Rate of Model Collapse in Recursive Training

TL;DR

This work analyzes how model quality degrades under recursive training where each generation is trained on data generated by the previous model, formalized as Θ_k = hatθ(X_{k,1},...,X_{k,n}) with X_{k+1,i} ∼ P_{Θ_k}. Focusing on fundamental distributions under ML/near-ML estimation, it derives explicit collapse rates: for discrete unigram models, the survival probability of a symbol satisfies Pr(Θ_{k,i} ≠ 0) ≥ 1 − exp(−λ_i/k), indicating slow forgetting; for the one-dimensional Gaussian model, the variance Σ_k^2 collapses to zero with Pr(Σ_k > ε) ≤ (σ_0/ε) exp(−k/(4n)), implying linear-in-n timing to collapse; for Gaussian mixtures, an approximate joint-ML estimator yields analogous exponential decay bounds. The results are extended to Bernoulli and Poisson processes, and validated by experiments showing similar scaling, suggesting that forgetting can be slow even with large per-generation sample sizes. Collectively, the paper provides explicit rates and martingale-based analyses that deepen our understanding of model-collapse dynamics under synthetic-data-driven recursion and inform practical considerations for data composition in iterative training.

Abstract

Given the ease of creating synthetic data from machine learning models, new models can be potentially trained on synthetic data generated by previous models. This recursive training process raises concerns about the long-term impact on model quality. As models are recursively trained on generated data from previous rounds, their ability to capture the nuances of the original human-generated data may degrade. This is often referred to as \emph{model collapse}. In this work, we ask how fast model collapse occurs for some well-studied distribution families under maximum likelihood (ML or near ML) estimation during recursive training. Surprisingly, even for fundamental distributions such as discrete and Gaussian distributions, the exact rate of model collapse is unknown. In this work, we theoretically characterize the rate of collapse in these fundamental settings and complement it with experimental evaluations. Our results show that for discrete distributions, the time to forget a word is approximately linearly dependent on the number of times it occurred in the original corpus, and for Gaussian models, the standard deviation reduces to zero roughly at $n$ iterations, where $n$ is the number of samples at each iteration. Both of these findings imply that model forgetting, at least in these simple distributions under near ML estimation with many samples, takes a long time.

Figures (13)

• Figure 1: Simplified recursive training procedure of shumailov2023curse. The underlying dataset is denoted by $S_0$. The model parameter $\Theta_k$ is trained based on dataset $S_{k}$ and sample $S_{k+1}$ is obtained by sampling from model with parameter $\Theta_k$.
• Figure 2: Multiple trajectories of recursive training across generations in the Gaussian and Bernoulli models, with $n=100$ samples per generation.
• Figure 3: Comparison of theoretical bounds and empirical observations for $\Pr(P_k \neq 0)$ at a particular generation in a Bernoulli recursive training process with $n$ samples at each generation. Results are shown for varying initial success probabilities $(p_0)$.
• Figure 4: Comparison of theoretical estimate and empirical observations for $\Pr(\Lambda_k \neq 0)$ at a particular generation in a Poisson recursive training process with $n = 10$ samples at each generation. Results are shown for varying initial arrival rate $\lambda_0$.
• Figure 5: Comparison of theoretical upper bounds with empirical observations of $\text{Pr}(\Sigma_k>\epsilon)$ in Gaussian recursive training process $(\epsilon = 10^{-1}, n = 10)$.

Theorems & Definitions (23)

• Definition 1
• Theorem 1
• Theorem 2
• Theorem 3
• Corollary 1
• Theorem 4
• Lemma 1
• proof
• Lemma 2
• proof