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Is Model Collapse Inevitable? Breaking the Curse of Recursion by Accumulating Real and Synthetic Data
Matthias Gerstgrasser, Rylan Schaeffer, Apratim Dey, Rafael Rafailov, Henry Sleight, John Hughes, Tomasz Korbak, Rajashree Agrawal, Dhruv Pai, Andrey Gromov, Daniel A. Roberts, Diyi Yang, David L. Donoho, Sanmi Koyejo
TL;DR
The paper studies whether model collapse is inevitable when generative models train on their own outputs, addressing the gap created by prior replacement-based analyses. It combines experiments across generative-model classes with a linear-model theory and finds that accumulating synthetic data alongside real data avoids collapse, while identifying settings that remain open for investigation.
Problem
The paper asks whether accumulating training data changes the model-collapse outcome, since prior studies mainly examined replacing real data with synthetic outputs.
Method
The authors compare replacing and accumulating data across transformers, diffusion models, and variational autoencoders, then analyze the distinction theoretically using sequences of linear models.
Results
Accumulating synthetic data alongside real data avoids model collapse across the tested model classes and modalities, whereas replacing data yields collapse.
Takeaways & Limitations
The findings suggest that data accumulation may be robust to model collapse and should be considered in analyses of generative models trained on web-scale data.
Takeaways & Limitations
The experiments use stochastic sampling from previous models and leave deterministic generation and other accumulation regimes for future work.
Abstract
from arXiv · showhide
The proliferation of generative models, combined with pretraining on web-scale data, raises a timely question: what happens when these models are trained on their own generated outputs? Recent investigations into model-data feedback loops proposed that such loops would lead to a phenomenon termed model collapse, under which performance progressively degrades with each model-data feedback iteration until fitted models become useless. However, those studies largely assumed that new data replace old data over time, where an arguably more realistic assumption is that data accumulate over time. In this paper, we ask: what effect does accumulating data have on model collapse? We empirically study this question by pretraining sequences of language models on text corpora. We confirm that replacing the original real data by each generation's synthetic data does indeed tend towards model collapse, then demonstrate that accumulating the successive generations of synthetic data alongside the original real data avoids model collapse; these results hold across a range of model sizes, architectures, and hyperparameters. We obtain similar results for deep generative models on other types of real data: diffusion models for molecule conformation generation and variational autoencoders for image generation. To understand why accumulating data can avoid model collapse, we use an analytically tractable framework introduced by prior work in which a sequence of linear models are fit to the previous models' outputs. Previous work used this framework to show that if data are replaced, the test error increases with the number of model-fitting iterations; we extend this argument to prove that if data instead accumulate, the test error has a finite upper bound independent of the number of iterations, meaning model collapse no longer occurs.
1 Introduction
The paper asks whether training generative models on their own outputs inevitably causes model collapse, especially when data accumulate rather than replace earlier data. It finds that accumulation avoids collapse across experiments and motivates an analytical explanation for the difference.
- Training models on their own generated outputs can progressively degrade performance until newer models become useless, a phenomenon termed model collapse.
- Prior analyses largely assume that each generation replaces previous training data, whereas real-world model training often uses increasing datasets.
- Accumulating synthetic data alongside original real data avoids model collapse, even under a pessimistic scenario of uncontrolled synthetic-data growth.
- Experiments test causal language models, molecular-conformation diffusion models, and image variational autoencoders under replacing and accumulating data regimes.
- Replacing data increases test error across iterations, while accumulating synthetic and real data avoids collapse across the tested models and modalities.
2 Accumulating Data Avoids Model Collapse in Deep Generative Models
Across language, molecular-conformation, and image-generation experiments, replacing synthetic data causes model collapse, whereas accumulating synthetic data with real data avoids it or substantially slows degradation. The accumulation pattern is consistent across tested model classes, though image-generation test error still increases gradually.
- Replacing data caused model collapse across causal transformers, diffusion models, and variational autoencoders, while accumulating data avoided collapse across the tested models and datasets.Here, avoiding collapse means bounded error over model-fitting iterations.
- 2.1 Transformer-Based Causal Language Modeling: For language models, replacing data increased test cross entropy across architectures, parameter counts, and sampling temperatures, whereas accumulation produced equal-or-lower test cross entropy.Lower temperature accelerated error growth under replacement, but the qualitative trend held at both tested temperatures.
- A size-matched synthetic-data ablation still showed performance degradation under replacement, although at a lower rate than in the standard comparison.This tests whether accumulation’s larger training sets alone explain the observed difference.
