Escaping Model Collapse via Synthetic Data Verification:  Near-term Improvements and Long-term Convergence
Authors: Bingji Yi
Qiyuan Liu
Yuwei Cheng
Haifeng Xu
Venue: ICLR 2026 Poster (2026)
Keywords: Model Collapse
Synthetic Data
Verifier-guided retraining
Primary area: learning theory

Abstract: 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.

Published as a conference paper at ICLR 2026
ESCAPING MODEL COLLAPSE VIA SYNTHETIC DATA VERIFICATION: NEAR-TERM IMPROVEMENTS AND
LONG-TERM CONVERGENCE
Bingji Yi'
Qiyuan Liu'
Independent Researcher
Department of Statistics
yibingji ji gmail. com
University of Chicago
qiyuanl i an@uchicaagon edu
Yuwei Cheng
Haifeng Xu
Department of Statistics University of Chicago
Department of Computer Science
University of Chicago
yweeichee.guuchhcasp..edd
MaiiEngxu@uchhee www.ugg..cccooooooo edu
ABSTRACT
Synthetic data has been increasingly used to train frontier generative models. How- ever, recent studies raise key concerns that iteratively retraining 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 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 in- jecting information through an external synthetic data verifier, whether a human or model collapse. Specifically, we
better model, synthetic retraining will not 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 Mm05LMM2133MM the XSUM task confirm these theoretical insights.
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Figure 1: Iterative VAE Retraining on MNIST. Left: Original MNIST images (real data). Middle: Samples from a VAE trained on 500 real images. Right: Samples after 40 rounds of synthetic retraining. The top branch (green) uses verifier-fittere synthetic data, producing clearer and more realistic digits: the bottom branch (red) retrains without verification, leading to severe degradation and mode collapse. The final column shows VAE trained all 60K real images (upper bound quality).
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INTRODUCTION
The use of synthetic data has gained significant traction due to its ability to reduce data collection costs and enhance privacy protection, with applications in computer vision (Wood et al., 2021), healthcare (Azizi et al., 2021; Santangelo et al., 2025), finance (Potluru et al., 2023) and recently large language models (LLMs) (Chen et al., 2024). A growing body of work has demonstrated that training with synthetic data can improve model performance in various applications such as image recognition (He et al., 2023; Tremblay et al., 2018), image generation (Doersch & Zisserman, 2019; Shrivastava et al., 2017; Tian et al., 2023) and language generation (Gunasekar et al., 2023; Guo et al., 2024; Zelikman et al., 2022). However, recent studies caution that recursively training models synthetic data alone can lead to a degradation of quality, phenomenon often coined model collapse (Shumailov et al., 2024; Dohmatob et al., 2024a; 2025; 2024b; Alemohammad et al., 2024; Gerstgrasser et al., 2024). This contrast between empirical success and pessimistic research findings gives rise to a natural question about what might have caused the discrepancy.
In response to the above question, important practical observation is that symthetic data rarely used in raw form in the aforementioned applications. Instead, practitioners often apply filtering steps to remove low-quality synthetic samples before retraining. For example, in natural language generation, synthetic sentences are often screened using grammar checkers or LLM--s-a-judge pipelines (Zheng al., 2023; Gu al., 2024); in computer vision, synthetic images can be filtered using pretrained discriminators or human annotation (He et al., 2023); in recommendation and preference learning, synthetic feedback often validated against external heuristics or known user signals (Tu et al., 2025; Iskander et al., 2024; Lupidi et al., 2024; Lampis et al., 2023; Zhang et al., 2024). Common across all these approaches the use of knowledgeable "discriminator" (henceforth the verifier ) whether machine or human that evaluates and filters out low-quality candidate synthetic samples (i.e., those not passing the discriminator's screening). This observation naturally raises the following research question:
Does verifter-base baafer filtering of synthetic data contribute to the observed empirical success of model improvements, and does it prevent model collapse in the long run?
There has been growing body of work studying the mechanisms of model collapse, including theoretical analyzes that are often examined in the context of classic parameter estimation problems (Dohmatob et al., 2024a; 2025; Gerstgrasser et al., 2024; Dey & Donoho, 2024; Xu et al., 2025; Suresh et al., 2025). However, these works have all assumed the use of synthetic data without filtering. Only few recent works start to examine how filtered synthetic data affects the performance of iterative retraining, but under idealized assumptions such access perfectly correct verifier (Amin et al., 2025) or highly structured errors in synthetic data (i.e., i.i.d. noise added to binary labels (Feng al., 2025)). A realistic and instructive framework for analyzing synthetic retraining of generative models remains still poorly understood. In this paper, we further push the boundary of this important agenda, and examine iterative retraining under imperfect verifiers that filter out low-quality synthetic data based on their (possibly biased) knowledge. We refer to this process as verifier-base synthetic retraining for convenience. Specifically, we seek principled understandings about the empirically observed short-term successes of verifier-based synthetic retraining and analysis about its long-term convergence under iterative retraining.
