Synthetic data are widely used for training modern AI systems. However, their statistical understanding remains limited. It is well recognized in statistics that repeated self-training on synthetic data can amplify errors inherited from pre-trained models, leading to model collapse. Verification plays a crucial role in selecting high-quality synthetic data to help prevent model collapse and improve model performance. We study verified self-training in a linear regression framework, where a pre-trained estimator generates synthetic data with injected generation noise and a verifier selects samples according to a verification threshold. Post-training on the accepted samples updates either all coordinates (full fine-tuning) or only an active subset (partial fine-tuning). Our analysis reveals three potentially counterintuitive effects. We establish that tighter verification does not always lead to lower prediction risk. We also find that a suitable level of injected generation noise can reduce prediction risk, while excessive noise can increase it. Moreover, full and partial fine-tuning can respond differently to the same verification rule, since pre-training error in the frozen coordinates affects both sample selection and post-training risk. We analyze these effects by deriving finite sample risk bounds for four separate regimes determined by the relative scales of the verification threshold, generation noise, and pre-training error. The bounds precisely characterize how verification affects prediction risk across these regimes. In particular, we quantify how the generation noise, the effective accepted sample size, and the covariance of the accepted covariates affect prediction risk. The analysis extends to iterative verified full fine-tuning, where sufficiently tight verification reduces the initial error contribution at a geometric rate.
Figures & tables
Figure 1: Excess prediction risk under verified post-training. Panel (a) displays risk against the verification threshold under partial fine-tuning, panel (b) shows risk against generation noise under full fine-tuning. In panel (c), we compare full and partial fine-tuning across verification thresholds when generation noise is small. For partial fine-tuning, the excess risk excludes the direct contribution from the frozen coordinates.
Figure 2: Post-training with verified synthetic data.
Figure 3: Full fine-tuning with s=d=20 , n=2000 , and m=3×106 . Simulation estimates of V1+V2 are plotted against the verification threshold δ , with σ~ fixed in each panel. Tighter verification reduces the variance in Regime I even when δ exceeds d/n . The variance is smallest in Regime II, increases in Regime III, and remains near d/n in Regime IV, where tight verification provides little improvement when generation noise is too small.
Figure 4: Full fine-tuning with s=d=20 , n=2000 , and m=3×106 . Simulation estimates of V1+V2 are plotted against the generation noise σ~ under (a) tight and (b) loose verification. In both cases, the variance is minimized at an intermediate generation noise level. Nearly deterministic generation leaves the variance near d/n , whereas excessive generation noise increases it.
Figure 5: Partial fine-tuning with s=10 , d=20 , n=2000 , and m=3×106 . Simulation estimates of V1+V2 are plotted against δ , with σ~ fixed in each panel. Regime I exhibits the same reduction as under full fine-tuning, while the frozen coordinates change the behavior in the Regimes II and III. The U-shaped graph in Regime II reflects the tradeoff between the error in the accepted responses and the frozen coordinate contribution, and the variance remains near s/n in Regimes III and IV.
Figure 6: Partial fine-tuning with s=10 , d=20 , n=2000 , and m=3×106 . Simulation estimates of V1+V2 are plotted against σ~ under (a) tight and (b) loose verification. In both cases, sufficiently large generation noise is needed to reduce the variance below s/n , while excessive generation noise increases it. The variance is therefore minimized at an intermediate generation noise level.
Regime
Scales
V1 terms
V2 terms
No verification
all generated samples
s/n
sσ~2/m
I
d/n≲δ
nsmin(1,(δ/σ~)4)
mpIsmin(δ2,σ~2)
II
δ≲d/n≲σ~
ns((nσ~2+d−sd−s)2+(δ/σ~)4)
mpIIs(δ2+nσ~2+d−sσ~2(d−s))
III
δ≲σ~≲d/n
ns((nσ~2+d−sd−s)2+(δ/σ~)4)
mpIIIs(δ2+nσ~2+d−sσ~2(d−s))gd−s,d(d/n/σ~)
IV
σ~≲δ≲d/n
s/n
mpIVsσ~2gd−s,d(d/n/δ)
Table 1: Leading active coordinate variance terms for partial fine-tuning (s<d) .
