Collapse or Thrive: Perils and Promises of Synthetic Data in a Self-Generating World
Authors: Joshua Kazdan
Rylan Schaeffer
Apratim Dey
Matthias Gerstgrasser
Rafael Rafailov
David L. Donoho
Sanmi Koyejo
Venue: ICML 2025 poster (2025)
Keywords: model collapse
synthetic data
model-data feedback loops
data-model feedback loops
generative models
generative modeling
kernel density estimation
supervised finetuning
Primary area: deep_learning->large_language_models
TL;DR: forecasting whether future frontier generative models will collapse or thrive due to synthetic data

Abstract: What happens when generative machine learning models are pretrained on web-scale datasets containing data generated by earlier models? Some prior work warns of “model collapse” as the web is overwhelmed by synthetic data; other work suggests the problem can be contained (i.e. collapse can be avoided) by managing how available data are used in pretraining. In this paper, we report experiments on three ways of using data (training-workflows), across three generative model task-settings (multivariate Gaussian estimation, kernel density estimation, and language-model fine-tuning) to further confirm the possibility of containment: (a) we confirm that the training-workflow of {\it replacing} all real data by successive generations of purely synthetic data indeed suffers model collapse in all task-settings studied; (b) we consider the training-workflow of {\it accumulating} synthetic data alongside real data and training on all data combined and confirming that, although the proportion of real data eventually becomes zero, models remain stable and their test losses do not diverge under this training-workflow; (c) we consider a training-workflow where real and synthetic data accumulate together but successive generations of pretraining are constrained to use fixed-size data subsets each generation. In this workflow, we observe slow and gradual rather than explosive degradation of test loss performance across generations. Our insights are particularly important when forecasting whether future frontier generative models will collapse or thrive, and our results open avenues for empirically and mathematically studying the context-dependent value of synthetic data.

Collapse or Thrive?
Perils and Promises of Synthetic Data in a Self-Generating World
Joshua Kazdan
Rylan Schaeffer 2 Apratim Dey' Matthias Gerstgrasser? Rafael Rafailov?
David Donoho Sanmi Koyejo²
Abstract
1. Introduction
What happens when generative machine learning models are pretrained on web-scale datasets con- taining data generated by earlier models? Some prior work warns of "model collapse" as the web is overwhelmed by synthetic data; other work suggests the problem can be contained by man- aging how available data used in pretrain- ing. We report experiments on three ways of using data erative model task-settings (multivariate Gaus- sian estimation, kernel density estimation, and language-men model fine-tuning) to further confirm the possibility of containment: (a) confirm that the training-workflow of replacing all real data by successive generations of purely synthetic data suffers model collapse; (b) we consider the training-workffor of accumulating synthetic data alongside real data and training on all data com- bined and confirm that, although the proportion of real data eventually becomes zero, models re- main stable and their test losses do not diverge under this training-workfloww (c) we consider a workfllow where real and synthetic data accumulate together but successive generations of pretraining are constrained to use fixed-sizz size data subsets each generation. In this forkflow, we observe slow and gradual rather than explo- sive degradation of test loss performance across generations. Our insights are important when forecasting whether future generative models will collapse or thrive, and our results open avenues for empirically and mathematically studying the context-dependent value synthetic data.
With each passing day, the internet contains more AI- generated content (Altman, 2024). What impact will this have on future deep generative models and their successors? Previous work forewarned that sequences of models trained exclusively on successive iterations of their own outputs can exhibit model collapse: model performance degrades with each iteration and later models become progressively more unusable (Shumailov al., 2023). Due to increasingly heavy usage of LMs produce written content for the inter- net (Bommasani et al., 2022; Reuel et al., 2024; Perrault & Clark, 2024; Kapoor et al., 2024), model collapse alarmists believe current trends, if unmitigated, will irredeemably pollute the pretraining data supply and degrade language models across time.
across three gen-
However, the model collapse literature contains variety of methodologies, assumptions, and models. Research sup- ports conflicting positions about such future scenarios (Taori & Hashimoto, 2023; Hataya et al., 2023; Martínez et al., 2023; Shumailov et al., 2023; Alemohammad et al., 2024; Martínez et al., 2023; Bohacek & Farid, 2023; Guo et al., 2024; Bertrand et al., 2024: Briesch al., 2023; Gillman et al., 2024; Wyllie et al., 2024; Dohmatob al., 2024a;b; Gerstgrasser et al., 2024; Seddik et al., 2024; Marchi et al., 2024; Padmakumar & He, 2024; Chen et al., 2024; Fer- bach et al., 2024a; Veprikov et al., 2024; Dey & Donoho, 2024). Concordance in the field is especially challenging because "model collapse" has been defined in various ways and studied under different assumptions concerning training- workflows and task-setrings. Shumailov al. (2023) defined model collapse a "degenerative process affecting genera- tions of learned generative models, in which the data they generate end up polluting the training set of the next gener- ation." " Dohmatob et al. (2024b) called model collapse the worsening of scaling law curves when training on synthetic as opposed to real data. In their theory sections, Shumailov et al. (2024) and Gerstgrasser et al. (2024) defined model collapse as divergent test loss after multiple iterations of training in certain task settings. Due to space constraints, we discuss Related Work more extensively in Appendix A.
'Equal contribution
Stanford Statistics "Stanford
Computer Science.
Correspondence to: Joshua
Kazdan <ikazdan @ stanforrdedde edu>, Rylan Scha-
effer
<rschacf@ec stanford.eddu>. <<anmm @ es. stanford.edus
Sanmi Koyejo
Proceedings of the 42dn International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
In this paper, we take model collapse by its literal mean-

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Collapse or Thrive? Perils and Promises of Synthetic Data in Self-Generating World
ing: that model performance deteriorates to the level of uselessness across iterations of an assumed data training- workflow under an assumed generative model task-setting, We bring new clarity to the discussion by studying how three different data training-workfloww shape trained model performance over multiple generations. The first training- workflow, initially studied by Shumailov et al. (2023), fully replaces previous data at each iteration with newly gener- ated synthetic data from the most recent generative model. In particular, under the replace training-workflow, real data are only used at the very first iteration. We conduct exper- iments in three different generative model task-settings from basic multivariate Gaussian modeling (MGM) to ker- nel density estimation (KDE) to supervised fine-tuning of language models (SFT) - and observe that in each setting, the replace madiing-wwrkffee
2. We characterize test-loss behavior in a middle-ground between the replace and accumulate workflows. This middle ground reflects the impact of limited training- compute in a data ecosystem that is being flooded with synthetic data.
We illustrate that when real data are in limited sup- ply, supplementing the training set by adding small amounts of synthetic data can improve test loss. On the other hand, when real data are abundant, supple- mentation with synthetic data only increases test loss.
2. Testing Two Model Collapse Claims in
Three New Generative Modeling Settings
induces collapse. We also consider the data training- workflow of accumulating all prior data both real and syntheric and find that on all three task-settings, collapse does not occur: the test loss on real data stays bounded at reasonable levels. Additionally, we introduce and study a more realistic "fixed compute" all real and synthetic data are retained, as in the accumulate setting. However, dataset is subsampled from the total available pool of data to train the next model. While the pool of data grows over time, the size of the subsampled training dataset remains constant regardless of the model iteration. This training-workllow reflects the reality that available compute might not keep pace with the rate of synthetic data gen- eration, foreing model providers to sample a fraction of future web-scale data for training. This training-workfll can be viewed as middle ground between the assumed fixed-saplee size replace training-workloww workflow and the increas- ing sample-size accumulate training- workflow. We find that test errors grow larger and more quickly than in the accumulate scenario, but more slowly than they grow in the replace scenario. These results are consistent across five different generative model task-settings. Third, we investi- gate whether the proportion cardinality of initial real data matters more for preventing model collapse and discover a non-trivial interaction between real and synthetic data: when real data are scarce, adding a small amount of syn- thetic data can reduce test loss, whereas when real data are ample, any synthetic data increases the test loss calculated on real data.
Gerstgrasser et al. (2024) recently made two claims:
1. Where model collapse has been documented, the col- lapse can be attributed to the replace workflow, where at each iteration, prior data (synthetic or real) are deleted en masse.
training-workflorr workflow. In this
2.
In the accumulate workflow where synthetic data accu- mulate alongside the original real data, model collapse is avoided.
If correct, these two claims suggest that model collapse is less likely to pose a significant threat to future deep genera- tive models since accumulating data over time is a more real- istic model of internet evolution. As a partner Andreessen- Horowitz elegantly explained, deleting data en masse is "not what is happening on the internet. We won't 't replace the Mona Lisa or Lord of the Rings with Al-generated data, but the classics will continue to be part of the training data set" (Appenzeller, 2024). Despite such plausible heuristics, the claims of Gerstgrasser et al. (2024) were not formally tested in three generative modeling task task-settings studied in influ- ential recent work (Shumailov et al., 2024): (1) Multivariate Gaussian Modeling (MGM); (2) Kernel Density Estimation (KDE); and (3) Supervised Fine-tuning of Language Models (SFT). In this section, we ask and answer: In these three new generative modeling settings, do Gerstgrasser et al. (2024) Claims I and 2 continue to hold? In all three settings, we find that the answer is yes.
To summarize our contributions:
2.1. Model Collapse in Multivariate Gaussian Modeling
1. We show that the hypotheses of Gerstgrasser et al. (2024) are more universally applicable than originally claimed: three generative model settings widely cited as evidence for the dangers of collapse under a replace maining-workflor (Shumailov et al., 2024) fail to col- lapse under the accumulate maining-wrkffee
Following Shumailov et al. (2023); Alemohammad et al. (2024); Bertrand et al. (2024), we study what happens when one iteratively fits multivariate Gaussians and samples from them. We begin with n real data drawn from Gaussian with mean y and covariance (()) x (") ..... xx(") r..i
N (( (()). For model fitting, we compute the unbiased
2