Our contributions. We start from theoretical investigations and analyze verifier-based synthetic retraining with verified synthetic data on the foundational linear regression model a canonical setting that has become central to the study of model collapse (e.g., (Dohmatob et al., 2024a; 2025; Gerstgrasser et al., 2024; Zhu et al., 2025; Garg et al., 2025)).² 1 We then verify our theoretical insights through thorough empirical studies in real-world generative settings. Our main contributions are summarized as follows:
Does verified synthetic data improve retraining and, if so, under what conditions? We show that it indeed can, provided the right conditions are met. Through new form of bias- variance trade-off under data filtering, we characterize the regimes in which verifier-baeed synthetic retraining leads to strict model improvement, rather than degradation, in the short
The theoretical analysis of (Feng al., 2025) is also hased linear models, specifically, linear classifiers.
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temn (Theorem 3.1). The conditions we identify highlight the mixed effect of synthetic sample size, the verifier's 's bias and selectivity during filtering, yielding practical insights regarding when verification of synthetic data is beneficial.
Can the retraining improvement in the short term be sustained in the long run? Our theoretical analysis shows that the answer is No - unless the verifier has no bias. Formally, we show that verifier-base synthetic retraining will converge to the verifier's knowledge center in the long term (Theorem 4.1). This result reveals how the verifier's quality affects the asymptotic dynamies of iterative retraining. Notably, while verifier selectivity influences short-term performance, it does not change the long-run converging point, though it does affect its convergence rate.
Do the insights from the above theoretical analysis apply to real generative models em- pirically? Indeed, both simulations of linear regression settings and training Variational Autoencoders (VAEs) on the real-world MNIST data demonstrate that our theoretical in- sights align with observed training dynamics in these settings. For example, Figure I shows how the generative model of VAE can start from a poor model trained on a small number of 500 images and gradually improve from being retrained on filtered synthetic imaged generated by itself, until after 40 iterations of retraining generating sharp images that are vi- sually comparable to a VAE trained directly on the entire MNIST dataset. More quantitative analysis of our empirical studies can be found in Section 5.
1.1 RELATED WORK
Understanding and mitigating model collapse. Recent work shows that heavy reliance on syn- thetic data in iterative training can cause model collapse-the degradation of performance when a model is repeatedly retrained on its own synthetic outputs (possibly mixed with real data).² Empirical evidence supports this phenomenon: Shumailov et al. (2024) show that recursive training on unfiltered synthetic data induces distribution shift and mode collapse, while Dohmatob et al. (2025) find that even small synthetic proportions can harm performance. In linear settings, Dohmatob et al. (2024a) analyze collapse mechanisms explicitly, and Dohmatob et al. (2024b) connect degradation to altered neural scaling laws.
To mitigate collapse, prior work broadly explores three strategies. First, accumulating data or gradually increasing the synthetic dataset size across iterations can suppress noise and bound errors (Gerstgrasser et al., 2024; Dey & Donoho, 2024; Xu et al., 2025; Kazdan et al., 2025; Barzilai & Shamir, 2025). Second, mixing synthetic data with real data stabilizes retraining (Bertrand et al., 2024; Fu et al., 2024; 2025), as performance progressively degrades without sufficient fresh real data (Alemohammad et al., 2024). Recent studies have even derived optimal mixing ratios to maximize this stabilizing effect (He et al., 2025; Garg et al., 2025). Finally, algorithmic interventions, such as the token re-sampling procedures proposed by Zhu et al. (2025), offer alternative pathways to avoid collapse.
Unlike prior work that relies unfiltered synthetic data, our framework incorporates external verifier to remove low-quality samples. Such verifiers may be human annotators or stronger teacher models. Filtering is widely used in iterative retraining and has shown empirical success in preventing model degradation and even improving performance (He et al., 2023; Tian et al., 2023; Guo et al., 2024; Zelikman et al., 2022; Zhang et al., 2024; Lampis et al., 2023; Haluptzok et al., 2023; Patwa et al.. 2024). Motivated by this, we develop a principled understanding of when improvement is possible Dosiblle-nmmely whether generative model can leverage the verifier's feedback, embedded in the selected synthetic subset, to achieve sustained gains.
