Model collapse

Model Collapse refers to a phenomenon where machine learning models gradually degrade due to errors coming from uncurated training on synthetic data, meaning the outputs of another model including prior versions of itself.

Shumailov et al. coined the term and described two specific stages to the degradation: early model collapse and late model collapse. In early model collapse the model begins losing information about the tails of the distribution – mostly affecting minority data. Later work highlighted that early model collapse is hard to notice, since overall performance may appear to improve, while the model loses performance on minority data. In the late model collapse model loses a significant proportion of its performance, confusing concepts and losing most of its variance.

Mechanism
Synthetic data, although theoretically indistinguishable from real data, is almost always biased, inaccurate, not well representative of the real data, harmful, or presented out-of-context. Using such data as training data leads to issues with quality and reliability of the trained model.

Model Collapse occurs for three main reasons – functional approximation errors, sampling errors, and learning errors. Importantly, it happens in even the simplest of models, where not all of the error sources are present. In more complex models the errors oftentimes compound, leading to faster collapse.

Disagreement over likely real-world impact
Some researchers and commentators on model collapse warn that the phenomenon could fundamentally threaten future generative AI development: As AI-generated data is shared on the Internet, it will inevitable end up in future training datasets, which are often crawled from the Internet. If training on synthetic data inevitably leads to model collapse, this could therefore pose a difficult problem.

However, recently other researchers have disagreed with this argument, showing that if synthetic data accumulates alongside human-generated data, model collapse is avoided. The researchers argue that data accumulating over time is a more realistic description of reality than assuming all existing data will vanish every year, and that the real-world impact of model collapse may not be as catastrophic as feared.

An alternative branch of the literature investigates the use of machine learning detectors and watermarking to identify model generated data and filter it out.

1D Gaussian Model
In, a first attempt has been made at illustrating collapse for the simplest possible model - a single dimensional normal distribution fit using unbiased estimators of mean and variance, computed on samples from the previous generation.

To make this more precise, we say that original data follows a normal distribution $$X^0 \sim \mathcal{N}(\mu,\sigma^2)$$, and we posses $$M_0$$ samples $$X^0_j$$ for $$j = 1, \dots, M_0$$. Denoting a general sample $$X^i_j$$ as sample $$j = 1, \dots, M_i$$ at generation $$i$$, then the next generation model is estimated using the sample mean and variance:

$$\mu_{i+1} = \frac{1}{M_i}\sum_j X^i_j; \quad \sigma_{i+1}^2 = \frac{1}{M_i-1}\sum _j(X^i_j-\mu_{i+1})^2.$$

Leading to a conditionally normal next generation model $$X^{i+1}_j|\mu_{i+1},\;\sigma_{i+1}\sim \mathcal{N}(\mu_{i+1},\sigma_{i+1}^2)$$. In theory, this is enough to calculate the full distribution of $$X^i_j$$. However, even after the first generation, the full distribution is no longer normal, it follows a variance-gamma distribution.

To continue the analysis, instead of writing the probability density function at each generation, it is possible to explicitly construct them in terms of independent random variables using Cochran's theorem. To be precise, $$\mu_1$$ and $$\sigma_1$$are independent, with $$\mu_1 \sim \mathcal{N}(\mu, \frac{\sigma^2}{M_0})$$ and $$(M_0-1)\sigma_1^2 \sim \sigma^2\Gamma\left(\frac{M_0-1}{2}, \frac12\right)$$, following a Gamma distribution. Denoting with $$Z$$ gaussian random variables distributed with $$\mathcal{N}(0, 1)$$ and with $$S^i$$ random variables distributed with $$\frac{1}{M_{i-1}-1}\Gamma\left(\frac{M_{i-1}-1}{2}, \frac12\right)$$, it turns out to be possible to write samples at each generation as

$$X^0_j = \mu + \sigma Z^0_j $$,

$$X^1_j = \mu + \frac{\sigma}{\sqrt{M_0}}Z^1 + \sigma\sqrt{S^1}Z^1_j $$,

and more generally

$$X^n_j = \mu + \frac{\sigma}{\sqrt{M_0}}Z^1 + \frac{\sigma}{\sqrt{M_1}}\sqrt{S^1}Z^2 + \dots + \frac{\sigma}{\sqrt{M_{n-1}}}\sqrt{S^1\times\dots\times S^{n-1}}Z^n+\sigma\sqrt{S^1\times\dots\times S^{n}}Z^n_j$$.