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Collapse Thrive? Perils and Promises of Synthetic Data in Self-Generating World
Replace
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Figure 1. Model Collapse in Multivariate Gaussian Modeling, Top: Previous work (Shumailov et al., 2023; Alemohammad al., 2024; Bertrand et al., 2024) proved model collapse occurs under the replace crainin--workfoon which iteratively fits means and covariances to data, deletes carlier data, and replaces it with samples from Gaussian with the fitted parameters (left). However, under the accumulate workflow where one doesn't delete data after each model-fitting iteration, model collapse does not occur (right). Note: We visualize the fit model-fitting iterations. Middle: If data replaced, then the
Gaussians zero-mean for easy comparison of the fit covariances fitted means drift away from the original data's mean, but if data instead accumulate, then the fitted means stabilize. Bottom: If data replaced, then the fitted covariances collapse compared the original data's covariance, but if past data not discarded, the fitted covariances stabilize quickly and collapse is averted.
mean and covariance of the most recent data:
Shumailov et al. (2024) proved that as 0o,
(1++1 a. Reelaaee
(I+1) def RReplace
(4)
(1(+1 E (W Replice)
N (n [()) + cc,
1=1
()++1) Replace
Σ (x plott
) - Alstt yr
where W2 denotes the Wasserstein-2 distance. This result simply states that the fit covariances will collapse to 0 and that the Wasserstein-2 distance will diverge as this model- data feedback loop unfolds. However, this result assumes that all data are replaced with new synthetic data after
For sampling, we draw synthetic data using the fit pa- rameters: x(!)
x())
(r) a.ddd N (P )
3

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Collapse Thrive? Perils and Promises of Synthetic Data in Self-Generating World
10
3 10?
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Datrect: Stctes Bell
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Figure 2. Model Collapse in Kernel Density Estimation. Left: We consider 4 standard datasets from sklearn: Blobs, Circles, Moons Swiss Roll. Center: For four datasets, deleting data en masse causes the negative log likelihoods (NLL) of held-out real data increase with each model-fitting iteration. Right: For all four datasets, the accumulate training-workllow workflow avoids diverging test loss on real data. Interestingly, for specific pairs datasets and numbers of samples per iteration, training on real and accumulated synthetic data can yield lower test-loss on held-out real data than would training real data alone.
each model-fitting iteration, which is likely unrealistic model of data evolution; rule we do not delete earlier content from the internet, replacing it masse with new model-generated content after fitting each state-of-the model. To study what happens if data instead accumulate across model-fittine fittiing iterations, we fit to all previous real and synthetic data, a mixture which the fraction of real data asymptotically approaches 0:
(+++1) Accumulate
(X
n(1 +
1=0
where x(11 def x
Accumulate is shorthand for the cen- tered datum. Data are then sampled using these fit Accu- mulate parameters rather than the fit Replace parameters.
(I+1)
Empirically, we find that replacing all data with new syn- thetic data after each model-fitting iteration causes model collapse (Fig. Left), whereas accumulating data across model-fitting iterations prevents model collapse (Fig. Right). More specifically, we find that data are deleted,
(++1) A Accumulate
1
n(t 1(t+1)
1=0 j=1

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Collapse or Thrive? Perils and Promises of Synthetic Data in Seff-Generating World
Cross Entropy on Real Dats. (TDast)
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Gemma 2 2B
Gemma 9B
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Figure 3. Model Collapse in Supervised Fine-tuning of Language Models. Fine-tuning Google's Gemma2 models on Nvidia's HelpSteer dataset demonstrates that model collapse occurs if previous data are replaced after each model- fitting iteration (left), whereas model collapse is avoided if new synthetic data instead accumulate with previous real and synthetic data (right).
the squared error between the fit mean ( and the initial mean (1)) diverges (Fig. 1, Middle Left), and the fit covari- Replace relative to the initial covariance
growing linearly n(t + 1):
()) collapses to 0 (Fig. 1, Bottom Left). In contrast, if data accumulate, the squared error between the fit mean and the initial mean plateaus quickly (Fig. 1, Middle Right), as does the fit co- variance relative to the initial covariance (Fig. 1, Bottom Right). For univariate Gaussians, we can mathematically characterize the limit distribution:
ance (5(2)
((++1) PReplace (x)
def
n
K
X
nh
j=1
(1+11 PAccumulate (x)
def
1
x
K
nh(1 + 1)
1= j=1
where K is the kernel function and h is its bandwidth param- eter. We consider standard Gaussian kernel. For sampling, at each iteration, we draw n new synthetic data points from the fitted KDEs. We evaluate the performance using the neg- ative log- likeliiiddood (NLL) real held-out test data; lower NLL indicates better performance. We use four standard synthetic datasets from sklearn (Pedregosa et al., 201 I): blobs, circles, moons, and swiss roll.
Theorem For notational efficiency, for a univariate Gaus- sian, let p(tt and g(t) denote Accemulaaee and {1) Then
Accummaate
(in(T//T) oB
IE [o,]
t100
π T/Vn
ob 1 sin(T/vn)
[[( 1o)
10000
t/ 4//n
Our results validate Claims 1 and of Gerstgrasser et al. (2024) in the KDE task-setting. (Fig. 2): replace causes rapid increase in NLL as iterations increase, indicating that the KDEs are becoming increasingly poor modeling the real data distribution. In contrast, under accumulate, the NLL of real test data remains relatively stable, demon- strating that this training- workflow avoids model collapse. Surprisingly, accumulating data can yield lower negative log likelihoods on held-out test data that decrease with ad- ditional model-fitiin iterations. While synthetic data valuable elsewhere e.g., Jain et al. (2024), Mobahi et al. (2020) their value here is intuitive: the synthetic data "fills in" the gaps between training data points and better approxi- mate the distribution. For the simplest 'out of the box' KDE, test losses may diverge (although very slowly) even under accumulate; but, as explained in the Appendices, with suffi- cient care in setting up the bandwidth selector of the KDE, accumulate test losses will not diverge. By contrast, replace test losses will diverge regardless of the bandwidth selector.
See Appendix B for the proof. This reveals two key dif- ferences when data accumulate: the covariance longer collapses, and the mean longer diverges. This also im- plies that the Wasserstein-2 distance no longer diverges. Altogether, if data accumulate, model collapse is mitigated.
2.2. Model Collapse in Kernel Density Estimation
Shumailov et al. (2024) introduced second generative modeling sett-ng kernel density estimation (KDE). We begin with n real data points drawn from an initial proba-
bility distribution p()); ( ..... x(()) ri..d. (()) We then iteratively fit KDEs to the data and sample new synthetic
data from these KDEs. In the replace training-workflow, we fit a KDE to n data from the most recently fit model, whereas in the accumulate training-workfloww workflow, we fit a KDE data from all previous iterations, with the available dataset
5