Filtering and selecting synthetic data. While rich line of empirical work demonstrates that these filtering strategies can improve model performance, theoretical understanding about iterative retraining with filtered synthetic data remains largely unexplored, with only a few recent exceptions. Amin et al. (2025) assume strong, reliable quality function and focus how an external labeler aids learning under this fixed filtering mechanism. Feng ct al. (2025) study a classification problem and identify sharp phase transition. However, modeling synthetic data merely as noisy labels abstracts away the structural dependencies between features and labels inherent to true generative processes. Finally, Ferbach et al. (2024); Wei & Zhang considers a conceptually similar problem of
"There is no widely agreed-upon formal definition; see (Schaeffer al., 2025) for discussion.
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learning discrete preference distribution from human feedback by using humans' preferred choices as a filtering strategy. In their population- analysis, curating synthetic data via an external reward function forces the model distribution to converge to the highest-reward level set, maximizing expected reward but ultimately collapsing in diversity.
Similar to many of the aforementioned studies above, our theoretical analysis also focuses linear models (Feng et al., 2025; Dohmatob et al., 2025; Gerstgrasser et al., 2024). However, our model allows inaccuracy of the verifier in terms of both bias and variance. Errors in the synthetic data primarily stem from the inaccuracy of the generative model itself rather than exogenous noise. We show that model's short-term performance varies smoothly with the verifier's bias, selectivity, and size of synthetic data, rather than exhibiting a sharp phrase transition from complete failure to perfect accuracy as in Feng al. (2025). In the long run, the model's performance converges the verifier's knowledge center whereas verifier's selectivity only affects convergence speed. Our results bridge short-term and long-term perspectives of iterative retraining, illustrating how varied verifier qualities give rise to distinct performances of iterative retraining.
Comparison with reward maximization frameworks. While both use external feedback to evalu- ate generated data, approach substantially differs in various aspects from frameworks such as preference matching (Ferbach et al., 2024; Wei & Zhang) and RLVR (Guo et al., 2024; Yu et al., 2025). We provide detailed comparison of these paradigms in Appendix B.
2 MODELING VERIFIER-BASED SYNTHETIC RETRAINING: THE LINEAR REGRESSION CASE
In this section, we formalize our model of iterative retraining with verified synthetic data, coined verifier-based synthetic retraining for convenience. Following recent works in this space (Dohmatob et al., 2024a; Gerstgrasser et al., 2024; Garg et al., 2025; Zhu et al., 2025; Garg et al., 2025), we focus on the foundational linear regression setting where the objective is to estimate a high-dimensional coefficient vector f in the following linear model
x 0 +
where { ~ N0( x E RR, and 0* IRP is the unknown parameter interest. We use the standard Mean Squared Error (MSE), i.e., MSB(0) = Eg 10 112, to evaluate estimators.
Modeling the verifier and data filtering rule. Suppose we have access verifier that possesses prior knowledge of 8*, modeled by a knowledge set. Specifically, the verifier's 's knowledge is described by spherical ball:
(,) = (0 E R 110 - 0oll sr}.
with fixed center 0. and radius r. We assume this knowledge set indeed contains the true parameter, i.e., O* E (0a), but the true parameter 0* is unknown. The verifier does not reveal Og or directly (see modeling motivations below). Instead, it only provides binary feedback indicating whether a given (real or synthetic) data point (I,, 31) is consistent with the knowledge 0* (,((.) or not. Specifically, the verifier outputs Yes if
y: x 0cl S r1|1::11 +
and No otherwise. Here ae is constant related the verifier's capability. This filtering rule is motivated by the following bound on expected errors: E [ys x 6el E[1, (0+ - 0.) &il] S
r r| + E r||x:|| + V 2
Since the true might be unknown in practice, da serves as
estimate of the true a.
We refer to A 1000 as the bias of the verifier, whereas r captures the selectivity of the verifier the smaller r is, less likely the verifier accepts a data point (I;, 31). The verifier only needs to provide Yes/No answers based the above selection rule in equation 1, but does not needs to know the parameter f.., r of the knowledge set. The motivation of this modeling primarily comes from practice, explained below.
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Motivation of binary feedback from verifiers. We adopt the binary feedback from verifiers mainly for practical reasons. In practice, eliciting simple yes/no feedback is far less noisy and more cost-effective than asking verifiers directly specify 9g or r. Indeed, in real applications verifiers may not even know these quantities explicitly, which would correspond to model parameters the verifier is stronger teacher model or how the human reasons if the verifier is human. This model choice is also aligned with the widely adopted comparison- learning from human feedback (RLHF) (Ouyang et al., 2022). Such binary feedback has become standard approach in preference alignment for large language models, where LLM raters and human evaluators provide pairwise or accept/reject judgments that effectively guide learning at scale (Wettig et al., 2024). Although simple, our theory and empirical evaluations both show that this single bit of information for each sample can successfully be injected into the retraining process to improve models.
feedback in reinforcement
Synthetic Retraining with Verifier-based Filtering We begin with an initial set of real data (X", Y), where X R and y _0 The initial estimator 0 is obtained via Ordinary Least Squares (OLS) 3 80 = (xx - 'x y We then proceed with iterative synthetic retraining via the generate-vertf--eetaal procedure outlined in Figure 2, the rigorous retraining Scheme 2 and Algorithm are provided in Appendix C.
