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In generative modeling, the goal is to find a way for a model to output samples of some distribution $p_X$ given a lot of samples $x_1, \ldots, x_n$. In particular, we want sampling from our model $G$ to satisfy

  • $G(z)$ is a new example
  • $G(z)$ looks like it was sampled from $p_X$.

GAN's approach this by finding a Nash equilibrium where $p_g=p_X$, where $p_g$ is the distribution implicitly defined by mapping the latent noise $z$ under $G$. How do we know that $G$ does not simply memorize the input data?

For example, if I train a GAN to output pictures of cats, how do I know that the output isn't just a modified picture of one of the cats that was used to train $G$? In the original Deep Convolutional GAN (DCGAN) paper, they have the following explanation DCGAN image which I don't find particularly convincing.

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  • $\begingroup$ IDK but I could see writing an objective function which selects against similar samples. Practicality speaking though a slightly perturbed image of it's changed in a valid way is a new sample. $\endgroup$ – FourierFlux Jan 28 at 7:08
  • $\begingroup$ Right, but... what objective? What are robust similarity metrics between images? Pixel-wise similarity performs poorly, and DeePSiM doesn't seem to have much theoretical justification (seems somewhat ad hoc). $\endgroup$ – Anon Jan 28 at 21:46
  • $\begingroup$ Also, I disagree with your statement "Practicality speaking though a slightly perturbed image of it's changed in a valid way is a new sample." One of the purposes have generative modeling is to automate a creative component of some larger task (e.g. creating images that are realistic but not just copies). If the model just copies it's training data, then the creative task hasn't truly been automated. For examples, if I want to make some model that can generate music or art, then a model that just copied the training data would result in me having unwittingly committed a copyright violation. $\endgroup$ – Anon Jan 28 at 21:50
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To my knowledge, there aren't any truly robust methods. One strategy that I have seen in involved for the nearest neighbor (often using Euclidean distance). From there, the practitioner conducts visual inspection to check that overfitting is not occuring. This tends to be a poor indicator though because nearest-neighbor approaches tend not to work in such high-dimensional space. Also, see here for failures of pixel-wise loss. Another strategy involves looking at the nearest-neighbors of embeddings of the images (both synthesized and original data) into some space that is designed to extract features. This sounds better but doesn't seem to have much empirical or rigorous theoretical justification, with only a heuristic explanation.

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What you're interested is GAN mode collapse and mode dropping. (You can call it overfitting too, it's just that the community has adopted these names). There are literally thousands of GAN papers devoted to solving the problem with varying success, but checking for mode collapse/dropping is still an area of active research.

Mode collapse means that the generator only learns to output a small number of distinct images / classes and very obviously fails to learn the distribution in any sense. Mode dropping means that the generator may appear to learn the distribution, but still drops modes (for example, suppose you train a GAN to generate images 1000 different species of birds... you're unlikely to notice if the GAN only produces images of 500 species and forgets the rest). We can also talk about inter-class mode collapse (GAN fails to learn all classes) and inter-class collapse (GAN can generate images from all classes, but only generates one / a few distinct images for each class).


General purpose metrics for GAN evaluation

The most popular metrics for measuring GAN performance are Inception Score and Frechét Inception Distance. A one sentence summary of Inception Score: do the images look like they're being drawn from many different classes? A one sentence summary of FID: does the distribution of perceptual features from generated images match that of ground truth images?

IS can detect inter-class mode collapse, whereas FID can also detect intra-class mode collapse. However these metrics aren't perfect by any means, and a high score / low distance isn't a guarantee that no collapse/dropping has happened.

Another metric is Classifier Augmentation Score, which basically uses GAN outputs to train a classifier, and measures the performance of the classifier. Intuitively, any form of overfitting / mode dropping would result in a poor classifier. However CAS is pretty expensive to compute.

Specific methods to detect mode collapse/dropping

In GLO, the authors propose to examine mode dropping in GANs by a "reconstruction" test. Basically, for some held out ground truth image $x$, we find the best noise vector $z$ which when passed through the GAN, produces something like $x$, and measure the cost by $\text{min}_z\ ||f(z)-x||_2^2$. The idea is that if your GAN can only produce one image of a cat, it will be unable to reconstruct the held out cat image. A similar idea was explored in more detail recently in Seeing what a GAN cannot generate.

The birthday paradox can be rephrased a bit more abstractly as: "If there are $n$ distinct outcomes, it's likely that you'll come across the same outcome twice after sampling only $\Theta(\sqrt{n})$ times". So, if a GAN only copies the 10,000 images from the training set, we can figure this out by just looking at a ~100 of them, as done by Arora and Zhang.

PRD proposes some direct method to measure the precision and recall of a GAN. (precision meaning high quality, non blurry samples, and recall meaning that we haven't dropped any modes in the data, and are fully covering the distribution). It's too complicated to summarize neatly, but the results look promising.

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  • $\begingroup$ I don't think mode collapse is the same thing as overfitting. I explicitly mention in my question that overfitting refers to when a generative model simply memorizes some data points in a way such that sampling from the model just outputs a (near) copy of one of the data points in which it was trained on, rather than generating a new data point that looks like it came from the same distribution. The problem of mode collapse is an entirely different problem where the generative model is only able to generate a small portion of the support of the true data. $\endgroup$ – Anon Feb 2 at 23:47
  • $\begingroup$ To clarify with an example, with the generative model in thispersondoesnotexist.com, how do I know that a generated person does not actually exist (i.e. that this model does not just spit out one of its training data points or something nearly identical)? Over fitting refers to this phenomenon, which seems difficult to detect. Mode collapse is just the phenomenon where a large number of samples from this generator seem to converge to the same image, which is easy to detect. $\endgroup$ – Anon Feb 2 at 23:52

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