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Preventing Model Collapse: A Fisher-Rao Perspective on the Dynamics of Training with Synthetic Data

Matteo Marchi, João Pedro Silvestre, Bahman Gharesifard, Paulo Tabuada

arXiv:2609.18878v1cs.LGcs.SYeess.SY

TL;DR

The paper asks how much fresh human data is needed to prevent model collapse, given that earlier Euclidean analysis can become uninformative at high dimensions. It models iterative training under the Fisher-Rao metric and proves that, above a threshold, trajectories converge to a stable Fisher-Rao ball; the resulting effective data requirement is greater than previously implied.

  • Problem

    The minimum human-to-synthetic data ratio needed to prevent model collapse remains unclear, and Euclidean bounds can become uninformative as dimensions grow.

    Source: §I · p. 1

  • Method

    The paper models iterative training as a closed-loop stochastic process and derives contraction and invariance bounds using the Fisher-Rao metric on the probability simplex.

    Source: §I · p. 1, §III · p. 3

  • Results

    Above a specific human-to-synthetic data ratio threshold, trajectories converge to a stable ball in the Fisher-Rao metric.

    Source: §III · p. 3, §V · p. 6

  • Takeaways & Limitations

    The effective amount of human data required to prevent collapse is greater than previously implied, with bounds that scale properly with the dimension of categorical distributions.

    Source: §I · p. 1, §V · p. 6

Abstract

from arXiv · show

Large Language Models (LLMs) are now routinely trained using synthetic data, since high-quality human data has been exhausted by the ever increasing needs of larger and larger models. However, recursive training on synthetic data frequently induces model collapse, a degenerative feedback loop where models progressively forget the true underlying data distribution. Training on a mixture of synthetic and fresh human data is a logical countermeasure and can prevent model collapse. However, it is an open question as to what is the exact minimum required ratio of human-to-synthetic data to maintain training stability. In this paper, we establish rigorous theoretical guarantees on the minimum rate of human data required to prevent model collapse. Although previous work established a formal lower bound for this ratio, such bound can be vacuous for very high dimensions, as the analysis relies on the usual Euclidean metric in R^n and is not adapted to the space of categorical probability distributions. Instead, in this paper we explicitly leverage the information-geometric structure of the probability simplex by analyzing the dynamics of the process under the Fisher-Rao metric. We derive quantitative contraction and invariance bounds that are stable and do not become trivial as the dimensions increase. Thus, we show that the effective required data ratio to prevent model collapse is different than previously implied.

I. INTRODUCTION

Growing training-data demands have made synthetic text attractive, but recursive synthetic training can cause model collapse, and the required human-data ratio remains an open question. The paper analyzes this question on the probability simplex using the Fisher-Rao metric, deriving bounds intended to remain meaningful as dimension grows.

I. INTRODUCTION

  • Training foundational models on trillions of tokens intensifies concerns about sustaining the supply of fresh, high-quality human text.The passage continues that this supply is projected to be exhausted, as cited in [7].Source: §I · p. 1
  • Recursive synthetic training can drive models away from the true data distribution, while small fixed proportions of human data may be insufficient to stop collapse.The paper frames the minimum human-data requirement as an unresolved question.Source: §I · p. 1
  • The paper models iterative training as a closed-loop stochastic process and uses the Fisher-Rao metric to derive dimension-stable contraction and invariance bounds.It reports that the effective human-data requirement is greater than previously implied.Source: §I · p. 1

A. Notation

This section establishes notation for Euclidean spaces, norms, probability distributions, and asymptotic growth. It also notes properties of the KL divergence and the relationship between the metric and geodesic Hellinger distance.

A. Notation

  • The notation includes R^n, R+, the probability simplex, the 1- and 2-norms, and the inner product.The passage also introduces natural numbers and nonnegative reals.Source: A. Notation · pp. 1–2
  • O(·) and Θ(·) denote standard asymptotic bounds for the limiting behavior of sequences.The section specifies their definitions using constants and an index threshold.Source: A. Notation · p. 2
  • The metric on the probability simplex induces the geodesic Hellinger distance.Source: A. Notation · p. 2
  • The KL divergence is nonnegative and equals zero if and only if its two distributions are equal.Source: A. Notation · p. 2

B. Generative Models

The paper represents a generative model as a map from parameters to a probability distribution over categorical outcomes. It also describes temperature-scaled softmax as the practical transformation from logits to probabilities.

B. Generative Models

  • A generative model maps a parameter vector to an output distribution over a finite list of categorical outcomes.Each distribution entry is the nominal probability of producing the corresponding outcome.Source: B. Generative Models · p. 2
  • The categorical outcomes are represented as one-hot vectors.The paper identifies this representation as one-hot encoding.Source: B. Generative Models · p. 2
  • A temperature function converts temperature-scaled raw logits into probabilities using the standard softmax form.The paper describes this form as ubiquitous across nearly all modern generative models.Source: B. Generative Models · p. 2

C. Iterative Training

The iterative-training model combines samples generated by the current model with samples from a fixed external distribution, then abstracts training as an update to model parameters. The section motivates the Fisher-Rao analysis by contrasting it with Euclidean bounds that become uninformative as the categorical dimension grows.