Figure 7: Test risk across verification thresholds and generation noise scales, with n=2048 , d=762 , and m=32,000 . Panel (a) varies the threshold at σ~≈1.58Rpre , while panel (b) presents the best average test risk ratio among the 11 examined thresholds for each generation noise scale. Each setting averages 40 runs. The direct fitting reference fits the linear model directly on all 7229 target human labels.
Figure 8: Test risk across verification thresholds and generation noise scales at a fixed accepted sample size, with n=2048 and d=762 . Across 16 attempted runs per setting, every fitted estimator uses 3048 accepted samples, and the star marks the minimum test risk ratio in each panel.
Synthetic data has been increasingly used to train frontier generative models. However, recent studies raise key concerns that iteratively retraining a generative model on its self-generated synthetic data may keep deteriorating model performance, a phenomenon often coined model collapse. In this paper, we investigate ways to modify the synthetic retraining process to avoid model collapse, and even possibly help reverse the trend from collapse to improvement. Our key finding is that by injecting information through an external synthetic data verifier, whether a human or a better model, synthetic retraining will not cause model collapse. Specifically, we situate our theoretical analysis in the fundamental linear regression setting, showing that verifier-guided retraining can yield near-term improvements, but ultimately drives the parameter estimate to the verifier's "knowledge center" in the long run. Our theory further predicts that, unless the verifier is perfectly reliable, these early gains will plateau and may even reverse. Indeed, our experiments across linear regression, Variational Autoencoders (VAEs) trained on MNIST, and fining-tuning SmolLM2-135M on the XSUM task confirm these theoretical insights.
Bingji Yi, Qiyuan Liu, Yuwei Cheng +1
Independent Researcher. Work done while visiting UChicago CS. · Work done while visiting UChicago CS. · Department of Statistics, University of Chicago. +1
Synthetic data has become a promising way to scale model training beyond limited human-generated data but it may also induce strong model collapse (Dohmatob et al., 2024), where any fixed fraction of synthetic data prevents model performance from improving under data scaling, leaving a non-vanishing excess risk floor. In this paper, we study how synthetic data affects the generalization of one-pass SGD in high-dimensional linear regression with model shift. We establish finite-sample risk bounds for mixed and two-stage training, separating standard bias and variance from source-mismatch effects, namely fluctuation and persistent drift under mixing and filtered initialization bias under two-stage. These bounds reveal a sharp contrast: mixed training induces strong model collapse, while two-stage training avoids the floor by using synthetic data only in the first stage, showing that collapse is not inevitable under a simple data curriculum. Under a random sketch model, we further obtain scaling laws for both protocols, with tight results for mixed training in the optimization-saturated regime. These laws show that larger models may amplify synthetic-induced degradation under mixing, and quantify how high-quality synthetic pretraining may reduce bias in two-stage training. Finally, we establish an exact finite-sample necessary-and-sufficient condition for two-stage training to strictly outperform real-only training under the same real-data budget and identical real-stage updates. Overall, our results highlight that synthetic data is neither inherently harmful nor beneficial; its effect depends critically on both its quality and the training protocol used to incorporate it.
Jichu li, Difan Zou
Renmin University of China · The University of Hong Kong
Synthetic data are increasingly used to scale LLM training, yet more synthetic data do not necessarily produce better models. Useful synthetic data must add information relevant to the target task without introducing errors that offset their benefit, and the value of an example can change as the training set grows. We develop a linear theory that characterizes this tradeoff and determines where synthetic data are useful, how much should be added, and the marginal value of adding one example to an existing set. The analysis shows the conditions when input coverage alone is sufficient and when synthetic errors must also be considered. Guided by these results, we introduce Training-Aware Target Coverage (TATC), a synthetic data selection method for LLM fine-tuning. TATC identifies candidates whose training effects are beneficial to the target task and selects among them to expand coverage of target-relevant directions not already represented by the available data. Experiments on text and image data verify the linear theory. With mathematical reasoning tasks, TATC selects synthetic solutions for fine-tuning Qwen2.5-Math-1.5B-Instruct and outperforms alternative synthetic-data selection methods on GSM8K across selection budgets. In summary, we provide a principled approach to synthetic data selection by quantifying and maximizing its value to the target task.
Yang Ba, Michelle V. Mancenido, Rong Pan
School of Computing and Augmented Intelligence, Arizona State University · School of Mathematical and Natural Sciences, Arizona State University