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Strong Model Collapse

Elvis Dohmatob, Yunzhen Feng, Arjun Subramonian, Julia Kempe

arXiv:2410.04840v2cs.LGstat.ML

TL;DR

The paper examines how synthetic data in training affects model performance and how model size shapes collapse. It analyzes real–synthetic data mixtures theoretically and empirically, finding that collapse persists when the synthetic fraction does not vanish and that its relationship to model size can reverse beyond the interpolation threshold.

  • Problem

    The paper asks how synthetic data mixtures affect model performance and whether larger models worsen or mitigate collapse.

    Source: §1 · p. 1, §1.1 · p. 2

  • Method

    The authors analyze real–synthetic data mixtures in supervised regression, extend their theory to random-projection models, and test predictions on language models and feed-forward networks.

    Source: §2.2 · p. 5, §6 · p. 13

  • Results

    Collapse generally persists when the synthetic-data fraction does not vanish, while larger models can suffer more before the interpolation threshold and become more robust beyond it.

    Source: §1.1 · p. 2

  • Takeaways & Limitations

    The authors emphasize preserving and labeling real data as AI-generated data becomes more prevalent.

    Source: §6 · p. 13

    Iterative data mixing requires substantial bootstrapping and real data, plus reliable separation of real and synthetic data, making it computationally expensive and difficult to implement.

    Source: §5.2 · p. 13

Abstract

from arXiv · show

Within the scaling laws paradigm, which underpins the training of large neural networks like ChatGPT and Llama, we consider a supervised regression setting and establish the existance of a strong form of the model collapse phenomenon, a critical performance degradation due to synthetic data in the training corpus. Our results show that even the smallest fraction of synthetic data (e.g., as little as 1\% of the total training dataset) can still lead to model collapse: larger and larger training sets do not enhance performance. We further investigate whether increasing model size, an approach aligned with current trends in training large language models, exacerbates or mitigates model collapse. In a simplified regime where neural networks are approximated via random projections of tunable size, we both theoretically and empirically show that larger models can amplify model collapse. Interestingly, our theory also indicates that, beyond the interpolation threshold (which can be extremely high for very large datasets), larger models may mitigate the collapse, although they do not entirely prevent it. Our theoretical findings are empirically verified through experiments on language models and feed-forward neural networks for images.

1 Introduction

The introduction asks how synthetic data affects scaling-law performance and whether model size worsens or mitigates collapse. It presents theoretical results and experiments showing persistent collapse with nonvanishing synthetic data, with model-size effects changing around interpolation.

1 Introduction

  • The work is motivated by model collapse as degraded real-world generalization when training data contain synthetic material, and by reports that larger datasets may stop improving performance as data becomes more synthetic.The authors pose whether strategic mixing can address collapse and whether larger models are more or less prone to it.Source: §1 · p. 1, §1.1 · p. 2

1.1 Main Contributions

  • With a nonvanishing synthetic-data fraction, model collapse generally persists unless synthetic data becomes a vanishing share; simple data weighting therefore cannot generally fix it.The paper defines synthetic data as coming from a distribution that deviates from the real-data distribution.Source: §1.1 · p. 2
  • Larger models generally suffer more when synthetic and real distributions differ substantially, but beyond interpolation they may become more robust, predicting double descent.The paper illustrates this model-size profile in Figures 1 and 2.Source: §1.1 · p. 2, Figure 1 · p. 2, Figure 2 · p. 3
  • Experiments validate the theoretical predictions in toy settings, fully trained shallow MNIST networks, and GPT-2 models trained on BabiStories.Source: §1.1 · pp. 2–3
  • The analysis uses linear regression and random-projection neural-network approximations with operator-valued free probability to derive a test-error decomposition whose extra term induces collapse.The decomposition is Etest ≃ B + V + ζ.Source: §1.1 · p. 3, Figure 2 · p. 3

1.2 Related Work

  • The introduction situates the work among studies of synthetic-data retraining and clean-data kernel and random-feature models, while noting few theoretical analyses of mixed real and synthetic training.It also notes that rigorous studies of synthetic-data curation or filtering remain lacking.Source: §1.2 · p. 3

2 Theoretical Setup

The setup models training data as independent samples from real and synthetic distributions, with their mismatch captured through label differences. It evaluates penalized linear and random-projection models on the real distribution while scaling data, dimension, and model size proportionately.