C. Iterative Training

  • Each training step adds samples from the current model’s temperature-adjusted distribution and from a fixed external human distribution.The resulting dataset is used to update model parameters, with the training procedure abstracted as a function.Source: C. Iterative Training · p. 2
  • The paper’s prior-work account states that, without fresh data, iterative training can drive distributions toward simplex corners or the uniform distribution, a form of model collapse.The paper describes the degeneration as occurring with high probability in the limit.Source: C. Iterative Training · p. 3
  • The earlier Euclidean analysis yields a convergence ball, but the paper says its bounds become progressively less informative as the simplex dimension grows.For n approaching infinity, the passage says such a ball may encompass most of the simplex regardless of the human-data ratio.Source: C. Iterative Training · p. 3
  • The Euclidean distance between the cited family of distinct categorical distributions vanishes as O(1/√n), whereas their Fisher-Rao distance remains π/2 regardless of n.The paper argues that Euclidean bounds consequently under-penalize distributional distance and can overstate the adequacy of a human-data ratio.Source: C. Iterative Training · p. 3

III. MAIN RESULT

The main theorem characterizes conditions under which iterative training converges to a Fisher–Rao ball, including a convergence rate and a minimum human-to-synthetic data threshold. Its guarantee is conditional on stated assumptions and an explicit ratio-related inequality.

III. MAIN RESULT

  • The assumptions require bounded perturbations, a human distribution away from the simplex boundary, and an equilibrium when the perturbation is zero.The theorem setup also specifies that every entry of the human distribution is strictly positive.Source: §III · p. 3
  • The theorem identifies a Fisher–Rao convergence ball, the rate of convergence to it, and a minimum human-to-synthetic data threshold.The result is stated under specified assumptions and a condition on the ratio-related parameters.Source: §III · p. 3
  • The theorem requires μδ > η and considers times t ≥ tκ, with κ ∈ [0, 1).It also assumes Assumptions 1–2 and a further inequality.Source: §III · p. 3

2) In the Fisher-Rao case (9):

The scaling analysis shows that human-data growth proportional to n can leave the Fisher–Rao error growing, even when the Euclidean estimate predicts decay. Without additional structure on the equilibrium distribution, μ_n = n^(5/2+γ), γ > 0, keeps the Fisher–Rao error uniformly controlled as dimension grows.

2) In the Fisher-Rao case (9):

  • Under δ_n ∼ n^-β0 and μ_n ∼ cn^p, the proof establishes Fisher–Rao error decay for p > 2β0.The proof derives the scaling from the bound and equilibrium-coordinate estimates.Source: 2) In the Fisher-Rao case (9) · p. 4
  • Without structural information about θ_e beyond simplex constraints, μ_n = n^(5/2+γ), γ > 0, keeps the Fisher–Rao error uniformly controlled as dimension grows.The passage gives this as a natural choice for obtaining, for example, O(1/n) decay.Source: 2) In the Fisher-Rao case (9) · p. 4
  • At μ_n ∼ n, the Euclidean estimate predicts error of order 1/n, but the Fisher–Rao error can still grow.The analysis therefore indicates that stronger growth of the human-to-synthetic ratio is required.Source: 2) In the Fisher-Rao case (9) · p. 4

IV. PROOF OF THE MAIN RESULT

The proof establishes an eventual coordinate-wise floor and uses Fisher-Rao bounds to obtain the stated trajectory bound. A KL-divergence Lyapunov function supports the analysis while trajectories remain inside the simplex.

IV. PROOF OF THE MAIN RESULT

  • The paper first proves that solutions of (5) cannot approach the boundary asymptotically, before proceeding through intermediate lemmas to the main result.Source: §IV · p. 4
  • The proof uses KL divergence from the equilibrium as a Lyapunov function, with the coordinate floor enabling Fisher-Rao analysis where the metric is undefined on the boundary.The floor is independent of the metric and follows from the dynamics.Source: §IV · p. 5
  • The proof establishes the stated trajectory bound with ϵ_FR matching (9).The passage says this establishes (8), without giving the bound's formula.Source: §IV · p. 6

V. CONCLUSION

The paper bounds the human-data requirement for preventing model collapse using Fisher-Rao dynamics. It reports convergence to a stable ball above a threshold and bounds that scale with the dimension of the categorical distributions.

V. CONCLUSION

  • Above a specific human-to-synthetic data-ratio threshold, trajectories converge to a stable ball in the Fisher-Rao metric.The conclusion frames this as a bound on the human data required to prevent model collapse.Source: §V · p. 6
  • The Fisher-Rao approach replaces prior Euclidean analysis of the training dynamics.The conclusion identifies the probability simplex's information-geometric structure as the basis for the approach.Source: §V · p. 6
  • The resulting bounds scale properly with the dimension of the underlying categorical distributions.The conclusion attributes this scaling to accounting for distortions on the probability simplex.Source: §V · p. 6

APPENDIX

The appendix presents technical lemmas used in the main result, including scalar inequalities and bounds on coefficients.

APPENDIX

  • The appendix says its technical lemmas support the main result.One displayed scalar inequality is established for all r > 0.Source: APPENDIX · p. 7
  • The constants in Lemma 8 are c1 = θ/(2 max_i θe,i) and c2 = 1/(2 min_i θe,i).These constants bound the expression in that lemma.Source: APPENDIX · p. 8
  • The appendix states a two-sided bound on g(r) in terms of min{r, 1}, max{r, 1}, and (log r)^2 for r > 0.The lower and upper coefficients are both 1/2.Source: APPENDIX · p. 8

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