condition changes the score (DeVries et al. 2019). The classification accuracy score trains a classifier on generated image–label pairs and tests it on real ones, so a collapsed or missing class shows up as errors on real images of that class (Ravuri and Vinyals 2019). Read them for evaluation protocols that a single number can report.

16.8.4 Conditions, Translation and Text

Modulation as a general layer. The modulation \(a(c) \odot \phi + b(c)\) of Section 16.6 has a family of variants that differ in how the feature map is normalized first and where the scale and shift come from. Conditional batch normalization computes them from a question’s embedding to steer an image network (Vries et al. 2017), and FiLM drops the normalization and makes the layer a general conditioning device (Perez et al. 2018). Adaptive instance normalization normalizes each image’s channels separately and takes the scale and shift from the statistics of a style image (Huang and Belongie 2017); StyleGAN’s style injection descends from it. SPADE makes the modulation spatial: a semantic layout supplies a separate scale and shift at each location, so the generator can paint a scene from a label map without normalization washing out the layout (Park et al. 2019).

Auxiliary classifiers. The auxiliary classifier critic of Odena et al. (2017) conditions the critic through a classification head instead of the projection head of Section 16.6, and trains the generator to make its samples receive the requested label. That loss rewards samples that are easy to classify, so it favors class prototypes over within-class diversity. Gong et al. (2019) remove the bias with a second classifier trained on generated pairs only, whose loss enters the generator’s objective with the opposite sign. Read them for the alternative to projection and its repair.

Translation systems. Section 16.6 explains why pix2pix adds a critic to its \(L_1\) loss. Without paired examples, CycleGAN gives up the conditional critic: each direction plays an unconditional game against its target collection, and a cycle-consistency loss asks that translating forward and back return the input. The cycle loss constrains the map without identifying it: composing a solution with any invertible map that preserves the target distribution gives another solution, and when the target carries less information than the source, the generator can hide the missing information in imperceptible signals (Chu et al. 2017). The two papers add the systems around these ideas (Isola et al. 2017; Zhu et al. 2017): an encoder–decoder generator with skip connections between matching resolutions (a U-Net), so that the source’s layout reaches the output directly; and a patch critic that scores overlapping patches instead of the whole image, which is the local comparison the ambiguity argument asks for. They are the place to start for image-to-image applications.

Text as the condition. A caption is a sequence of tokens, not one class. GigaGAN feeds it to the generator through cross-attention from image features to the caption’s tokens, as text-to-image diffusion models do (Kang et al. 2023). By 2026 diffusion and autoregressive models dominate general text-to-image synthesis. GigaGAN is the reference for what a single-pass adversarial generator achieves at that task, and for the architectural changes it needed to scale.

16.8.5 Fixed Discrepancies and Learned Ones

The chapter’s critics are learned. Could a fixed discrepancy, with no second network, serve as well? Generative moment matching networks minimize the fixed-kernel MMD of Section 16.2, with neither a critic nor an inner optimization (Li et al. 2015). IMLE pairs each data point with its nearest generated sample, which penalizes any data mode left without a nearby sample (Li and Malik 2018). Sliced Wasserstein generators average one-dimensional transport costs along random projections (Deshpande et al. 2018), and Sinkhorn divergences interpolate between optimal transport and MMD (Genevay et al. 2018).

On natural images, several of these became competitive only after a learned component was added. MMD-GAN learns the kernel adversarially (Li et al. 2017), and sliced-Wasserstein generators add a discriminator to choose informative projections, because random directions carry little signal in high dimensions (Deshpande et al. 2018). This agrees with the estimation argument of Section 16.2: a discrepancy can separate distributions in principle and still be hard to estimate from samples. Sriperumbudur et al. (2012) give the convergence rates of the empirical estimators behind that argument, and Bińkowski et al. (2018) analyze the learned kernels of MMD-GAN and, from the same analysis, propose KID, the score of Section 16.5.

16.8.6 Adversarial Terms in Recent Systems

Section 16.7 explains why an adversarial term survives in one-step generators, autoencoders and vocoders. The systems below show the range of reasons; the numbers are those their papers reported between 2024 and 2026.