2.1 Data Distributions

  • The training set combines independent iid samples from real distribution P1 and synthetic distribution P2, with total size n = n1 + n2.Source: §2.1 · p. 4
  • The setup represents synthetic-real mismatch through differences in conditional labels, modeled by a covariance matrix Δ, and generally assumes diagonal matrices and equal feature covariances.The equal-covariance assumption keeps the marginal distribution of inputs the same across P1 and P2.Source: §2.1 · p. 4
  • Synthetic-data quality is measured by c2(Δ) = (1/d) tr ΣΔ, with smaller values indicating less disparity from the real distribution.Source: §2.1 · p. 4

2.2 Models and Performance Measure

  • The models include penalized linear regression and random-projection features z = Sx, whose output dimension m controls model size; only the read-out weights are learned in the simplified neural-network model.Performance is evaluated on the real distribution.Source: §2.2 · pp. 4–5
  • In the random-projection scaling limit, data size, dimension, and model size grow proportionately; the study also notes that synthetic data deforms the clean-data scaling law toward collapse.The limit tracks ratios including n2/n, d/n, m/d, and m/n.Source: §2.2 · p. 5

3 A New Bias-Variance Decomposition and the Emergence of Strong Model Collapse

The paper decomposes test error into bias, variance, and an additional synthetic-data term, showing that a nonvanishing synthetic fraction can leave performance above the real-data baseline as training sets grow. In random-projection models, model size changes the collapse profile, and experiments show that the optimal size depends on synthetic-data quality and proportion.

3.1 Classical Linear Models

  • Theorem 1 decomposes classical linear-model test error on real-distribution data into bias, variance, and an extra term ζ responsible for model collapse.With ∆ = 0, the extra term vanishes and the classical bias-variance decomposition is recovered.Source: §3.1 · p. 6
  • When ϕ → 0+, ζ ≃ p2^2 tr ∆, so the scaling-law error plateaus unless the synthetic-data fraction vanishes.For low-quality synthetic data with c2 = Ω(1), the classical-model test error is Ω(p2^2 c2).Source: §3.1 · pp. 6–7
  • Figure 3 shows that even a small amount of low-quality synthetic data makes test error depart from the ideal real-only scaling diagonal.The figure reports a close match between experimental curves and Corollary 1 predictions.Source: §3.1 · p. 7, Figure 3 · p. 7

3.2 Random Projections Model

  • Theorem 2 extends the bias-variance-plus-collapse-term test-error decomposition to models trained in a random-projection feature space.The model-size role is represented by the projection output dimension m.Source: §3.2 · pp. 7–8
  • With no synthetic data, the random-projection collapse term vanishes and the model recovers the real-data bias-plus-variance result.The result also generalizes the classical linear-model theorem as ψ → ∞.Source: §3.2 · p. 8

3.3 Some Consequences of Theorem 2

  • In the random-projection Gaussian setting, bias, variance, and the collapse term all diverge at the interpolation threshold m = n.The bias curve resembles double descent in the unregularized limit.Source: §3.3 · p. 8
  • The collapse profile depends on model design choices, including model size, and Figure 4 varies width, synthetic-data quality, and mixing ratio.Its test-error curves include error bars from five independent runs.Source: §3.3 · p. 8, Figure 4 · p. 8
  • With few synthetic samples, larger models are optimal for high-quality data, while lower-quality data favors an intermediate size; with more synthetic data, intermediate sizes are optimal.When synthetic samples substantially outnumber real samples, the Pareto frontiers are higher, indicating more serious collapse; very small models are never optimal.Source: §3.3 · p. 9

4 Experimental Results

Experiments test the theory beyond Gaussian data using MNIST neural networks and GPT-2 language models. Across these settings, synthetic data slows or collapses scaling, while model-size effects depend on the training regime and token count.