O*
Generate
(xk+1, y
Verify
(X++1) y
Retrain
Ge+1
Figure 2: Generate- Verify- Retrry-Retrain pipeline.
Since learning proceeds through the conditional y X* synthetic retraining requires specifying the covariate design X*; labels y are then generated conditionally via the model under verifier constraints. In principle, one could construct X arbitrarily: however, for mathematical clarity, below we describe a targeted though arguably natural design. In particular, we choose to align the synthetic covariates with a fixed orthonormal set (Vi,..., up} and construct X a block-structured form by repeating each vf as rows:
x Ul.....
T..
copics of Us
copics of 02
copies of Up
After verifier filtering, each orthogonal direction vy retains exactly ny samples with no S pn1 pnz S
<
Notably, the estimation of parameter g+++1 using only synthetic data from the model with 6h: though with filtering, leads to a Markovian transition J + gx++1 The above block design essentially helps "diagonalize" the transition operator 0+ + م The conceptual benefit of this covariance design choice is that we remove the rotational variability that arbitrary designs would introduce across iterations and decouple the dynamics along orthogonal directions. In practice, this design mirrors curating data along approximately orthogonal latent spaces topics (e.g., topical axes like polities, sports, mathematics). However, the choice of covariant X is not unique: alternatives (e.g., canonical basis, isotropic random directions) can yield similar qualitative conclusions, with potentially different constants or rates. We expect our theoretical insights to generalize to any reasonable design of the covariant X though the rigorous proofs may be less tractable for some designs.
3 ON THE NEAR-TERY IMPROVEMENT UNDER SYNTHETIC RETRAINING
This section investigates the verifier's role in synthetic retraining: does it help, when does it help, and why does it help? We focus on one round and show that verifier-guided retraining can improve performance under mild assumptions. The underlying mechanism is a verifier-induced bias-variance trade-off: filtering synthetic data reduces variance but may introduce bias.
For ease of presentation, assume Rank(X") = p. If (ERak(('') X*) < P. all our results are equally applicable by working the subspace of X"9
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3.1 SOURCE OF IMPROVEMENT: BIAS- VARSAACE TRADE-OFF
To understand why verifier-based retraining can improve upon the initial estimator 0", we must examine the fundamental biai-variance trade-off introduced by the filtering process. The initial estimator 6 is unbiased but suffers from high variance due to the limited real sample size no. When we generate synthetic data and apply the verifier, we effectively discard inconsistent samples. This filtering reduces the estimation variance. However, because the verifier itself may be imperfect, this filtering injects systematic bias. Therefore, synthetic retraining yields strictly better model precisely when the variance reduction achieved by filtering outweighs the injected bias and the sampling noise of the synthetic data itself.
3.2
CHARACTERIZING IMPROVEMENT IN ONE-ROUND RETRAINING
The following theorem rigorously quantifies this trade-off for the linear regression model, demon- strating exactly when the one-step estimator @ improves or degrades upon the initial baseline. By characterizing the MSE of 0 it reveals how synthetic sample size, verifier bias, and verifier selectivity determine the final outcome.
Theorem 3.1. Let {Hz Yi= denote the singular values of X and assume each of them satisfies
Hj = O mo)+ + Then there exist constants m1,j, ma.j E R and m2,j E (0, for j = 1.... P. as
well as a constant > such that:
P MSE(())= Σ
ma. + m2.5
m2.
m
O no 4/3
Y11
11
Synthetic Variance
Verification
holds with probability at least 1 - pexp (-Lmg 2/33 size.
where nı denotes the post-verification sample
While the explicit forms of the constants are deferred to Appendix C, their roles are highly intuitive: mij and ma, capture the directional bias between the verifier's knowledge center 0. and the ground
truth 0* along the j-th singular direction (vanishing if 02 = 0*), while m2,j < quantifies the variance reduction along that direction. Theorem 3.1 mathematically guarantees when verifier-guided retraining improves the model. Since the scaled baseline error is M)E((00))
Ei=1 11 can directly compare it to Equation 3. When the verifier is highly accurate (m1.j, ms, 0), the verification error term becomes dominated by Σ/= / Because m22 < 1, this verification error is strictly smaller than the real-data error. Thus, whenever the verified synthetic sample size ni is sufficiently large to drive the synthetic variance down, MSE((") strictly improves upon the baseline.