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Collapse Thrive? Perils and Promises of Synthetic Data in Self-Generating World
Multivariate Gaussian Modeling
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Figure 4. Model Collapse Under Fixed Compute Budget. We compare deleting data after each model-fitting iteration (replace) and accumulating data after each iteration (accumulate) with a new needccompute data paradigm accumulate- subslalpee-.aaapeee In accumulate- subsample, real and synthetic data accumulate but then subsampled so that each model is trained constant number of data. Accumulate-subsannple' test loss on real data deteriorates more quickly than accumulate's loss but more slowly than replace's 's loss, and frequently converges, albeit a higher plateau than accumulate. These results hold across five task-settings: multivariate Gaussian modeling, language model instruction finetuning, kernel density estimation, linear regression and language model pretraining.
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Collapse or Thrive? Perils and Promises of Synthetic Data in Self-Generating World
2.3. Model Collapse in Supervised Fine-tuning of Language Models
4. Cardinality of Real Data vs Proportion of Real Data in Mitigating Model Collapse
We now turn to the third setting for studying model col- lapse introduced by Shumailov et al. (2024): supervised fine-tuning of language models. Our chosen 'real data' is Nvidia's HelpSteer2 (Wang et al., 2024) instruction- following dataset; we will iteratively fine-tune a language model before sampling new text data from it. We chose Google's Gemma2 models (Team et al., 2024); they are rela- tively high performing and relatively small. For replace, we fine-tune the t-th language model only data generated by the (t- - 1) language model; for accumulate, we instead fine- tune the t-th model on the initial real data plus all synthetic data sampled from models 1, .,. 1 - 1. Thus, the amount of data for replace is constant ~ 12.5K, whereas the amount of data for accumulate grows linearly 12.5k -t. We again find that deleting data after each iteration leads to collapse whereas accumulating data avoids collapse (Fig. 3).
We conclude by turning to a key open question: Which matters more for avoiding model collapse: the cardinality of real data or the proportion of real data? Relatedly, how does the value of synthetic data for reducing test loss on real data depend on the amount of real data?
These questions are highly pertinent to researchers sampling from web-scale data in order pretrain or finetune language models. To explore these questions, we fine-tune Google's s Gemma 2 2B model using the HelpSteer2 dataset. We generate 100k completions from the fine-tuned model and filter out those exceeding 512 tokens, leaving 55,000 usable samples. We then construct datasets with varying mixes of real and synthetic data and fine-tune Gemma 2B on each. The final test loss for each configuration is recorded (Fig. 5).
This experiment provides several insights. First, both the number and proportion of real data have an impact on the test loss following SFT. To assess this, we first transformed the number of real datapoints n as wt/r, in keeping with intuitions from classical statistics on how the log likelihood scales with the number of data points. Then, based on observation of the data, we computed
3. Collapse Under a Fixed Compute Budget?
Thus far, we have focused on two training-workee workflows: re- place and accumulate. As discussed in Sec. 2, replace is unrealistic as a model of internet evolution: we cannot delete and regenerate the internet after pretraining each successive model. Accumulate might also be unrealistic because accu- mulate trains each successor model on ever-expanding data and increasing compute. For the sake of predicting likely outcomes for future generative models, we ask and answer: Does model collapse occur when data accumulate, while each model training uses a fixed compute budget?
real data
log real data + synthetic data
to best capture the relationship between the fraction of real data and the log likelihood. We measured I²² values of 0.59 for the transformed number of real data and 0.34 for the proportion of real data. We next computed F-statistics for the one-term versus two-term models involving each of these covariates, which gave us p-values of 6.9 10-25 and 4.6 x 10 25 respectively. These statisties suggest that both the proportion and the cardinality of real data have a statistically significant effect on the test loss, and explain a sizable fraction of the variance in the test loss.
We call this training-worff workflow secumulate-subsamme since data accumulate but at each model-fitiee fittiiiiin iteration are sub- sampled to fixed dataset size. We study the same three gen- erative modeling task-settings as above, plus two new task- settings from other prior work (Mobahi et al., 2020; Dohma- tob et al., 2024a; Gerstgrasser et al., 2024): linear regres- sion and pretraining language models on a GPT3.5/GPT4- generated dataset of kindergarten-level text (Eldan & Li, 2023).
Second, we find a difference in the effect of synthetic data on test loss comparing high- versus low- real-data regimes. In our experiments, when the number of real data is 1024 or lower, we find that there is a small but non-zero amount of synthetic data that improves the test loss when it is included. This optimal number of synthetic datapoints appears to be near 1024. This suggests that practitioners fine-tuning with insufficient amounts of real data should consider supple- menting with synthetic data to improve model quality. On the other hand, when real data are plentiful (more than 1024 datapoints), we find that more synthetic data almost always degrades final model quality when the number of real data is held constant. When there are more than 1024 real data- points, datasets containing only real data prove to be more valuable than datasets that contain ten times more real data mixed with synthetic data.
Across all five generative models, accumulate-subsane s test loss on real data lies between the test losses of Re- place and Accumulate (Fig. 4). Specifically, accunulate- subsample exhibits higher test loss than accumulate but lower test loss than replace, showing that the fixed compute budget imposes some performance penalty. In a qualitative difference, test losses on real data typically plateau for both retemulate-uubsamppe and accumulate. while test losses for replace typically diverge in an apparently unbounded manner. Thus, modifying accumulate to the more compute- realistic workflow accumulate-sale subsaappeee the threat of model collapse is still avoided.
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Collapse or Thrive? Perils and Promises of Synthetic Data in Self- Generaing World
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Figure 5. The Value of Synthetic Data in Supervised Fine-tuning of Language Models. Fine-tuning Google's Gemma 2 2B on Nvidia's HelpSteer 2 dataset on different combinations of real and synthetic data. We observe that when the number of real data is small, supplementing with synthetic data can improve test loss. With sufficient real data, adding synthetic data degrades performance.
These results raise interesting questions about the role of synthetic data in SFT that merit exploration. We can some- times achieve better results by removing all synthetic data from the training set than by doubling the amount of real data. When constructing datasets subject to cost constraints, these results suggest that removing synthetic or low-quality data can sometimes bring more value than collecting greater volumes of high-quality data. These experimental results bear comparison with prior theoretical work of Dohmatob et al. (2024b), cf. Corollary 3.3 et seq.
an optimal amount of synthetic data that should be added.
To us, the most realistic viewpoint on internet evolution assumes synthetic data accumulating from host of models alongside continuing influx of real-world data. Combining it with our experimental and analytical work, model col- lapse seems unlikely. Our experiments take a pessimistic viewpoint, in the sense that our experiments pay no attention to the quality of data, whereas in practice, engineers heavily filter data based on various indicators of data quality, e.g., (Brown et al., 2020; Lee et al., 2023; Wettig et al., 2024; Penedo et al., 2024; Li et al., 2024b; Sachdeva et al., 2024); for a recent review, see Albalak et al. (2024).
5. Discussion
We studied the risks of model collapse and identified two pathways to containment. We demonstrated in three new generative modeling settings that accumulating data over time avoids model collapse, whereas replacing data over time induces model collapse. In five generative modeling settings, we demonstrated that using a modified accumulate workflow- where each model trains on a fixed-size sample drawn from all available real and synthetic data model performance deteriorates, but still tends to plateau. The con- sistency of results across model types and datasets suggests that this distinction is a general phenomenon, and is not specific to any particular model, dataset, or learning algo- rithm. Lastly, we explored the value of synthetic data for reducing the test loss on real data and found two different regimes: when real data are plentiful, adding synthetic data can be harmful, but when real data are scarce, there exists
A promising future direction might combine synthetic data generation with filtering techniques to enable performant and efficient pretraining at scale. As we saw in kernel den- sity estimation (Fig. 2) and in language model pretraining TinyStories (Fig. 4), training on real and synthetic data accumulated over several iterations can yield lower loss on real test data than training on real data alone. Identifying un- der what conditions synthetic data can lower test loss during pretraining would be invaluable to industry practitioners.
Our results in Section 4 suggest that removing low-quality synthetic data from model training sets can improve test loss more than gathering additional high-quality data. De- veloping efficient identification and removal techniques for detrimental data could streamline the model fine-tuning pro- cess and produce better alignment.
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Collapse Thrive? Perils and Promises of Synthetic Data in Self-Generating World
Acknowledgements
Redacted for double blind reviewer.
Impact Statement
This paper advances the understanding of model collapse, critical issue in the era of synthetic data proliferation. By systematically analyzing different data evolution paradigms, our findings provide actionable insights into mitigating the risks associated with training generative models on recur- sively generated data.
The societal implications of our work are substantial. If left unaddressed, model collapse could lead to the degra- dation of Al-generated content, reducing the reliability of machine learning models deployed across industries, includ- ing healthcare, finance, and education. Our results highlight that while synthetic data can be beneficial under controlled accumulation, indiscriminate use may lead severe model degradation. This has ethical ramifications for Al deploy- ment, biased deteriorating models could reinforce misinformation, amplify biases, or reduce transparency in decision-making systems.
Additionally, our study underscores the importance re- sponsible data curation and compute-awar training strate- gies. As AI systems become increasingly self-referential, policymakers and practitioners must develop strategies to balance synthetic and real data, ensuring sustainable model performance. Our work informs these discussions by demonstrating that synthetic data, if managed appropriately, can enhance rather than harm AI models.
While our findings primarily contribute to advancing ma- chine learning theory, encourage continued research into the ethical governance of Al-generated datasets, particularly as reliance on synthetic data expands. Future work should explore mechanisms for filtering low-quality synthetic data and designing robust training paradigms to prevent long- term degradation AI capabilities.