4 Experimental Results

  • The experiments extend the theoretical framework beyond Gaussian data to MNIST neural networks and language modeling.The theory is developed for high-dimensional linear regression and random-projection models.Source: §4 · p. 9

4.1 Experiments on Neural Networks on MNIST

  • For MNIST training sets of 10,000–50,000 samples, larger fully trained networks are more susceptible to collapse, with all tested models remaining in the interpolation regime.The models were trained exclusively on synthetic data (p2 = 1).Source: Figure 6 · p. 9, §4.1 · p. 10
  • As synthetic-data proportion rises, scaling slows and eventually plateaus in both MNIST random-feature and fully trained networks; collapse subsides only as p2 approaches 0.The experiments use MNIST with regression loss.Source: Figure 7 · p. 10, §4.1 · p. 10

4.2 Language Modeling

  • In GPT-2 language modeling, even moderate amounts of high-quality synthetic data delay scaling-law progression and are expected to lead to plateaus or very small exponents.Figure 8 evaluates loss on the TinyStories test set.Source: §4.2 · pp. 10–11, Figure 8 · p. 11
  • With synthetic-only BabiStories training, larger GPT-2 models have lower test loss below 1 × 10^10 tokens, but amplify collapse when token counts exceed 3 × 10^10.The larger-token regime likely exceeds the interpolation threshold; smaller models then show lower loss and reduced overfitting.Source: Figure 8 · p. 11, §4.2 · p. 11

5 Can Strategic Data Mixing Schemes Prevent Model Collapse?

The section tests whether weighted single-step mixing or iterative bootstrapping can mitigate collapse. Single-step mixing fails when synthetic data remains, while iterative mixing can recover a scaling rate under demanding conditions but still underperforms real-only training.

5.1 Weighted Single-Step Data Mixing

  • Weighted mixing assigns constant sample weights within each real-data and synthetic-data group, with α controlling their relative contribution.The endpoints α → 0 and α → 1 correspond to training only on real data and only on surrogate data, respectively.Source: §5.1 · p. 11
  • Any fixed positive synthetic-data fraction leaves a positive lower bound on test error, and the optimal weight tends to using only real data under the stated conditions.The authors describe this as effectively model collapse and report empirical confirmation in Figure 9.Source: §5.1 · p. 12, Figure 9 · p. 12

5.2 Dynamic / Multi-Step Data Mixing

  • The iterative scheme repeatedly combines real samples with synthetic samples from the current source, then uses the trained model to generate the next synthetic source.The proposed bootstrapping aims to enhance synthetic-data quality over iterations.Source: §5.2 · p. 12
  • With a non-vanishing real-data share, t on the order of log(n/d) iterations recovers a scaling law proportional to the stated reference scaling.Figure 10 compares iterative-mixing experiments with theoretical predictions and a clean curve using the p1n real samples.Source: §5.2 · p. 13, Figure 10 · p. 13
  • That recovery requires substantial bootstrapping, abundant real data, and distinguishing real from synthetic samples across iterations, making implementation computationally expensive and challenging.The text presents these requirements as practical costs of iterative mixing.Source: §5.2 · p. 13
  • Iterative mixing consistently underperforms the clean real-only baseline, even when the initial synthetic data is high quality.The authors interpret this as synthetic data being largely neutralized rather than effectively leveraged.Source: §5.2 · p. 13, Figure 10 · p. 13

6 Discussion

The discussion frames strong model collapse as persistent under non-vanishing synthetic-data fractions and notes that experiments support the theory across LLMs and feed-forward image networks.

6 Discussion

  • Theoretical and empirical results characterize collapse as persisting when the synthetic-data fraction does not vanish, including under mixtures of real and synthetic data.The work also extends analysis to more complex mixing settings and random-projection models.Source: §6 · p. 13
  • The discussion emphasizes preserving and labeling real data, while identifying activation functions, depth, and optimization hyperparameters as future directions.It proposes Gaussian equivalents and operator-valued free probability as possible tools for extending theory to wide, fully trained networks.Source: §6 · p. 13

A Experimental Details

The appendix describes experiments using a random-projection regression model, two-layer neural networks, and GPT-2 language modeling. The toy random-projection setup varies synthetic-data quality and model width, while the language-model setup trains on generated stories.