This result highlights why verifier-based retraining is practically useful: in modern machine learning systems where real data collection costly but generative models or simulators are available, a moderately accurate verifier filter synthetic samples to effectively amplify limited real-world evidence and substantially reduce estimation error. Conceptually, this offers a sharp departure from classical model collapse literature, which typically models iterative synthetic data purely variance-inflating noise source (Shumailov et al., 2024; Alemohammad et al., 2024; Dohmatob et al., 2024a). Here, we prove that verification transforms synthetic data into variance-reducinn resource, provided the verifier's 's bias is sufficiently controlled. As we will demonstrate empirically in Section 5, this mechanism is not confined to the linear model; manifests clearly in complex models such as VAES and LLMs.
4 ITERATIVE RETRAINING CONVERGES TO THE VERIFIER'S KNOWLEDGE CENTER
Having established that single round of verifier- based retraining can improve estimation through a bias variance trade-off, a natural question arises: can this improvement be sustained over mul-
'That is, each dimension is well-represented in the original data. This holds easily when, e.g., the feature data is drawn i...d. from full-rank distribution.

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riple rounds, and what is the eventual outcome? To contextualize long-term analysis within the broader literature on model collapse, we first formalize these widely discussed empirical phe- nomena within our linear regression setting, Specifically, we define Model Ingradation/Collan as lim -0 MSE(O*) MSE(("), and Model Improvement as lim -00 MSE(O*) < (S).
Our key finding is that both behaviors can occur in long-term iterative retraining. The outcome depends critically on three factors: the growth rate of synthetic data, the verifier's bias, and the verifier's capability (i.e., its ability to reduce variance). Over time, iterative retraining injects increasingly more verifier knowledge into the estimator, while the contribution from the original data gradually decays. As result, the verifier and the generative model family eventually dominate the limit behavior, driving the estimator f* toward fixed point, which corresponds to the verifier's knowledge center e-.
This dynamic gives rise to three distinct phases of long-term behavior: (1) Unbiased verifier: If the verifier is unbiased (i.e., 0. 0*), iterative retraining yields continuous improvement and the estimator converges to the true parameter. (2) Mildly biased verifier: With small bias, iterative retraining can improve performance in the short term by reducing variance, but performance eventually plateaus or deteriorates as verifier bias accumulates. (3) Strongly biased verifier: With large bias, iterative retraining leads to degradation and may even cause collapse in the limit. Among these, the mildly biased case the most practically relevant. highlights a cautionary message: while synthetic retraining can initially boost accuracy, perfectly unbiased verifier is unrealistic; consequently, this inherent bias will ultimately prevent sustained improvement.
Formally, the following theorem characterizes the long-term behavior of the estimator 6 in linear regression under verifier-based synthetic retraining.
Theorem 4.1. There exist synthetic retraining processes (e.g., Algorithm 3) and a constant 0 < <<< such that:
2(k-))-1
Ell0² p2""1"" - 0ell² =op
(4)
n2
1=0
where S 112 ....... mg.... denote the number of verified synthetic samples per direction at each iteration. In particular, if lim Ti cc, then lim Elj0'
1-
The proof of Theorem 4.1 is provided in Appendix C, utilizing concentration bounds and supermartin- gale inequalities establish convergence. Here we focus on the main intuition and highlight the key novelty of our analysis.
The central observation establish Theorem 4.1 is that the iterative retraining procedure (see Figure 2) induces a Markov process: the next state @ depends only on the current state 0* Formally, the update can be expressed as
g++ = T(0*) + M/++1
(5)
where T(.) is a deterministic mapping determined by verifier filtering, and M-+11 a sub-Gaussian noise term due to the randomness of synthetic samples at iteration k + 1. Crucially, show that T(-) a contraction mapping, and that the variance of the noise decays at the rate (ar(m++1) x 1/nx+1- This perspective allows to view equation 5 discretized stochastic differential equation (SDE). As rig > oc, the noise term vanishes and the dynamics are dominated by the deterministic contraction T(O*), which drives the recursion toward its fixed point-the verifier's 's knowledge center 0g. The presence of the verifier therefore essential: it is precisely what transforms the update rule into a contraction, guaranteeing convergence.
By contrast, in prior work on model collapse without a verifier (e.g., Gerstgrasser et al. (2024); Xu et al. (2025)), the update reduces to the identity mapping. In that case, increasing the synthetic sample size can suppress noise accumulation and ensure bounded error (i.e., MSE(O*) < oo), but there is no contraction and hence no convergence or sustained improvement. The knowledge extracted from the verifier is precisely what elevates T(-) beyond an identity mapping. Our analysis is the first to formally show that the verifier fundamentally alters the long-term dynamics: it continuously injects knowledge, iteration by iteration, that the estimator moves closer to 0. over time.