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[Online; Collapse or Thrive? Perils and Promises of Synthetic Data in Self-Generating World
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A. Related Work
The limitations of using Al-generated images to train other image models have been well-documented since 2022 (Hataya al., 2023). Shumailov et al. (2023) most prominently sounded the alarm about synthetic data for training language models by showing that model trained repeatedly on its own outputs exhibits severely denigrated quality. Shumailov's 's theory and empirical work were quickly extended to many new settings (Alemohammad et al., 2024; Bertrand et al., 2024; Dohmatob et al., 2024b;a; Marchi et al., 2024).
Within the model collapse literature, a variety of data dynamics have been studied, which vary in how "real" data are discarded or retained, how "synthetic" data are generated, and how each is (or is not) incorporated into future training sets (Martínez et al., 2023; Mobahi et al., 2020; Dohmatob al., 2024a). common feature of many of these that least some real data are discarded, often because total dataset size is kept constant across model-fitting iterations. However, (Gerstgrasser al., 2024) note that this may not represent real-world dynamics, and that model collapse avoided when data accumulate. What is not clear, however, is whether this claim holds universally, including in the specific settings studied in other prior work. We help close this gap by extending Gerstgrasser's empirical and theoretical analyses to several of these settings.
Where model collapse can be seen as studying a worst-case scenario, it has also been observed that some kinds of synthetic data have a positive effect. (Dohmatob al., 2024b) and (Jain et al., 2024) find that certain amounts of synthetic data can improve model performance, and (Ferbach et al., 2024b) suggest that with curation, self- self-consuming loops can improve alignment with human preferences. A growing literature how to filter and hamness synthetic data has achieved impressive results variety of benchmarks (Zelikman et al., 2024; Li et al., 2024a; Yang al., 2024), raising interesting questions about the limits of when unfiltered synthetic data can help. In this vein, we answer question posed by (Gerstgrasser et al., 2024): does the proportion or the raw amount of real data in a mixed training set have greater impact on test loss? In the process, we find that proportionally small amounts of synthetic data can improve test loss when real data is scarce.
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Collapse Thrive? Perils and Promises of Synthetic Data in Self-Generating World
B. Iterative Gaussian Model Fitting: Mathematical Results and Proofs
B.1. Setup
Lemma 2. Using the notation of Theorem 1, we can express Pt Er= 0r-1 _ + Po.
Proof. Note that Xi, Me-1 1 + 12,1 where Zizt ~ (0, 1). Therefore,
#4
X
Tt
P4-1
21
P
0--1
Mt-1 0(-1
Therefore, be Ei=1 Tr-1 + no.
Lemma 3. Under the setup described in Theorem 1, E
(1
10000
sin(s///m)
nk
4/ v/
Proof. Using the recursive expression for pr in Lemma 2, we can rewrite
X ir mx)
(Xir Xx xx, +x,-1a1)
(Xir X²)² + +n(x, - 11)
= Σ ( + (Mr-1 + - - -11)2)
In the last line, define 83 Di=1 2x The term
2
2k
(14-1 + 0--17--111-
1
0--
k:
k=r
So
21
Zk
9 =
0 p-
S2 +
0--15p
ok
h
1=1
k=r
tot
8 +
1-11
1
1
Σ
0k-1
k
kmm+1
We now compute the conditional expectations of the terms in this sum. Where F; denotes the ith filtration,
52
0
r r<t t
0-
n-1
r r=t.
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Collapse Thrive? Perils and Promises of Synthetic Data in Seff-Generating World
For r we find that
1 (1 (1-
(1
1
E
Σ
0k-1
Fi =
o (1-
k
t
k=
On the other hand, when r < b
E
12-1
(1-
Σ
2k
0
2x 1F
k=++1
0-1
2
1
1 +
0,-1
- -
Σ
0h-1
f²
k-++1
Therefore,
E ttoo = (t 1)o2-1 + +0-11
(1-
+ o
t
+
+
(1-7)²
t
n
1
2
= 0
1 - 1 +
+
n
tn
t2
tn
2n
= 1/-11
t
tn
follows that
F--1l = oi-1 (1- 12 (-12)) < o
for all t. Thus, { a supermartingale, and
0.s. 0²
because of is bounded below by 0. Therefore, we still have convergence. Next, letting = B[o,], we have
m4 m4-1 (1 (1- 1
(1- 12
1.2n
SO
(3)
kmn
By theorem of Fuler, this is equal to
sin (T TD)
00 π /n
(4)
Observe that by performing variable replacement and using L'Hospital's 's rule, it is clear that lim,-+o0 E[o?] = =0.-
Finally, we are able to compute E[4 - )1 Corollary 4. The expected error in the mean
El(4 1o0 od
1
II kml
1
k2n (
)
(5)
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Collapse Thrive? Perils and Promises of Synthetic Data in Self-Generating World
Proof. Using the recursion from Lemma and the expression for the variance in Lemma 3, we can rewrite
E
Elo
[i
1n)
n
k=
1
0
22n f=1
pn
1
II 8=1
(2n
{2) )
1
Therefore,
lim Bl( -mo)²] o/ (1 (1- /in(TT/V vn. t10
/ a/Vn
C. Iterative KDE Fitting: Mathematical Results and Proofs
In this section, prove that the NLL diverges when iteratively fitting KDE's regardless of whether accumulates or replaces data from previous iterations.
Theorem In the replace setting described in Section 2.2, as long as one holds the bandwidth constant, the NLL asymptotically diverges.
Proof. Define fo as the density function for the data distribution from which the original data I1, ..., Tn are sampled. Define K₁ to the Gaussian kernel function with fixed bandwidth h. One can rewrite the fitted distribution at iteration as
Dj Kn D:-1
where * denotes the standard convolution of densities.
By simple recursion, it is clear that D: K Do- When Gaussian kernels with bandwidths a and b are convolved, basic calculation shows that the resulting effective bandwidth is v + 62, Consequently, by inductive argument, the effective bandwidth of K is htT.. Therefore,
lim Kt Do lim Kav? Do 8-000
t-000
because as the bandwidth goes to oo, the likelihood of any point goes to 0. Hence, regardless of the choice of test data, the negative log likelihood diverges to -o.
The same conclusion holds when one accumulates rather than subsampling data:
Theorem 6. For any non-trivial kernel (i.e. kernel whose Fourier transform is not 1), 2.2, the NLL diverges.
Proof. We adopt the same notation as Theorem 5, except this time denotes general kernel K that doesn't 't necessarily need be Gaussian. In this instance, it more convenient work in frequency space, where convolution in probability space corresponds to multiplication.
Define so the Fourier transform (FT) of fo. also called the characteristic function. Let denote the FT of K. Then
6t k P
16

---

Collapse Thrive? Perils and Promises of Synthetic Data in Self-Generating World
where denotes standard complex multipicattion Define ó, de a, 1 Ei= di Using this notation,
that AP: ò₃ PP- Define d, 32/50, and let
d0
de R. ac-1
(6)
at ((t 1)0r-1 + de) /t.
We that 04 L4,x (a+-1) is affine map with slope ((-1)) 1) k)/t and intereept O. Suppose that the characteristic function of the density converges to Poc. Then the map d, has fixed point, As long as k of 1, this fixed point must satisfy the equation
X ((t 1) + k)Lp
= 0 = (t - 1) + x)/g 1
= 0 = (-1 + x) + >
O.
Note that if PPo O, its inverse FT is a function that has 0 probability density everywhere in probability space. Equivalently, the variance of fe diverges to co.
Although the NI.. eventually diverges in the accumulate case, is clear from the expression for az that this divergence occurs very slowly.
For Gaussian kernel, both the replace and accumulate case offer an interesting shared insight. Throughout the iterative fitting process, regardless of whether accumulate replace, the bandwidth monotonically grows. Therefore, when one starts this process with very small bandwidth smaller than the optimal bandwidth for the density being fit, one could initially observe a decrease in the negative log likelihood as the bandwidth approaches its optimum.
Finally, model collapse, while inevitable with a fixed bandwidth, can be avoided in all cases by shrinking the bandwidth at a sufficiently fast rate. Since practitioners typically optimize their bandwidth according to the amount of the data that they have, the bandwidth should have the form c(In))15 where e a constant. In this setting, model collapse is avoided entirely.
Theorem 7. Under the accumulate training workflow consider density estimation as in Section 2.2 with a Gaussian kernel. Let the bandwidth at the tth model-fitting iteration be e (tn)-1/5 for a constant c. Then the asymptotic variance of the limiting KDE finite.
Proof. Let K K/(tn)-1/o denote the kernel at the ith model-fitting iteration. Let fo denote the original distribution, and define ft to be the distribution of the KDE at the tth iteration.
We write
fc = 8 i=1
Σ fi-1 * Kottn)-1/5
(1- 1)
1 + 7, h/ * Ke(
h-1 * K K((m)--1 Σ
1=1
!)
1
fi-1 + t fi- * K 1/((n)-1/3
Ko + - -K(en))
where Ko is the identity kernel, or equivalently the Gaussian kernel with bandwidth. Therefore, we find that
ft fo f ((1-1) Ko + K Kotin) 118
17

---

00. Collapse Thrive? Perils and Promises of Synthetic Data in Self-Generating World
Define W; to be random variable that is K -1ss with probability and Ko with probability 1 1.4 We rewrite X₁, random variable drawn at the ith fitting iteration as
X, Xo
W.
All of Xo. W1, -. W; are independent. The variance is given by
Var(X,) = Var(XXo +
W;)
1=1
-Va((Xo) +
(in)2/5
i=1
-Var(Xo) + 22/
$7/5"
Ast + 00,
Var( - VVr(Xo) +
<
2//
14
t=1
Therefore, when the kernel size is appropriately adjusted, the variance of the KDE under accumulate converges.
18

---

Collapse Thrive? Perils and Promises of Synthetic Data in Self-Generating World
D. Experimental Results: Sweep Configurations
D.I. Model Collapse in Multivariate Gaussian Modeling
To study model collapse in multivariate Gaussian modeling, we ran the following YAML sweep:
program: src/ fet_gauuusisn fit fit.gaussianss.
project: rerevisiting -model -oodel-collapse- -fitt -gaussians
method: grid
parameters:
data dim:
values: 1. 3, 10, 31, 100 ]
values: [10, 32, 100, 316, 1000]
aum_iterations:
values: 100
seed:
values: 00, 1,
2. 3. 4, 5,
6.
7. 8, 9,
10,
11.
12.
13. 14,
15, 16, 17, 18, 19, 20,
21,
22,
23.
24, 25,
26,
27.
28
29,
30, 31, 32, 33, 34, 35,
36,
37,
38,
39,
40,
41
42
43, 44,
45. 46. 47. 48, 49, 50.
51.
52.
53.
54,
55.
56.
57.
58.
59,
60. 61. 62. 63,
64. 65.
66.
67.
68,
69,
70.
71.
72.
73.
74.
75. 76. 77. 78. 79, 80,
81.
82,
83,
84,
85,
86,
87.
88, 89,
90, 91, 92, 93, 94, 95, 96,
97,
98, 99
setting:
values: I "Accumulate" Accumulate Subsamplee"" "Replace".
digma.squaree
values:
1.0,
Seeds were swept from 0 to 99, inclusive.
D.2. Model Collapse in Kernel Density Estimation
To study model collapse in kernel density estimation, we ran the following YAML sweep:
program src/ fit_kde/s fit_kdes py
project: rerevisiting -oodel- -fit -ddes
method: grid
parameters:
data_config:
parameters:
dataset_name:
values: b'blobs"]
lataset_kwarg
parameters:
n.features:
values: [2]
kernel:
values: [" ["gaussian")
kernel.bandwidth
values: 10.1, 0.5, 1.0]
num uum-samples per iterrtiit::n:
values: [10, 32, 100, 316, 1000]
num.iterations:
values: 100
19

---

Collapse Thrive? Perils and Promises of Synthetic Data in Self-Generating World
seed:
values: 0, 1, 2,
3,
4,
5,
6,
7, 8, 9,
10,
11,
12,
13, 14,
15. 16, 17, 18, 19, 20,
21.
22.
23. 24, 25.
26.
27,
28, 29,
30, 31, 32, 33, 34. 35,
36,
37,
38, 39, 40,
41,
42, 43, 44.
45. 46, 47, 48, 49, 50,
51,
52,
53, 54, 55,
56,
57, 58, 59,
60, 61, 62, 63. 64, 65,
66,
67, 68, 69, 70,
71,
72,
73, 74,
75. 76, 77, 78, 79, 80,
81,
82, 83, 84, 85, 86, 87, 88, 89,
90, 91, 92, 93, 94, 95, 96. 97, 98, 99
setting:
values: " "Accumulate",
"Replace",
program: src/ fit kdes fit kdes py
project: rerevisiting coldee-o fit -kde method: grid
parameters:
data config:
parameters:
dataset-name:
values: [" ("circles"]
dataset.kwargs:
parameters:
noise:
values: [0.05]
kernel:
values: [" ["gaussian") "]
kernel.bandwidth
values: [0.1, 0.5, 1.0] num_samples pelle_per_iteration
values: [10, 32, 100, 316. 1000]
num iiterrtttoon:
values: 100
seed:
values: [ 0, 1, 2. 3, 4, 5,
6,
7, 8, 9, 10,
11.
12.
13, 14,
15, 16, 17, 18, 19, 20, 21,
22,
23, 24, 25, 26,
27,
28, 29.
30, 31, 32, 33, 34, 35, 36,
37, 38, 39, 40. 41,
42.
43, 44.
45. 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56,
57, 58, 59,
60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71. 72, 73, 74,
75, 76, 77, 78. 79, 80, 81. 82, 83, 84, 85, 86, 87, 88, 89.
90, 91, 92, 93, 94, 95, 96, 97, 98, 99 setting:
values: "Accumulate",
" A'ccuuulate- -ecillatt-Suubaaee Subsample", "Replace",
program: src/ fit kdes / fit fiettkkes... py
project: rerevisiting -codel- - fit kdes method: grid parameters:
20

