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A Theoretical Perspective: How to Prevent Model Collapse in Self-consuming Training Loops
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Introduces recursive stability and gives the first generalization bounds for self-consuming training loops (STLs), showing a constant real-data proportion prevents collapse for transformers in in-context learning and analyzing synthetic augmentation trade-offs.
Problem
Empirical results on self-consuming training loops (models trained recursively on mixed real and self-generated data) are inconsistent, with some models collapsing and others not, and lack theoretical explanation of when STLs generalize.
Approach
Defines recursive stability (output deviation across generations under a one-point perturbation of the initial real data) and combines it with uniform stability and a conditional-i.i.d. analysis to bound generalization error on mixed datasets plus a cumulative distribution shift term controlled by the real-data proportion alpha.
Key results
- Generalization error for transformers in STLs with constant real-data proportion is bounded by O(n^-1 log^2 n + n^-1/2 log n + n^-1/4).
- Recursive stability of an L-layer transformer scales as (1-alpha)^i times B^((i+1)L)/(2n+1), so a constant alpha keeps it O(1/n).
- Optimal synthetic augmentation coefficient lambda* increases as the real dataset size n decreases.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Maintain a non-negligible constant proportion of real data | Self-consuming training loops · STL (general) | generalization error: the expression \left(1-(1-\alpha)^i\right) \alpha^{-1} remains bounded | Helped |
| Use a constant real-data proportion to control recursive stability | Transformer ICL · Transformer ICL in STL | recursive stability: \gamma_n^i=\mathcal{O}(1 / n) | Helped |
| Increase the synthetic augmentation coefficient \lambda | Accumulating synthetic data · Synthetic data augmentation in STL | cumulative distribution shift error: n^{-\frac{1}{4}} \log ((1+i \lambda) n) M \log (1/\delta) | Hurt |
| Increase the synthetic augmentation coefficient \lambda | Accumulating synthetic data · Synthetic data augmentation in STL | generalization error on mixed distributions: n^{-1} \frac{\rho^2}{(1+i \lambda)^2} \log ((1+i \lambda) n) i!\widetilde{B}_W^{(i+1) L} \log \frac{1}{\delta}+n^{-\frac | Helped |
| Increase the optimal synthetic augmentation coefficient as real dataset size decreases | Accumulating synthetic data · Synthetic data augmentation in STL | optimal augmentation coefficient: λ* increases as the size of the real dataset n decreases | Helped |
| Use no real data in the recursive training loop | Gaussian mixture model · Gaussian mixture model in STL | recursive stability: \gamma_n^i \lesssim i n^{-1/2} \log \frac{nd}{\delta} | Hurt |
| Use synthetic data to augment the real dataset | Transformer-based STL · Synthetic data augmentation in STL | overall generalization error: n^{-\frac{1}{4}} \log ((1+i\lambda) n) M \log (1/\delta); n^{-1} \frac{\rho^2}{(1+i\lambda)^2} \log ((1+i\lambda) n) i!\ | Mixed |
Surprising findings
- Synthetic augmentation trades off distribution shift against mixed-distribution generalization, so more synthetic data is not uniformly beneficial.
Limitations
- Results are theoretical bounds with assumptions (smoothness, Lipschitz, bounded norms) and no large-scale experiments are reported in the text.
- Bounds use a TV-distance term assumed to be of the same order across generations.
Experimental setup
Metrics: generalization error bound recursive stability parameter TV distance
Baselines: Gaussian model collapse analyses Bertrand et al. STL analysis Fu et al. diffusion TV bounds
Builds on: uniform stability SGD stability model collapse analyses (Shumailov et al.) conditional i.i.d. analysis of synthetic data
Closest papers by meaning
- Self-Correcting Self-Consuming Loops for Generative Model Training (ICML 2024 · similarity 0.92)
- Collapse or Thrive: Perils and Promises of Synthetic Data in a Self-Generating World (ICML 2025 · similarity 0.91)
- On the Stability of Iterative Retraining of Generative Models on their own Data (ICLR 2024 · similarity 0.90)
- Towards Theoretical Understandings of Self-Consuming Generative Models (ICML 2024 · similarity 0.90)
- Stability and Diversity of Networked Self-Consuming Generative Ecosystems (NeurIPS 2026 · similarity 0.89)
- Self-Consuming Generative Models with Curated Data Provably Optimize Human Preferences (NeurIPS 2024 · similarity 0.89)
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