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Figure 3: Top: Error changes of the one-step retraining estimator 0 versus estimator 0 only using
original real data, measured by log( ||6" -0" [I)): theory's prediction (a) and empirical comparisons (b).
||6--0*
Bottom: Iterative synthetic retraining over 60 rounds with biased verifier (A 1) and an unbiased verifier (A = 0).
This contribution also clarifies a common misconception: even with perfect verifier (0, - 0") and infinitely many synthetic samples in one iteration, convergence cannot occur in a single step. As shown in Theorem 3.1, while infinite samples remove the synthetic variance term, the verification error term persists. Thus, convergence requires the iterative action of the verifier, which gradually aligns the estimator with the truth.
5 EXPERIMENTS
In this section, we validate our theoretical predictions across three settings: a linear regression simulation that is consistent with our analytical assumptions, alongside Variational Autoencoders (VAEs) on MNIST and fine-tuning a pretrained Smo amolIMe 10oo1M221133 35M (Allal al., 2025) on a large-scal news summarization task, which together illustrate practical behavior under iterative retraining and filtering. Across all settings, the empirical results closely align with our theoretical predictions. Experimental code used to generate the results is publicly available at https: //t:://giii //ggtgttthuub com/ Liugiyuanhhh/Vern fied- Synthetic--t Data.
5.1 LINEAR REGRESSION ON SIMULATED DATA
Setting. We consider the linear model y = T + E, with E NO, 1), E RR, and x E RP, We first fit an OLS estimator on the real dataset (X",, yo), and then perform multiple rounds of synthetic retraining, where the synthetic covariate design is aligned with the right singular vectors of X
One-step Synthetic Retraining. Figures 3a and 3b empirically validates Theorem 3.1 by comparing the real-data estimator / with the one-step synthetic estimator 01 where color regions reflect different levels of error reduction, measured by log( -100-00) 1-
Theoretical predictions (Figure 3a) align closely with empirical results (Figure 3b), validating the sharpness our bounds. Using 100 real and 800 synthetic samples, set f* 1s and the verifier center 9. 8" + A u, where is random unit vector and A controls bias magnitude. The radius (see equation 1) determines verifier selectivity. These results confirm that synthetic retraining outperforms the baseline under small bias (green
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PID va Model Model-Fitting Iteration
FSD Model-Fitting Iteration
Trae segg
Werder traking
320
Liear groach
390
200
6BC
300
Moddal Pitting Storation
Model Pitting Iteration
(a) Filtering and training sample size.
(b) Verifier quality.
Figure 4: FID results across retraining rounds. (a) Effect of filtering and retained sample size. (b) Effect of verifier quality, varied by training data size. Together, the plots highlight how both sample selection and verifier strength shape generative performance.
region) but degrades under excessive bias (red region), consistent with the predicted short-term bias-variance trade-off.
Iterative Synthetic Retraining, Figure 3e validates Theorem 4.1, showing that under a biased verifier (A = 1), the retrained estimator converges the verifier's knowledge center 0, rather than the true parameter 0* In this 60-round experiment, sample size increases linearly from 100 to 5500, and results show that a more selective verifier (smaller r) accelerates convergence. Figure 3d presents the unbiased case (A = 0), where triffierbbase retraining consistently outperforms unfiltered baselines. Together, these findings illustrate how the verifier's contraction effect sustains error reduction and prevents model collapse. Additional experiments confirm that our conclusions are robust to random covariate designs (Appendix E. 1) and different verifier shapes (Appendix E.2).
5.2
VARIATIONAL AUTOENCODERS (VAEs) ON MNIST
Extending beyond linear regression, we evaluate our theory on real-worke image generation tasks using Variational Autoencoders (VAEs) on the MNIST dataset.
Setting and evaluation metries. Specifically, we adopt Conditional VAEs (CVAEs) to leverage class conditioning and avoid verifier-induced imbalance; otherwise, easily generated digits would dominate the retained synthetic data. To test the bias-variance trade-off and verifier information injection, we initialize the CVAE with only 500 real images (a challenging small dataset scenario). A discriminator, trained on varying amounts of real data alongside an equal number of synthetic samples, serves the verifier. It assigns a reality probability to each synthetic sample, from which retain the top 10% per digit. Motivated by our one-step analysis, this 10% threshold optimally balances sample quality against diversity. The CVAE is iteratively retrained verified data for 40 iterations until performance stabilizes. The synthetic sample size ng follows either a fixed or linear growth schedule. We evaluate generative performance using Fréchet Inception Distance (FID) (Heusel et al., 2017) and the evidence lower bound (ELBO) (Kingma & Welling, 2014). Appendices D and E.4 provide implementation details and initial sample size ablations.