A.1 Toy Setting: Random Projections Model

  • The toy random-projection experiment uses Gaussian data with power-law feature eigenvalues and varies projection width to study model size and synthetic-data quality.It uses d = 600, n1 = 300, n2 = 200, and total n = 500, with synthetic quality c2 ranging from 0 to 1.Source: §A.1 · p. 18

A.2 Two-layer neural networks

  • The two-layer neural networks are trained with stochastic gradient descent for 400 epochs, using a held-out validation set to choose the evaluated checkpoint.Training uses batch size 128 and learning rate 0.1.Source: §A.2 · p. 18

A.3 Language Modeling

  • The language-model experiments use GPT-2-small to generate synthetic BabiStories continuations from the dataset's original prompts.Generated stories are filtered for poor quality before being collected into the synthetic dataset.Source: §A.3 · p. 18
  • The GPT-2 training setup uses a 768-dimensional embedding, 12 attention heads, and a 512-token context length.Training also uses learning rate 5 × 10^-3, dropout 0.05, weight decay 0.1, and 2,000 warm-up iterations.Source: §A.3 · p. 18

B.1 Classical Linear Model in Over-Parametrized Isotropic Setting

The appendix develops asymptotic test-error results for over-parameterized regression and details bias-variance and random-projection calculations used in the analysis. It also connects the collapse result to iterative regression settings and specifies fixed-point and free-probability tools underlying the derivations.

B.1 Classical Linear Model in Over-Parametrized Isotropic Setting

  • In the over-parameterized isotropic limit, the excess test error for large ϕ is approximately (1 − 2/ϕ)p2c2, and it vanishes only when the synthetic fraction p2 tends to zero.The text concludes that strong collapse occurs in both under- and over-parameterized isotropic settings.Source: §B.1 · pp. 18–19

B.2 Connections to Classical Model Collapse in Regression

  • The appendix relates iterative regression with multiple data-generation stages to a proportionate-scaling setting and gives a simplification when each stage has enough samples to fit its training data.That condition is maxℓ ϕℓ ≤ 1, with equal noise levels across stages.Source: §B.2 · p. 19

C.1 Classical Linear Model

  • For the classical linear model, the appendix expresses test error through bias-variance components and derives trace-form expressions for the relevant terms.The derivation considers different real and synthetic feature covariance matrices before giving a specialized expression for k = 2.Source: C Raw Bias-Variance Decomposition · p. 19, §C.1 · p. 19

C.2 Random Projections Model

  • For random projections, the model uses a Gaussian random feature map and decomposes test error into variance and bias terms.The variance terms are handled through fixed-point computations, while the bias expression includes the mismatch between the real and synthetic targets.Source: §C.2 · pp. 20–21

D.1 Classical Linear Model

  • The classical-model derivation introduces fixed-point equations for e1 and e2, then defines positive solutions u1 and u2 for the subsequent analysis.The resulting quantities feed into the trace expressions used in the proof.Source: §D.1 · p. 22

D.2 Random Projections

  • The random-projection derivation defines quenched quantities from the projected random matrices and uses them to express the bias terms.The resulting components are analyzed using operator-valued free-probability theory.Source: §D.2 · p. 23

D.3 Proof of Proposition 2

The proof derives test-error terms using operator-valued free probability and fixed-point equations for mixed-data settings. It then applies these calculations to establish the stated proposition and corollaries.

D.3 Proof of Proposition 2

  • The proof uses operator-valued free probability and fixed-point equations to derive limiting trace expressions for the variance and other terms in Proposition 2.The derivations include special cases as one data-source proportion tends to zero and formulas for arbitrary proportions.Source: §D.3 · pp. 23–25

D.4 Proof of Proposition 3

  • A more general result established in Section F.1 implies Proposition 3 as a special case.Source: §D.4 · p. 27

E Proof of Theorem 1 and Corollaries

  • The general mixed-proportion case is technically harder because it requires deterministic equivalents for traces and resolvent derivatives of sums of random matrices; the proof uses operator-valued free probability.Source: E Proof of Theorem 1 and Corollaries · p. 27

E.1 Proof of Theorem 1

  • The proof recovers the classical formula for the bias term.Source: §E.1 · p. 28

E.2 Proof of Corollary 1

  • The corollary’s asymptotic expansion includes a synthetic-data contribution alongside the σ²d/n term.The displayed expression is 2c2 + O(ϕ²) ≃ σ²d/n + p2 2c2 + O(ϕ²).Source: §E.2 · p. 28

E.3 Proof of Corollary 2

These proofs apply the results for mixed data sources to a specialized covariance setup, then examine iterative mixing and a further asymptotic result.