Tokenizers. Most latent image generators run on an autoencoder, and the adversarial term in that autoencoder decides how much texture the decoder can produce. VQGAN combined reconstruction and perceptual losses with a patch critic (Esser et al. 2021), and the Stable Diffusion autoencoder kept the combination (Rombach et al. 2022). The ViTok study measures both sides of the trade-off that Section 16.7 derives: fine-tuning its decoder adversarially lowered reconstruction FID to 0.50 at some cost in PSNR and SSIM, while the FID of images generated in its latent space improved only from 5.5 to 4.9 (Hansen-Estruch et al. 2025). Its successor ViTok-v2 scales the autoencoder to five billion parameters and replaces both the adversarial loss and the LPIPS perceptual loss with a fixed loss on self-supervised DINOv3 features (Hansen-Estruch et al. 2026). Read together, the two papers show what a critic adds to a tokenizer at moderate scale and what replaced it at large scale.

Distillation with real data. A distilled student that only matches its teacher can be no better than the teacher. DMD2 adds an adversarial loss against real images, which lets its ImageNet \(64 \times 64\) student reach FID 1.28, below its teacher (Yin et al. 2024). LADD reuses the teacher diffusion model as the critic’s backbone and finds that synthetic teacher samples train it well enough to drop the distillation loss, leaving only the adversarial objective (Sauer et al. 2024). The FLUX.1-schnell model card states that its twelve-billion-parameter rectified-flow transformer was trained this way to generate in one to four steps (Black Forest Labs 2024). Whole-image FID separates these few-step methods poorly. The SDXL-Lightning report also computes FID on image patches, which responds to local texture, and finds larger differences there (Lin et al. 2024): the ambiguity that the adversarial term addresses sits in fine detail, where a whole-image score averages it away.

Few steps without a critic. An adversarial term is not the only route to few-step generation. MeanFlow trains a one-step generator from scratch, with no teacher, distillation or critic, and reports FID 3.43 on ImageNet \(256 \times 256\) (Geng et al. 2025). Continuous-time consistency models report FID 1.88 on ImageNet \(512 \times 512\) with two network evaluations and no adversarial loss (Lu and Song 2025). Both replace the critic with a regression target designed to stay informative at low step counts.

Vocoders and video. RAF applies the pairing loss of Section 16.3 to waveforms; a BigVGAN-base trained with it, with twelve percent of the parameters, outperforms the full BigVGAN trained with the least-squares loss in perceptual quality (Lee and Choi 2026). WaveFM reaches single-step synthesis without a critic, by flow matching followed by consistency distillation (Luo et al. 2025). For video, adversarial post-training fine-tunes a diffusion generator against real data with a zero-centered gradient penalty from the family of Section 16.4, and produces two seconds of \(1280 \times 720\) video at 24 frames per second in one forward evaluation (Lin et al. 2025).

Likelihood models as critics at scale. Direct discriminative optimization, the construction of Section 16.7 that uses the likelihood model itself as the critic, fine-tunes pretrained EDM diffusion models and lowers FID from 1.79 to 1.30 on CIFAR-10 and from 1.58 to 0.97 on ImageNet \(64 \times 64\) (Zheng et al. 2025).

Discussions

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Bińkowski, Mikołaj, Danica J. Sutherland, Michael Arbel, and Arthur Gretton. 2018. “Demystifying MMD GANs.” International Conference on Learning Representations. https://arxiv.org/abs/1801.01401.

Black Forest Labs. 2024. FLUX.1 [Schnell] Model Card. Https://huggingface.co/black-forest-labs/FLUX.1-schnell.

Brock, Andrew, Jeff Donahue, and Karen Simonyan. 2019. “Large Scale GAN Training for High Fidelity Natural Image Synthesis.” International Conference on Learning Representations. https://arxiv.org/abs/1809.11096.

Chong, Min Jin, and David Forsyth. 2020. “Effectively Unbiased FID and Inception Score and Where to Find Them.” Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 6070–79. https://arxiv.org/abs/1911.07023.

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Lin, Shanchuan, Xin Xia, Yuxi Ren, Ceyuan Yang, Xuefeng Xiao, and Lu Jiang. 2025. “Diffusion Adversarial Post-Training for One-Step Video Generation.” Proceedings of the 42nd International Conference on Machine Learning.

Lu, Cheng, and Yang Song. 2025. “Simplifying, Stabilizing and Scaling Continuous-Time Consistency Models.” International Conference on Learning Representations.

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Sauer, Axel, Tero Karras, Samuli Laine, Andreas Geiger, and Timo Aila. 2023. “StyleGAN-T: Unlocking the Power of GANs for Fast Large-Scale Text-to-Image Synthesis.” Proceedings of the 40th International Conference on Machine Learning.

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