Results. Since our verifier emphasizes perceptual realism over likelihood calibration, we report FID as the primary metric and defer qualitatively similar ELBO results Appendix D. As shown in Figure 4a, synthetic retraining with strong verifier (trained on 60K real images) yields rapid early FID improvement before plateauing, whereas unverified retraining (dashed curves) leads to severe degradation. This validates our theory: early gains arise from the short-term bias-variance fixed-thoure
trade-off (Theorem 3.1), and the verifier's contraction effect drives convergence biased- point that substantially improves upon the initial CVAE (Theorem 4.1. The eventual plateau reflects two limitations: training the initial generator on only 500 real images inherently restricts overall diversity, and the standard MLP verifier lacks diversity- selection bias by disproportionaaely rejecting harder-to- -e-eeerate modes. For reference, a baseline CVAE trained 60K real images achieves 17.56 FID, whereas the best synthetic model reaches 21.17 after 40 iterations. Figure 4b further shows that with 20K synthetic samples per round, stronger verifiers (trained more real data) produce larger FID improvements, while weaker ones cause early plateau or even degradation. Qualitative results are shown in Figure 1, with consistent MNIST-specific FID results reported in Appendix E.3.
mechanisms, introducing

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5.3 LARGE-SCALE NEWS SUMMARIZATION
Extending beyond image generation, we evaluate our theory on natural-language tasks using the XSUM news-summarization dataset (Narayan et al., 2018).
Setting and evaluation metrics. We use the the pretrained Smo1LM2-1355 model (Allal et al., 2025) as our generator, first fine-tuning it on 12.5% of the XSUM training set for one epoch using full-parameter training. While following the base setup of Feng et al. (2025), we depart from their single-round evaluation by considering a multi-iteration generate-verffy-retrai regime to capture how performance evolves over repeated cycles. Given the low-entropy nature of news summarization, we follow the common practice of employing greedy decoding for both generation and evaluation.
In each iteration, the model generates synthetic sum- maries for the training corpus, which are scored using ROUGE-1 against ground-truth references. Acting as an oracle verifier, we select the top 12.5% of these summaries to form the synthetic retraining dataset. The model is then retrained on this subset, and test ROUGE-1 scores are recorded after each iteration.
ROUGE-1 score Iteration
0.165
Fioer
Unitural
0.160
ROUGE-1
Results. Figure 5 reports ROUGE-1 scores across
0.150
15 rounds for both filtered and unfiltered retraining regimes, while all other experimental conditions re- main identical. Under verifier filtering, performance improves monotonically during early iterations be- fore eventually stabilizing. In contrast, the unfiltered baseline fluctuates around its initial score without meaningful improvement. These observed dynamics mirror our MNIST findings and theoretical predic- tions, demonstrating that our framework scales to natural- tasks.
0.145
0
12
Iteration
Figure 5: ROUGE-1 score vs. iteration on the XSUM dataset. Filtered retraining yields consistent early improvements, whereas unfil- tered retraining shows no significant gain.
DISCUSSION
Our study provides a theoretical and empirical characterization of verifier-guided synthetic retraining. We show that the process yields short-term gains by reducing variance through verifier filtering, but in the long run the estimator converges to the verifier's knowledge center. This explains both the promise and the risk of such methods: a high-quality verifier can inject reliable external knowledge, while a biased verifier inevitably steers the model away from the truth. Viewed through the lens of information elicitation, our framework formalizes how external signals are incorporated recursively into training and why the outcome reflects the verifier's 's information.
Meanwhile, we also acknowledge the limitations of our Primaatl our analytical testbed relies on a well-specified parametric setting (linear regression) that assumes the existence of global, ground-truth optimal parameter 0*. This idcalized assumption mathematically abstracts the complex, often competing attributes of a "good" generative model-such as sample diversity and generation quality-into singular distance metric. In practice, while models like LLMs parametric, they are approximating true data data-generating process that is unknown and highly likely non-parametric. For complex domains like natural language, singular "true model" may even exist; therefore, defining or evaluating an optimal model remains an open challenge. While our empirical extensions VAES and LLMs validate the theory qualitatively, formal generalization richer models such as exponential families or simple neural network architectures are interesting future directions. Other future venues include developing sharper bounds for nonlinear models, exploring effectiveness of alternative synthetic design strategies beyond block orthogonalization, and studying verifier dynamics in large language models (LLMs) and vision models.