E.3 Proof of Corollary 2

  • For λ = 0 and 0 < ϕ < 1, the fixed-point equations can be solved explicitly, and Theorem 1 predicts the test error shown in Figure 9.The explicit formulas are omitted because they are complicated.Source: §E.3 · p. 28

E.4 Proof of Corollary 3

  • When p2 is bounded away from 1, the large-iteration scaling is a constant multiple of the scaling for n units of clean data.Source: §E.4 · p. 30
  • The other iterative term increases with t and is lower-bounded by c2 + E, recovering model collapse depending on c2.Source: §E.4 · p. 30

E.5 Proof of Corollary 4

  • The proof derives the second part of Corollary 4 by expanding the preceding expression around ϕ = ∞.Source: §E.5 · p. 30

F.1 Proof of Proposition 3

The proof gives a more general proposition under proportionate scaling, using fixed-point equations to obtain its formula. Special parameter choices recover earlier propositions, and the proof of Theorem 2 follows from Proposition 3 and Section C.2.

F.1 Proof of Proposition 3

  • Proposition 4 states the general result under proportionate scaling, without requiring equal covariance matrices.The proof defines its quantities through a system of fixed-point equations and associated matrices.Source: §F.1 · p. 30
  • Setting τ = γ = 1 and ω = 0 recovers Proposition 2, while Σ1 = Σ2 = Σ recovers Proposition 3.Source: §F.1 · p. 31

F.2 Proof of Theorem 2

  • The proof of Theorem 2 follows directly from Proposition 3 and the computations in Section C.2.Source: §F.2 · p. 32

F.3 Recovering Theorem 1 from Theorem 2

The appendix shows how the general result recovers Theorem 1 and analyzes the general regularized case through equations for its auxiliary scalars.

F.3 Recovering Theorem 1 from Theorem 2

  • For any regularization strength λ > 0, taking the stated limits in Theorem 2’s formulas recovers Theorem 1.Source: §F.3 · p. 32

G.1 The General Regularized Case

  • In the general regularized case, u and ω′ solve a pair of linear equations; as γ → ∞, θ ≃ κ and ω′ → 0.The limiting u is given by ϕI2,2(κ)/(1 − ϕI2,2(κ)) ≃ df2(κ)/n /(1 − df2(κ)/n).Source: §G.1 · pp. 32–33

G.2 Unregularized Limit

This section derives analytic limiting formulas as λ approaches zero from fixed-point equations for e and τ. The limits depend on the parameter regime, including a separate case when θ0 = 0.

G.2 Unregularized Limit

  • The quantities e and τ are unique positive solutions of a pair of fixed-point equations, which are rewritten using η and yield the relations in (164).The equations include e′ = λ/(1 − ϕη), τ′ = λ/(1 − η/γ), and τ′e′ = λγθ.Source: §G.2 · p. 34
  • The equations imply η ≤ min(γ, 1/ϕ), and their product relation gives a quadratic equation for η in terms of λ and θ.The quadratic is written as ϕη^2 − (ϕγ + 1)η + γ − λ = 0.Source: §G.2 · p. 34
  • Provided θ0 > 0, as λ → 0, τ′ tends to (1 − ψ)γθ0 for ψ ≤ 1 and zero for ψ > 1, while e′ tends to zero for ψ ≤ 1 and (ψ − 1)θ0/ϕ for ψ > 1.Source: §G.2 · p. 34
  • When θ0 = 0, τ′ and e′ tend to zero, while τ tends to 1 − 1/γ and e tends to 1 − ϕ ≥ 0.The passage states this case occurs only if ψ < 1 and γ > 1, or ψ ≥ 1 and ϕ ≤ 1.Source: §G.2 · p. 34
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