---

Collapse Thrive? Perils and Promises of Synthetic Data in Self-Generating World
data_config:
parameters:
dataset_name:
values: ["moons"]
dataset.kwargs:
parameters:
noise:
values: [0.05]
kernel:
values: ["gaussian"] "J
kernel bandwidth:
values: [0.1. 0.5, 1.0]
values: [10, 32, 100, 316, 1000]
num iterations:
values: 100
seed:
values: [ 0, 1, 2, 3, 4, 5,
6, 7, 8, 9. 10, 11,
12, 13, 14,
15. 16, 17, 18, 19, 20,
21.
22, 23, 24, 25. 26.
27,
28, 29,
30, 31, 32, 33, 34, 35,
36,
37,
38, 39, 40, 41, 42, 43, 44.
45. 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59,
60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74,
75, 76, 77, 78, 79, 80,
81, 82, 83, 84, 85, 86, 87, 88, 89,
90, 91, 92, 93, 94, 95, 96. 97, 98, 99 setting:
values: " "Accumulate",
" AAcumulate-Suuaampee "Replace".
program:
src/ fitiikkee///ee fites/fiitkkees py
project: rerevisiting -oodel -ddes
method: grid
parameters:
data_config:
parameters:
dataset name:
values: [" ["swiss_roll")
dataset.kwargs:
parameters:
noise:
values: [0.05]
kernel:
values: ["gaussian"]
kernel.bandwidte
values: [0.1, 0.5, 1.0]
values: [10, 32, 100, 316, 1000]
num iterations:
values: [ 100 ]
seed:
values: [ 0, 1, 2. 3, 4, 5, 6,
7.
8,
9,
10,
11,
12.
13, 14.
15, 16, 17, 18, 19, 20, 21, 22,
23,
24,
25,
26,
27
28,
29.
21

---

Collapse Thrive? Perils and Promises of Synthetic Data in Self-Generating World
30, 31,
32,
33,
34,
35,
36, 37.
38, 39, 40, 41,
42,
43, 44,
45, 46,
47,
48
49, 50,
51, 52, 53, 54, 55, 56,
57,
58, 59,
60. 61. 62. 63. 64. 65, 66, 67, 68, 69, 70, 71. 72, 73, 74,
75, 76, 77. 78, 79. 80, 81, 82, 83, 84, 85, 86, 87, 88, 89.
90, 91, 92, 93, 94, 95, 96, 97, 98, 99
setting:
values: " "Accumulate". "Accumulate -Subsample",
"Replace",
Seeds were swept from 0 to 99, inclusive.
D.3. Model Collapse in Linear Regression
To study model collapse in linear regression, we ran the following YAML sweep:
program: src/ fit linear cetieaarrrereesioonss aii.niiaaiirrraaat.aa.aaaa reggeessirrgrssoons py
project: rerevisiting -model -in method: grid
parameters:
data_dim:
values: [ 100, 10, 31, 3,
values: [10, 32, 100, 316, 1000]
num_itrrations:
values: 100 ]
seed:
values: [ 0, 1, 2. 3, 4, 5,
6.
7. 8 9, 10,
11,
12.
13, 14.
15. 16, 17, 18, 19, 20, 21,
22
23, 24, 25,
26
27
28
29.
30, 31, 32, 33, 34, 35, 36,
37,
38, 39, 40.
41.
42,
43,
44
45, 46, 47, 48, 49, 50, 51.
52,
53, 54, 55, 56,
57,
58, 59.
60, 61. 62, 63, 64, 65, 66.
67,
68, 69, 70,
71,
72.
73, 74.
75. 76, 77, 78, 79, 80, 81,
82,
83, 84, 85, 86,
88, 89.
90, 91, 92, 93, 94, 95, 96, 97, 98, 99
setting: values: "Accumulate",
" Secumullae-Suusamppll "Replace",
sig sigma.squared:
values: [ ]
0.1, 1.0, 10.
Seeds were swept from to 99, inclusive. Note: We ran this sweep as separate sweeps; to understand why, see this GitHub issue.
22

---

Collapse Thrive? Perils and Promises of Synthetic Data in Self-Generating World
E. Additional Experimental Results for Model Collapse Hyperparameters
Due to space limitations in the main text, we can oftentimes only present a subset of runs corresponding to a subset of hyperparameters. We present additional figures with wide range of hyperparameters here for completeness.
E.1. Additional Results for Model Collapse in Linear Regression
Replace
Accumuite-Suheee Subsemale
Accumultte
Num. Samples per Ineration
Moun Squared Rive (Test)
10"
Laneu Regenoscoo
32
10-1
1001
10-9
316
1000
0-3
(11
25
73
50
100
25
73
Model-Fitting Fitting Iteration
Fitting Ineration
Model Mode--Fating Iteration
Figure Linear Regression Dimension and variance o² 0.10.
Replace
Secumbate-Sbeeappll
Accumulate
Num. Samples per Toeration
Mean Sqarral Brro (Trot)
101
Lineat Regenescoon
10-1
1000
10
10
25
50
75
100
25
50
100
25
50
75
100
Fitting Iteration
Tittng
Model Fitting Iteration
Figure 7. Linear Regression for Data Dimension d = and variance o² 1.00.
Replace
Accummiete Subsample
Accummiate
Num. Sampless per Ineration
Moan Squared Ervor (Test)
101
10
Liuesc Reggn
10P
100
1
316
1003
-0-
10-3
25
75
50
100
25
73
Madel-FFtting Fitting Iteration
Madel Madel-Uisting Ineration
Model Fitting Iteration
Figure Linear Regression Dimension and variance a² 10.0.
23

---

Collapse Thrive? Perils and Promises of Synthetic Data in Self-Generating World
Reploco
Acrummistt
Num. Samples per Ieeration
Moan Squardi Ervo (Thot)
10
10
10"
Lizesr Rogroee Mouroookk
32
1001
[0-1
316
0-2
1000
01-3
25
75
100
50
100
25
25
Modol-Fatinng Iteration
Moddl-Fisting Iteration
Model-Fisting Iteration
Figure 9. Linear Regression for Data Dimension and variance o² 0.10.
Reploce
Aceummlete Subammale
Sentmmlte-Sabeengle
Accummiste
Num. Samples per Ireration
Mren Sqpuerd Fovor (Tost)
10
10
Lizosr Regressioa
32
100
10"
316
10.1
1000
25
75
50
100
25
75
Model-FFin Fitting Iteration
Model-Fistine Fisting Iteration
Model-Fisting Iteration
Figure 10. Linear Regression for Data Dimension d and variance o² 1.00.
Reploce
Aceummltte
Sentmmlte-akeennlle Subsample
Accummiate
Num. Sumples per Toeration
Mour Square Ereor (Teat
10
10/
Licroa Regrosson
32
100
10"
316
10
1000
-0-
25
75
100
50
100
25
75
Pitting Iteration
Modd-FFiting Treration
Model-Fisting Iteration
Figure 11. Linear Regression for Data Dimension d and variance o² 10.0.
Replace
Accumulate Scuumiate-ubssample
Accumniate
Num. Samples per Toeration
Mran Squand Error (TToat
Lauou Regression
100
316
0-11
1000
10-2
10
25
75
100
50
100
25 Model-Fisting Iteration
75
Modal-Patting
Iteration
Model-Fistnng Iteration
Figure 12, Linear Regression for Data Dimension and variance σ² 0.10.
24

---

Collapse Thrive? Perils and Promises of Synthetic Data in Self-Generating World
Replore
Acrummlate
Num. Somples per Ieeration
Moin Squared Eiro (Thot)
32
Lizesr Regrookkn
100
316
10"
1003
25
75
100
50
100
25
25
Modol-Fittnng Iteration
Model-Fisting Iteration
Model-Fisting Iteration
Figure 13. Linear Regression for Data Dimension 10 and variance o² 1.00.
Reploce
Aceumulte Subsmmale SentmbiteSabeenple
Accummiate
Num. Samples per Ieeration
Mren Sqqurre Feror (Test)
Lizesr Regresstaa
100
316
10"
1003
0-1
10-2
25
75
50
100
25
75
Model-Fiting
Iteration
Moddl-Fisting Iteration
Moddl-Fisting Iteration
Figure 14. Linear Regression for Data Dimension 10 and variance 0² 10.0.
Repluce
Aceummlate SentmnlateSteeunlle
Accummlate
Num. Sumples per Toeration
Mour Squard Ereor (Teat)
100
10"
Licra Regresson
316
1000
10-9
25
75
100
50
100
25
75
Mittii Iteration
Modd-FFistng Ireration
Model-Fisting Iteration
Figure 15. Linear Regression Dimension 32 and variance o² 0.10.
Replace
Accumultte Sccumbate-aussample
Accumniate
Num. Sumples per Tooration
Mran Sqpand Ertor (Thoat
100
10
Lauou Regression
316
1000
0-1
25
75
100
50
100
25
75
Model-Fatting Iteration
Model-Fisting Iteration
Model-Fisting Iteration
Figure 16. Linear Regression for Data Dimension = 32 and variance o² 1.00.
25