- 2.2 Diffusion Models on Molecular Conformation Data: For molecular diffusion models, test loss increased across eight iterations with replacement but remained relatively constant when data accumulated.Replacement caused most performance worsening in the first synthetic-data iteration, with little further degradation afterward.
- 2.3 Variational Autoencoders on Image Data: For VAEs trained on CelebA, replacement rapidly increased test error and drove generations toward a single mode, while accumulation slowed error growth and retained major variation axes.Accumulated generations still lost some minor details, including glasses and accessories.
- 2.3 Variational Autoencoders on Image Data: In image generation, accumulating data did not produce fully bounded test error: error increased with iterations, albeit much more slowly than under replacement.The authors identify differing conditions and architectures as an open question because prior work reported faster deterioration even with accumulation.
3 Accumulating Data Avoids Model Collapse in Linear Models
An analytical linear-regression framework contrasts replacing versus accumulating synthetic data across model-fitting iterations. Replacing data causes unbounded test-error growth, while accumulation yields bounded error under the stated assumptions.
- 3.1 Notation and Preliminaries: The framework models sequences of linear predictors trained on synthetic data generated from earlier models, comparing replacement and accumulation strategies.It assumes an original full-column-rank design matrix and independent Gaussian noise terms across iterations.
- 3.1 Notation and Preliminaries: Model collapse is defined here as test error diverging to infinity as model-data iterations continue.The paper notes that other work uses “model collapse” for different test-error or distributional phenomena.
- 3.1 Notation and Preliminaries: The analysis is presented for ordinary linear regression but can extend to ridge regression and kernel methods.The paper focuses on linear regression as a simple model for studying model collapse.
- 3.2 Precise Test Error Characterization Under Accumulating Data: Theorem 1 expresses every accumulated-data estimator in terms of the true parameter, the original design matrix, and iteration-specific noise.The result applies for every iteration n ≥1.
- 3.2 Precise Test Error Characterization Under Accumulating Data: When data are replaced, test error grows linearly with iterations; when data accumulate, Theorem 2 gives an iteration-independent upper bound.The replacement growth rate depends on σ2, d, and T, while the accumulation result assumes T ≥ d + 2 samples per iteration.
- 3.2 Precise Test Error Characterization Under Accumulating Data: Accumulation limits each iteration’s added noise because iteration i contributes only fraction 1/i of the training dataset, unlike replacement.Discarding previous data makes newly introduced noise more influential across iterations.
- 3.2 Precise Test Error Characterization Under Accumulating Data: Numerical simulations closely match the analysis: replacement error keeps growing, whereas accumulated-data error rapidly plateaus.For replacement, the prefactor is σ2d/(T − d − 1).
4 Discussion
The discussion argues that accumulating synthetic data alongside real data avoids the unbounded-error form of model collapse studied here, while identifying important open questions and terminology differences.
- 4 Discussion: Across causal transformers, molecular diffusion models, image VAEs, and linear regression, accumulating synthetic data with real data avoids the paper’s form of model collapse.The conclusion is framed as a strong suggestion rather than a universal guarantee.
- 4 Discussion: The authors call for studies of additional real-data inflow, synthetic-data schedules, human filtering, and deterministic generation.Their experiments generate synthetic datasets by stochastic sampling from the previous model.
- 4 Discussion: “Model collapse” is used inconsistently across studies, covering divergent test error, modal collapse, uniformity, and artifact amplification.The authors caution that claims across articles require careful comparison.
A Summarization and Discussion of Prior and Related Work
Prior work largely studied synthetic-data feedback loops under replacement, while this paper distinguishes that setting from accumulating data and argues the discrepancy warrants closer examination.
- Prior empirical and theoretical work: Prior empirical studies primarily analyzed iterative training in which each generation’s synthetic data replaces earlier data, enabling fixed-size comparisons across generations.The paper contrasts this setup with real-world training practices in which data volumes grow over time.
- Prior empirical and theoretical work: Prior theory found that test error grows with iteration count in synthetic-data regression, while also studying regularization as a possible mitigation.The cited theoretical framework motivates the paper’s linear-regression analysis.
- Accumulating data: The paper’s setting differs from independent synthetic augmentation because each iteration’s synthetic data are generated by a model trained on all preceding data.This iterative, learned-source construction is presented as closer to the authors’ model of internet evolution.
- Accumulating data: Earlier work only partially examined accumulation, including one StyleGAN2 experiment that interpreted accumulation as slowing rather than preventing collapse.The authors argue that closer examination supports a different interpretation, while noting discrepancies remain an open direction.