Note, that these are not joint distributions, as $$Z^n$$ and $$S^n$$ depend directly on $$Z^{n-1}_j$$, but when considering $$X^n_j$$ on its own the formula above provides all the information about the full distribution.

To analyse the model collapse, we can first calculate variance and mean of samples at generation $$n$$. This would tell us what kind of distributions we expect to arrive at after $$n $$ generations. It is possible to find its exact value in closed form, but the mean and variance of the square root of gamma distribution are expressed in terms of gamma functions, making the result quite clunky. Following, it is expanding all result to second order in each of $$1/M_i$$, assuming each sample size to be large. It is then possible to show that

$$\frac{1}{\sigma^2}\operatorname{Var}(X^n_j) = \frac{1}{M_0}+\frac{1}{M_1}+ \dots + \frac{1}{M_{n-1}}+1 + \mathcal{O}\left(M_i^{-2}\right).$$

And if all sample sizes $$M_i = M$$ are constant, this diverges linearly as $$n\to\infty$$:

$$\operatorname{Var}(X^n_j) = \sigma^2\left(1+\frac{n}{M}\right); \quad \mathbb{E}(X^n_j) = \mu.$$

This is the same scaling as for a single dimensional Gaussian random walk. However, divergence of the variance of $$X^n_j$$ does not directly provide any information about the corresponding estimates of $$\mu_{n+1}$$ and $$\sigma_{n+1}$$, particularly how different they are from the original $$\mu$$ and $$\sigma$$. It turns out to be possible to calculate the distance between the true distribution and the approximated distribution at step $$n+1$$, using the Wasserstein-2 distance (which is also sometimes referred to as risk):

$$\mathbb{E}\left[\mathbb{W}^2_2\left(\mathcal{N}(\mu,\sigma^2),\mathcal{N}(\mu_{n+1},\sigma^2_{n+1})\right)\right]=\frac{3}{2}\sigma^2\left(\frac{1}{M_0}+\frac{1}{M_1}+ \dots + \frac{1}{M_{n}}\right)+\mathcal{O}\left(M_i^{-2}\right),$$

$$\operatorname{Var}\left[\mathbb{W}^2_2\left(\mathcal{N}(\mu,\sigma^2),\mathcal{N}(\mu_{n+1},\sigma^2_{n+1})\right)\right]=\frac{1}{2}\sigma^4\left(\frac{3}{M_0^2}+\frac{3}{M_1^2}+ \dots + \frac{3}{M_{n}^2} + \sum_{i\neq j}\frac{4}{M_iM_j}\right)+\mathcal{O}\left(M_i^{-3}\right).

$$

This directly shows why model collapse occurs in this simple model. Due to errors from re-sampling the approximated distribution, each generation ends up corresponding to a new step in a random walk of model parameters. For a constant sample size at each generation, the average distance from the starting point diverges, and in order for the end distribution approximation to be accurate, or for the distance to be finite, the sampling rate $$M_i$$ needs to increase superlinearly, i.e. one needs to collect increasingly more samples over time, perhaps quadratically. However, even in that case the expected distance after $$n$$ steps remains non-zero and the only case in which it does in fact end up being zero is when sampling is infinite at each step. Overall, this only shows us how far on average one ends up from the original distribution, but the process can only "terminate", if the estimated variance at a certain generation becomes small enough, effectively turning the distribution into a delta function. This is shown to occur for a general gaussian model in the subsection below.

N-D Gaussian Model
Furthermore, in the case of multidimensional model with fully synthetic data, exact collapse can be shown.

Linear Regression
In the case of a linear regression model, scaling laws and bounds on learning can be found.

Statistical Language Model
In the case of a linear softmax classifier for next token prediction, exact bounds on learning with even a partially synthetic dataset can be found.

Impact on large language models
In the context of large language models, research found that training LLMs on predecessor-generated text—language models are trained on the synthetic data produced by previous models—causes a consistent decrease in the lexical, syntactic, and semantic diversity of the model outputs through successive iterations, notably remarkable for tasks demanding high levels of creativity.