---

Collapse Thrive? Perils and Promises of Synthetic Data in Self-Generating World
Replore
SeurmllteSSbsampa
Aorummlate
Num. Somples per Ieeration
Moin Squarei Eirv (That)
100
10
Lizesr Rogroent Regroockn
316
1000
25
75
100
50
100
25 Mlste-Fisting Iteration
25
Modol-Fattnng Iteration
Model-Fisting Iteration
Figure 17. Linear Regression for Data Dimension 32 and variance o² 10.0.
Reploce
Acoumilate Sennanitee-abssnnle
Accummiate
Num. Samples per Iteration
Mren Sqpurrd Fvvor (Test)
316
Lizssr Begressia
1000
10a
10
25
75
50
100
25
75
Model-Fittng Iteration
Model-Fisting Iteration
Model-Fisting Iteration
Figure 18. Linear Regression for Data Dimension 100 and variance 0² 0.10.
Reploce
Accummiate
Num. Sumples per Toeration
Mour Square Ereor (TTat)
316
Licss: Regresson
1008
10/
10-1
25
75
100
50
100
25
75
Pittiig Iteration
Modd-FFittng Treration
Model-Fisting Iteration
Figure Linear Regression Data Dimension 100 variance σ² 1.00.
Replace
Acoummlete-Saeeet Subsample
Accumnlate
Num. Samples per Toeration
Mran Squand Ertor (Thoat
316
Lacsu Regression
1000
10a
25
75
100
50
100
25
75
Model-Fitting Iteration
Moddl-Fistnng Iteration
Model-Fisting Iteration
Figure 20. Lincar Regression for Data Dimension 100 variance o² 10.0.
26