- Avoiding model collapse: Existing avoidance strategies in the replacement regime include sufficiently accurate initial models, enough real data, or carefully selected real-data mixtures.These approaches address replacement-based collapse rather than the accumulating-data setting studied here.
B Proofs of Mathematical Results
The paper proves its accumulation result by characterizing recursively fitted linear parameters, then deriving the resulting test error under stated Gaussian-design and full-rank assumptions.
- Assumptions and setup: The analysis assumes the original data matrix has full column rank so that X⊤X is invertible.A separate lemma establishes full rank almost surely for Gaussian rows when T ≥ d + 2.
- Test-error derivation: Theorem 2 substitutes the parameter representation into the test-error expression and takes expectations over the random quantities.The derivation uses isotropic features, Gaussian iteration noise, and the full-rank lemma.
- Test-error derivation: The resulting test-error formula applies to ridgeless regression with isotropic features in the data-accumulation setting.The paper presents this formula after completing the theorem derivation.
- Parameter characterization: Theorem 1 expresses the fitted parameters at every iteration in terms of the true parameter and accumulated iteration-specific noise.The proof proceeds by induction, using a base case and an inductive step.
- Parameter characterization: The proof tracks each noise term’s coefficient through the recursive sums, showing earlier terms receive progressively different multiplicities.The counting argument identifies coefficients from E1 through En+1 before completing the induction.
Implementation Details
The experiments use Hugging Face Transformers for model training and vLLM for dataset generation, with appendix figures expanding the main learning-curve visualizations.
- Software: Model training uses Hugging Face Transformers, while synthetic dataset generation uses vLLM.These tools implement the language-model experiments and data-generation pipeline.
- Visualization: Appendix Figures 9–12 provide larger learning-curve views with epoch or gradient-step x-axes and linear-linear or log-log scales.The figures expand the corresponding main-paper visualization.
Ablations
The ablations test whether accumulation’s advantage depends on dataset size, generation temperature, training amount, first-iteration quality, or unequal training-set sizes.
- Dataset-size control: Replacing data still degrades performance when the synthetic dataset grows to match the accumulation regime’s training-set size.The degradation rate is lower, but the qualitative replacement failure remains.
- Dataset size and epochs: Varying dataset size and training epochs produces no qualitative change in the main results.The ablation includes training on one-fifth of TinyStories for one or three epochs.
- First-iteration quality: Changing the amount of first-iteration training produces no qualitative differences in later-iteration behavior.This control addresses whether an unusually poor first synthetic dataset drives subsequent outcomes.
- Learning curves: The appendix learning curves compare replacement and accumulation across individual model-fitting iterations using multiple axis scales.The figures include linear and log-log presentations, with some comparisons accounting for differing gradient-step counts.
D Additional Details on VAE Experiments
The VAE experiments compare replacing versus accumulating image data across iterations. Replacing collapses reconstruction capability, whereas aggregated data preserves accurate reconstructions with fine details.
- Each VAE data iteration uses 100 training epochs before generating 163K new training images from sampled latent representations.
- Replacing data causes the VAE to collapse onto a single representable mode.Reconstructions from the test set show that the model trained only on the prior iteration cannot represent other classes.
- Accumulating data preserves the VAE’s generative capabilities and produces little to no degradation in reconstructed images.Aggregated-data reconstructions remain accurate, including smaller details such as glasses and hats.
E Linear Regression: Replacing Data with Increasing Sample Size
The linear-regression analysis contrasts replacing data with accumulating data and with a matched-size Replace-Multiple baseline. Accumulation yields bounded test error, while the replacement baselines continue to grow.
- Replacing Data: Replacing data with fixed training cardinality makes test squared error scale linearly with model-fitting iterations n.
- Accumulating Data: Accumulating T × i samples, including T real samples and T synthetic samples from each prior model, upper-bounds test squared error independently of iterations.
- Replace-Multiple: Replace-Multiple fits the i-th model on T × i samples from the preceding model, matching Accumulate’s per-iteration training-data amount.
- Replace-Multiple: Under Replace-Multiple, test squared error grows logarithmically with the number of iterations.
- Comparison: Replace-Multiple does not match Accumulate’s total compute because it draws T × i recent-model samples per iteration, versus T synthetic samples for Accumulate.
F Additional Linear Regression Numerical Results
The numerical linear-regression results illustrate how data-iteration policies affect test error. Replacing data produces growing error, while accumulating data avoids unbounded degradation.
- Replace: With replacement, test squared error grows linearly with the number of model-fitting iterations n.
- The figure contrasts replacement, Replace-Multiple, and accumulation settings for sequences of linear models fit to generated targets.