Reviews:
[{"review_id": "CqETeppvtc", "reviewer": "Reviewer_NdVu", "summary": "The manuscript studies three ways of using synthetic data, both empirically and theoretically. \n\nThe manuscript starts by examining Gerstgrasser et al. 2024's two claims with the proposed three generative modeling settings in section 2, where some well-established settings and tasks are directly used, making the exact contribution of sec 2 unclear. \n\nSection 3 considers a slightly different setting, namely training with a fixed compute budget, and then studies the value of synthetic data for reducing test loss on real data depending on the amount of real data.", "questions": "", "limitations": "", "rating": 1, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "While we appreciate your review, you are mistaken about your claim that we primarily replicated experiments from other works, and the questions that you asked were largely answered in the paper. To explain line-by-line.\n\n> It seems like the manuscript just reproduces these experiments [from Bertrand, Alemohammad, and Shumailov] in a unified setting with some take messages.\n\nThis is false.\n\nShumailov: Shumailov proves that model collapse occurs for KDEs and Gaussian fitting in the REPLACE SETTING.  A main point of our paper is that if you repeat these experiments IN THE ACCUMULATE SETTING, then collapse does not occur.  We are the first to do this, and thereby directly compare and contrast the replace and accumulate settings.\n\nAlemohammad: This paper again tests model collapse for generative image models in the replace setting.  We don’t test generative image models at all in this paper, and we don’t use the replace setting.\n\nBertrand: Bertrand tests model collapse for language models in a variation of the accumulate setting.  In this case, the proportion of real data is fixed but non-zero throughout all iterations.  We allow the fraction of real data to go to 0 in our version of the accumulate setting, which marks a significant conceptual difference from Bertrand’s setup.  \n\n**All experiments in this paper are original, except for the pretraining experiment, which we extend from Gerstgrasser et. al. to include more model-fitting iterations.**\n\nOverall, our Section 2 makes a novel and important contribution by identifying seemingly contradictory claims in the model collapse literature and comparing them in a head-to-head manner to resolve them.\n\n>The evidence in Sec 4 is not very rigorous to answer the key open question mentioned in the first paragraph of Sec 4.\n\nWe provide an F-test of whether the proportion or the cardinality of real data dictates model performance.  This is a standard and fully rigorous statistical evaluation. The high statistical significance (p-values of $6.9e-25$ and $4.6e-25$ strongly supports our conclusions. If you disagree, please let us know why and what you would recommend instead.\n\n> It is unclear how to get the value of $1,024$ to support the claim that \"when the number of real data is 1024 or lower, we find that there is a small but non-zero amount of synthetic data that improves the test loss when it is included.\"\n\nThis value is our result, shown in Figure 5, where the curves for real data samples ≤1024 show improvement with synthetic data addition, while curves for samples >1024 do not show this benefit. We are not making a theoretical claim about the number 1024; we are reporting the empirical result we observed across multiple experimental runs.\n\nWe believe these clarifications help address the reviewer's concerns and demonstrate both the novelty and rigor of our work. Additionally, the review does not discuss any strengths of the paper, and you left most of your review sections blank.  We hope that you can provide more information about why you gave such a low score and also provide specific constructive criticism for how we can improve the manuscript.", "claims_and_evidence": "It is hard to distinguish between the authors' contributions and the existing papers. E.g.,\n* The manuscript argues that \"Following Shumailov et al. (2023); Alemohammad et al. (2024); Bertrand et al. (2024), we study what happens when one iteratively fits multivariate Gaussians and samples from ...\" and \"Shumailov et al. (2024) proved that as t → ∞\". It seems like the manuscript just reproduces these experiments in a unified setting with some take messages. Similar comments can be applied to Sec 2.2 and Sec 2.3.\n* The evidence in Sec 4 is not very rigorous to answer the key open question mentioned in the first paragraph of Sec 4. E.g., \n    * it is unclear why we can compute the $\\log (real / (real + synthetic))$ to best capture the relationship between the fraction of real data and the log-likelihood.\n    * it is unclear how to get the value of $1,024$ to support the claim that \"when the number of real data is 1024 or lower, we find that there is a small but non-zero amount of synthetic data that improves the test loss when it is included.\"", "methods_and_evaluation_criteria": "Looks ok. The main concern here is that evaluations and settings in Sec 2 reuse the existing work.", "theoretical_claims": "The reviewer checks the theorem in the main text and does not spot the issues.", "experimental_designs_or_analyses": "Yes. See comments above.", "relation_to_broader_scientific_literature": "NA", "essential_references_not_discussed": "NA", "other_strengths_and_weaknesses": "## update after rebuttal\nStrengths:\n- The paper is well-structured. It considers problems like Multi-variate Gaussian Modeling, Kernel Density Estimation, Linear Regression, and Language Modeling.\n\nWeaknesses:\n- The reviewer acknowledges the additional experiments provided by the manuscript. However, the reviewer does not think it has made sufficient contributions to make it above the bar of ICML acceptance.\n- The reviewer believes in adding extensive results on various configurations/settings of recursive training image diffusion models and language autoregressive models. For example, settings include but are not limited to different numbers of training samples in the original real dataset, the different ratios between real and synthetic datasets, and different types of real-world datasets.", "other_comments_or_suggestions": "## update after rebuttal\nIt would be great if the authors could include more numerical results. \nFor example, in section 4, fine-tuning different base models on various real datasets with different dataset sizes would be very helpful.\n- The training hyper-parameters used in the experiments are unclear. It would be great if the authors could report these and provide a comprehensive ablation study.", "questions_for_authors": "See comments above.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "yhOTWeyQMF", "reviewer": "Reviewer_Y9pa", "summary": "The paper investigates model collapse (a phenomenon observed when generative models are trained with output generated from such models) in 3 different settings (replace, accumulate, and accumulate-subsample). They find that model collapse occurs in the replace and the accumulate-subsample setting (but slower). They do not observe model collapse in the last setting (accumulate). Additionally, they investigate how the portion of real and synthetic data in a dataset influences model performance in a supervised fine-tuning regime.\n\n## Update after rebuttal: ##\n\nI would like to thank the authors for their response. However, my concerns about novelty have not been lifted and some of my questions have not been addressed. Therefore, I will keep my initial score.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "### On Novelty\n\n> My biggest concern with the paper is its novelty. It builds heavily on Gerstgrasser et al. and section 2 (a large part of the paper) only provides new results for different kinds of generative models with equal conclusions.\n\nWe disagree with this characterization. Our Section 2 identifies and resolves a fundamental tension in the literature where competing papers have reached contradictory conclusions about model collapse. This systematic evaluation across multiple generative settings provides valuable scientific evidence about the generality of these phenomena.\n\nMoreover, we make several fundamental contributions:\n1. We discovered that in KDE settings (Figure 2), certain combinations of real and synthetic data actually outperform models trained on real data alone - a counterintuitive result not previously reported for KDEs. KDEs are appealing because they are a widely used statistical model that are straightforward to experiment with and analytically tractable to study.\n2.  We provide novel theoretical results (Theorem 1) on the limiting distribution for univariate Gaussians under the accumulate training workflow, proving that both the covariance and mean errors converge to non-zero constants - in stark contrast to the replace setting where they diverge. This theoretical result mathematically characterizes why the accumulate workflow avoids collapse.\n3.  Our work is the first to systematically compare all three workflows (replace, accumulate, accumulate-subsample) across five different generative settings, providing a more comprehensive understanding of when and how model collapse occurs.\n\n### On Accumulate-Subsample and Prior Work\n\n> The second part of the paper (accumulate-subsample) provides empirical insights that mixing real and generated data slows down model collapse. This has been observed and discussed in literature as early as Bertrand et al., Alemohammad et al. and even for the case of language models in Briesch et al.\n\nWe should do a much better job explaining how our approach contributes to the literature. While others do study mixing real and generated data, our work pushes the field forward in important ways:\n\nReason #1: We frame our analysis from an extremely practical real-world engineering constraint: finite compute. Other papers study model-data feedback loops in ways that aren't clearly motivated by practical considerations. This framing is important.\n\nReason #2: Other papers add complications that make drawing clear insights difficult. For instance, Bertrand et al. 2023 studies _iterative retraining_, where each new model is initialized from the previous model's parameters and the each new model’s optimizer is initialized from the previous model's optimizer’s state This helps mitigate model collapse because each model is then limited in how far it moves from the preceding model, but this iterative retraining poorly matches real-world conditions as frontier models aren't (to the best of our knowledge) initialized from their predecessors' parameters.\n\nReason #3: The real+synthetic data loops of previous papers e.g., Bertrand and Alemohammad both focus on real data plus synthetic data drawn only from the most recent model. But in reality, if one is scraping the web, we know of no way to filter the collected data for data generated by the most recent generation of models. Our Accumulate-Subsample setting is meant to more faithfully model reality: synthetic data are added online, and are vacuumed up with real data to train the next model.\n\nReason #4: Briesch et al. 2023 is difficult to interpret because the results of their v1 and v2 Arxiv manuscripts (specifically, the key Figure 4) appear to reach slightly contradictory conclusions that the authors don't address.\n\nIn the revised manuscript, we will better contextualize our contributions relative to these prior works, explicitly highlight the practical relevance of our approach, and more clearly explain how our findings extend beyond what was previously known.\n\n### Definitions and Realistic Assumptions\n\n> I also want to note that the literature is not unanimous on the claim that accumulating data prevents model collapse. That is an ongoing discussion. (e.g. Dohmatob et al. B).\n\nWe agree that the field doesn't have a consensus. This is partially because the field has defined model collapse in multiple and sometimes conflicting ways (a point made by a recent position paper https://arxiv.org/abs/2503.03150) and partially because different papers make assumptions that are unrealistic.\n\nThis is why we feel it is so important to construct experiments in a way that emulates reality as faithfully as possible. In our opinion, the field has drawn conclusions from experimental setups that are oftentimes not well grounded in real-world considerations.", "claims_and_evidence": "The claims of the paper (see summary) are supported by a wide range of experiments and in some cases theoretical analyses. The experiments are convincing and support the claims. I do not agree with the authors that model collapse is avoided in the accumulate-subsample case (judging from their experimental data) but rather slowed down.", "methods_and_evaluation_criteria": "The authors evaluate their claims for multivariate gaussian models, kernel density estimation, and supervised fine-tuning of language models for the replace and accumulate setting. They also provide theoretical insights in those cases. For the accumulate-subsample setting, they provide experiments for the 3 aforementioned cases as well as linear regression and pretraining language models. For the last part (section 4) experiments for supervised fine-tuning are presented.", "theoretical_claims": "Theoretical results are provided for the results in section 2. I did not check the detailed proof in the appendix for correctness.", "experimental_designs_or_analyses": "The experimental design seems to be extensive and convincing.", "relation_to_broader_scientific_literature": "- My biggest concern with the paper is its novelty. It builds heavily on Gerstgrasser et al. and section 2 (a large part of the paper) only provides new results for different kinds of generative models with equal conclusions. The second part of the paper (accumulate-subsample) provides empirical insights that mixing real and generated data slows down model collapse. This has been observed and discussed in literature as early as Bertrand et al., Alemohammad et al. and even for the case of language models in Briesch et al. The last section 4 is also close to the results of Dohmatob et al. A (as the authors note correctly).\n- I also want to note that the literature is not unanimous on the claim that accumulating data prevents model collapse. That is an ongoing discussion. (e.g. Dohmatob et al. B).", "essential_references_not_discussed": "I suggest that the authors discuss other work more that investigates mixing real and generated data in the model collapse setting (see above).", "other_strengths_and_weaknesses": "While I found the results and experiments in section 4 interesting, it seems to me that this section is another topic than the main paper. It felt out of place reading this section after the paper so far. To me section 4 is more about how to mix data in a dataset but leaves out the context of model collapse.", "other_comments_or_suggestions": "References: \n\n- Alemohammad, S., Casco-Rodriguez, J., Luzi, L., Humayun, A. I., Babaei, H., LeJeune, D., ... & Baraniuk, R. Self-Consuming Generative Models Go MAD. In The Twelfth International Conference on Learning Representations. (2024)\n- Bertrand, Q., Bose, J., Duplessis, A., Jiralerspong, M., & Gidel, G. On the Stability of Iterative Retraining of Generative Models on their own Data. In The Twelfth International Conference on Learning Representations. (2024)\n- Briesch, M., Sobania, D., & Rothlauf, F. (2023). Large language models suffer from their own output: An analysis of the self-consuming training loop. arXiv preprint arXiv:2311.16822.\n- Dohmatob, E., Feng, Y., Yang, P., Charton, F., & Kempe, J. (2024, July). A tale of tails: model collapse as a change of scaling laws. In Proceedings of the 41st International Conference on Machine Learning (pp. 11165-11197). \n- Dohmatob, E., Feng, Y., Subramonian, A., & Kempe, J. (2024). Strong model collapse. arXiv preprint arXiv:2410.04840. \n- Gerstgrasser, M., Schaeffer, R., Dey, A., Rafailov, R., Sleight, H., Hughes, J., ... & Koyejo, S. (2024). Is model collapse inevitable? breaking the curse of recursion by accumulating real and synthetic data. arXiv preprint arXiv:2404.01413.\n- Gonzalo Mart´ınez, Lauren Watson, Pedro Reviriego, Jose Alberto Hern ´ andez, Marc Juarez, and Rik ´ Sarkar. Towards understanding the interplay of generative artificial intelligence and the internet. arXiv preprint arXiv:2306.06130, 2023.", "questions_for_authors": "-\tQ1: The paper focuses heavily on test loss in their experiments. Did the authors also consider other metrics and evaluations that are important for generative models  (e.g. diversity biases as mentioned in Martinez et al., Briesch et al., and Guo et al. or)?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "napoOX2QCJ", "reviewer": "Reviewer_uuKC", "summary": "This paper examines the phenomenon of model collapse in generative learning, where models are trained on data that includes synthetic generations from previous iterations. The authors investigate three training paradigms - Replace, Accumulate, and Accumulate-Subsample - and find that Replace leads to model collapse, Accumulate prevents it, and Accumulate-Subsample slows degradation but does not fully eliminate it. The paper provides empirical evidence across multiple generative model settings including multivariate Gaussian estimation, kernel estimation and language model fine tuning.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "### Definitions of Model Collapse\n\nYou raise an excellent point. A recent review (https://arxiv.org/abs/2503.03150) identifies multiple definitions of model collapse. We focused on test loss divergence as it addresses an existential threat: all future generative models becoming useless. We'll expand our discussion to acknowledge alternative definitions and clarify our specific focus.\n\n### Novelty in Accumulate-Subsample Paradigm\n\nWe appreciate the opportunity to clarify what distinguishes our work from prior research:\n\n1. Practical Framing: We specifically examine accumulate-subsample through fixed compute per model-fitting iteration, which directly models real-world ML development constraints. This framing is absent from the papers cited but is absolutely critical. One can study all manner of different workflows for data, but their importance depends on how closely they model the real world.\n\n2. Training Methodology: Bertrand et al. (2024)  studies _iterative retraining_, where each model is initialized from the previous model and each optimizer is initialized from the previous model’s optimizer state. The iterative retraining approach in Bertrand et al. mitigates model collapse because models are constrained from wandering too far from their predecessors. However, this doesn't reflect how frontier models are developed in practice. In comparison, we model the scenario where successive model generations are trained independently.\n\n3. Data Evolution: We consider synthetic data accumulation across multiple model generations, whereas prior work typically only considers synthetic data from the most recently trained model e.g., Bertrand & Alemohammad.\n\nWe will improve our manuscript to clarify.\n\n### Related Work\n\n> How does this work differ from Dohmatob et al. (2024), which already studied real-vs-synthetic data trade-offs?\n\nOur work differs substantially from both Dohmatob papers. One examines linear regression specifically, while the other investigates scaling laws. Both are largely theoretical. In contrast, our work is more empirical and spans multiple generative paradigms (Gaussian modeling, KDE, language model fine-tuning) and examines different data treatment workflows across these diverse settings.\n\n> Please move the related work section to the main text.\n\nIf accepted, we will move Related Work into the main text after the Introduction. We placed it in the Appendix due to the 8-page limit.\n\nWe will expand our related work to include additional relevant papers discovered during the review process e.g., Gillman et al. 2024 and better clarify the positioning of our contributions.\n\n### Understanding Accumulate-Subsample\n\n> Can the authors justify why accumulate-subsample prevents collapse beyond empirical results?\n\nThe accumulate-subsample setting is difficult to study analytically and we were not able to prove satisfactory results. Intuitively, accumulate-subsample slows collapse because each new training batch contains a fraction of real data that provides an \"anchor\" to the true distribution.\n\n### Benefits of Synthetic Data\n\nOur contribution isn't showing that synthetic data can help performance - we agree this possibility is well-known across many papers. Rather, one contribution is that such a result holds for kernel density estimation - a  simple yet fundamental statistical model - without sophisticated data manipulations. Another contribution is assessing the role of cardinality or proportionality of synthetic data , which to our knowledge was previously unexplored.\n\nUpon careful review of Seddik et al. (2024), we couldn't find where they demonstrate synthetic data improves performance.\n\n### Realistic Data Quality\n\nOur experimental design deliberately aimed to model how synthetic data proliferates on the internet in practice:\n\n- Content on the internet is not systematically filtered. Vast amounts of web data is garbage, spam, AI-generated slop, etc.\n- By studying the \"pessimistic\" scenario without filtering, we establish a baseline understanding of collapse dynamics. Filtering can only improve the quality of models.\n- That said, we agree data quality is an important factor, and we view our work as complementary to research on data quality research. Indeed, we write \"Our experiments take a pessimistic viewpoint, in the sense that our experiments pay no attention to the quality of data, whereas in practice, engineers heavily filter data based on various indicators of data quality, e.g., (Brown et al., 2020; Lee et al., 2023; Wettig et al., 2024; Penedo et al., 2024; Li et al., 2024b; Sachdeva et al., 2024); for a recent review, see Albalak et al. (2024).\"\n\n> There is no long-term stability analysis, making it unclear how performance evolves after many more iterations.\n\nWe ran most experiments for 100 iterations across most settings. Running language modeling pretraining for longer proved beyond our limited compute budget. We're open to extending our experiments.", "claims_and_evidence": "main claims:\n- Replace leads to model collapse, where test loss diverges across iterations. This is demonstrated in multivariate Gaussian modeling, kernel density estimation (KDE), and language model fine-tuning experiments.\n- Accumulate prevents collapse, as models trained on both real and synthetic data remain stable.\n- Accumulate-Subsample reduces but does not eliminate degradation, acting as a middle ground between Replace and Accumulate.\n\nIssues:\n- The paper presents accumulate-subsample as a new paradigm, but prior work (e.g., Dohmatob et al. 2024, Bertrand et al. 2024) already studied real-vs-synthetic data trade-offs in iterative training. \n- The paper claims that synthetic data can improve performance in some settings, but this was already explored in Dohmatob et al. (2024) and Seddik et al. (2024).\n- Definition of Model Collapse: The paper presents collapse as \"test loss divergence\", ignoring alternative definitions such as scaling law deviations (Dohmatob 2024), loss of rare modes (Shumailov 2024), and knowledge degradation (Peterson 2024).", "methods_and_evaluation_criteria": "Although the experimental setup is well-structured, with multiple model classes tested across generative settings, but the experiments focus on data quantity (real vs. synthetic ratios) but ignore data quality. This is an important step which is ignored.\n\nEvaluation criteria are relevant but incomplete, as long-term performance degradation is not analyzed rigorously beyond several iterations specifically for LLMs experiments.", "theoretical_claims": "Theoretical results are only for Gaussian and KDE models. No theoretical proof for why Accumulate-Subsample prevents full collapse.\nAnd prior work by Bertrand et al. (2024) already provided mathematical stability conditions for iterative training on synthetic data.", "experimental_designs_or_analyses": "The main issues I see with the experiments is that:\n- The paper does not analyze synthetic data quality or filtering, which are key considerations in real-world. For example most text that enters the internet does have some human evaluation on top of it. So I think all the setups in the paper are to some extend unrealistic.\n\n- There is no long-term stability analysis, making it unclear how performance evolves after many more iterations.", "relation_to_broader_scientific_literature": "- The paper overstates its originality and does not fully acknowledge prior research on model collapse.\n- A more comprehensive literature review is needed to position the contributions in relation to existing studies.", "essential_references_not_discussed": "The paper does discuss and cite many of the relavent work but I believe it does not do just in citing the previous findings.", "other_strengths_and_weaknesses": "- The main contributions lack novelty relative to prior work.\n- The definition of model collapse is too narrow.\n- The paper does not address synthetic data filtering or quality.", "other_comments_or_suggestions": "See below.", "questions_for_authors": "1 - How does this work differ from Dohmatob et al. (2024), which already studied real-vs-synthetic data trade-offs?\n\n2 - Why does the paper not discuss alternative definitions of model collapse (e.g., rare mode loss, scaling law deviations)?\n\n3 - Can the authors justify why accumulate-subsample prevents collapse beyond empirical results?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "IbAqAymVgP", "reviewer": "Reviewer_APMk", "summary": "This paper investigates model collapse under different data regimes: 1) when we train the model on the latest synthetic data (the setting used in Shumailov et al.) 2) when we accumulate the synthetic data at each iteration (keeping the original data in the first iteration) 3) when we accumulate and subsample to obtain a fixed dataset size at each iteration. The authors show that in 2) the test losses do not diverge, and in 3), the test losses do not diverge or the divergence is slow.\nAnother contribution is the investigation of the importance of the size of the real data vs its proportion in the training set.\n\n\nUpdate after rebuttal:\nI keep my rating of 3.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your thoughtful review and recommendation. We appreciate your recognition that our paper is well-written and that our claims are properly validated.\n\n### Theoretical Results for Accumulate-Subsample\n\nYou raise an excellent question. The accumulate-subsample workflow presents significant analytical challenges because of subsampling the data at each model-fitting iteration.  We welcome any suggestions you might have for tackling this theoretical challenge, as we believe it represents an important open problem for the field.\n\n### Clarification of Relationship to Dohmatob et al. in Section 4\n\n> Could the authors be more clear on how the theoretical results in Dohmatob et al. differ from the insights shown (experimentally) in Section 4?\n\nDohmatob et al. 2024 show in Corollary 3.3 that in a simplified mathematical setting, the test error on real data scales as $(T_{real} + T_{AI})^{-(1 - 1/\\beta)} + k^{-(\\beta -1)}$. This means that in this setting, AI-generated data can help improve performance.  Directly comparing our empirical results to their theoretical results is difficult because our generative models differ. Additionally, their result decouples the amount of data $T_{real} , T_{AI}$ from the tail cutoff $k$, whereas decoupling these two factors in real data would be difficult (and potentially not possible). We see our results as complementary, but different, and want to give appropriate credit to prior work. We will clarify these distinctions in the revised paper.\n\n### Realistic Settings\n\n> Can the authors provide results for a more realistic setting than the one used in the gaussian and kde experiments. The SFT one conducted in the paper is a good example of realistic scenario.\n\nWe appreciate this feedback. To explain why we used these classical statistical models, at least three previous papers have studied multivariate Gaussians, making this setting crucial for consistent comparison against their results. These settings offer analytical tractability for theoretical results while still demonstrating the key phenomena.\n\nWe’d also like to highlight that our paper contains results from pretraining sequences of language models on real text (TinyStories), as shown in Figure 4.\n\nCould you please clarify what \"realistic\" properties you're looking for that our current experiments don't address?  Is it a matter of the particular generative model? If so, what model and data combination would you like us to run experiments for?\n\nWe'd be happy to expand the language model experiments with additional details in the revision or to conduct new experiments if you have specific suggestions.", "claims_and_evidence": "All the claims are validated with experiments, and some of them with theory.\n\nThe empirical evidence encompasses three data settings: 1) multivariate gaussian modeling 2) kernel density estimation 3) SFT of LLMs\n\nThe theroretical evidence encompasses the univariate gaussian modeling setting (accumulate workflow), KDE (replace and accumulate workflow).  However, no result is given for the accumulate + subsample setting.", "methods_and_evaluation_criteria": "The datasets choice mimicks the one adopted in Shumailov et al.\nThe gaussian modeling and KDE experiments feel more like sanity checks (for which some theoretical results can be proven) than realistic settings where model collapse might appear.", "theoretical_claims": "Yes, no issue.", "experimental_designs_or_analyses": "Yes, no issue.", "relation_to_broader_scientific_literature": "In the first part of the paper, the objective of the analysis is to extend some insights which were given in Gerstgrasser et al. ([1]) (namely that the accumulate workflow might not lead to divergence) to the experimental setup used in Shumailov et al. ([2]). \nSection 4 looks at the importance of the cardinality of the real data vs its proportion. This section is empirical, but previous theoretical results in [3] shared common insights on the potential benefits of synthetic data up to a certain point, when real data is scarce.  \nAnother important reference is [4] which also studies model training on real and synthetic data (with the slight difference that there is no accumulation in [4], as the synthetic data from the latest iteration is mixed with the real data).\n\n\n[1] Gerstgrasser, M., Schaeffer, R., Dey, A., Rafailov, R., Korbak, T., Sleight, H., Agrawal, R., Hughes, J., Pai, D. B., Gromov, A., Roberts, D., Yang, D., Donoho, D. L., and Koyejo, S. Is model collapse inevitable? breaking the curse of recursion by accumulating real and synthetic data, 2024.\n\n[2] Shumailov, I., Shumaylov, Z., Zhao, Y., Papernot, N., Anderson, R., and Gal, Y. Ai models collapse when trained on recursively generated data. Nature.\n\n[3] Dohmatob, E., Feng, Y., Yang, P., Charton, F., and Kempe, J. A tale of tails: Model collapse as a change of scaling laws. In Forty-first International Conference on Machine Learning, 2024.\n\n[4] Bertrand, Q., Bose, A. J., Duplessis, A., Jiralerspong, M.,and Gidel, G. On the stability of iterative retraining of generative models on their own data. The Twelfth International Conference on Learning Representations, 2024.", "essential_references_not_discussed": "Most of the works related to this paper are cited, but the related work section in the appendix could be improved to make the objective and distinctness of each work more clear, namely the experimental setup considered (how is the synthetic data mixed with real data) and the theoretical results.", "other_strengths_and_weaknesses": "The paper is well-written and easy to follow.", "other_comments_or_suggestions": "NA", "questions_for_authors": "Do the authors have some intuition on possible preliminary theoretical results for the accumulate-subsample workflow, even in a very simplified setup?\n\nCould the authors be more clear on how the theoretical results in Dohmatob et al. differ from the insights shown (experimentally) in Section 4?\n\nCan the authors provide results for a more realistic setting than the one used the gaussian and kde experiments. The SFT one conducted in the paper is a good example of realistic scenario.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: The reviewers appreciated this paper's comprehensive investigation of how different training workflows impact model collapse across multiple generative settings. While some concerns were raised, for example, about novelty relative to prior work, others recognized the paper's empirical contributions and theoretical analysis demonstrating that accumulating both real and synthetic data prevents model collapse, whereas replacing real data with synthetic data leads to divergence. Overall, this paper provides valuable insights through rigorous experiments across diverse settings and offers a useful framework for understanding when synthetic data can be beneficial versus detrimental. The exposition of these findings makes this a worthwhile contribution to the understanding of synthetic data for training.
Decision: Accept (poster)