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Acknowledgment. This work is supported in part by the A12050 program at Schmidt Sciences (Grant G- 2-24-66104). Bingji Yi conducted this research while visiting UChicago CS. We also thank Cong Ma from UChicago, Hongning Wang and Bo Li from Tsinghua, Pinyan Lu and Gavin Tang from Shanghai University of Finance and Economics for helpful discussions and suggestions.
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APPENDIX OVERVIEW
This appendix contains: Appendix A (1-D Gaussian toolkit), Appendix B (comparison with reward maximization frameworks), Appendix C (reduction and full proof for linear regression), Appendix D (additional details on VAE experiments), Appendix (additional simulations and experiments), Appendix F (use of large language models)
A ONE-DIMENSIONAL GAUSSIAN TOOLKIT
In this section, we provide a toolkit for analyzing the one-dimensional Gaussian mean estimation problem with verifier-filtrree synthetic data. This toolkit serves the foundation for our analysis of the linear regression models. We will establish several key lemmas and theorems that characterize the MSE of the mean estimator under the one-dimensional Gaussian model. These results will be instrumental in proving Theorem 3.1 and Theorem 4.1 in Appendix C.
A.1 SETUP AND NOTATIONS
We consider the one-dimensional mean estimation problem where the real data Xi,..... X independently and identically distributed (i.i..) from Gaussian distribution:
X iid. N(H,
with known variance 2²
In our setting, verifier exists and encodes external knowledge that the true mean lies in interval [a, b] (i.e,u E [a, b)). Therefore, x X1+ X... is the empirical mean of real data, which minimizes MSE no extra information is supplied. We are interested in whether data verification could effectively inject new information and improve over X Consider the following synthetic data generation and filtering procedure:
Generate n synthetic data XI..... X1 N(X9,02)
Retain X [a, b] as X1 X and estimate μ using X
mf Σ X
We will compare the estimator X with X and formally characterize when data verification enhances degrades model performance i.e., when E(XI n1)² < E{x" -1²² not. Our key finding is that X introduces the core bias-variance trade-off that underpins model improvement or degradation. We will characterize the MSE of X which reveals how key quantities such as the real and synthetic sample size, the verifier's bias and variance will decide performance of the filtering strategy. These insights provide intuition for extending verifier-guided re-training to more complex settings.
We first review some notation and key results for the truncated normal distribution, which will be used in the subsequent sections. Consider a mieddimeniional normal distribution X NN(,, o²) and let X' be its truncated version restricted to the interval [a, 6). The distribution of X' is the called the truncated normal distribution, denoted as X' ~N(zlu,, a²) ((ac=<b)) The mean and variance of the truncated normal distribution X' are given analytically:
Φ b-
@ 1 b-H 1
E X'XI [n] =
= + omm
$ b-
Φ 8-#
b- n Φb b-
a-
(
Var(X'Iu) = o²
Φ b-it
Φ b-
+ a-
σ mm
(6)
where o(x) and (xx) denote the standard normal density and cumulative distribution functions,
respectively. Standardizing X via Z = X-# and setting
a a-
b
3
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the expression in equation 6 become:
E Z mi (a, (8)
Var (4) = m2 8)
(8)
where Z ~ ( is the standardized truncated normal distribution. For convenience, Thus, my and m2 correspond the first and second central moments of the standardized truncated normal distribution. In addition, also define the third central moment of the standardized truncated normal distribution:
we write N frune (a, 8) :=
my (cx B) = E( BZ')3
(B² ())(3) (a² 1)d(a) 3(6(3) - ((a)) ap((a)
($( {(a))
(Φ(3) (
2(0(3) o(a))
(9)
(b(3) (a))3
In particular, 0 < m2(a, 8) < 1 for any < 3 and my = 0i
A.2 CHARACTERIZATION OF E(X 2.2 BIAS- VARIAACE TRADE-OFF, AND MODEL IMPROVEMENT
Theorem A.1. Assume that ny >> nl 100. Then there exists constant K, depending only and 3, such that
m2(a, 3)
ma(a, 3) + m3(a, 3 3)m1
E(X!
- 4)2
mm 3)
02
71
710
Syntheric Variance
Verification Biac+ Variance
< K
+
(10)
1/3
3/2
111710
holds with probability at least 1 - exp (- 1/3)
Proof of Theorem A. It will be convenient to reparameterize the sample mean estimators by center- ing them around the true mean. Specifically, we define the residuals:
E1 ~ ~1~~
(11)
Note that X is the mean of n, i.i.d. samples from the truncated normal distribution (xx ((2)) (accrcb) The MSE of X can be decomposed as follows:
E[((X 1191 = g Exixx' (X''-14) -1)
Ego
Var( (X1 1, (E[X!
X9 -9)-1)² x9-11)²
m2(a - E1, =1)
+ Exo [(80 - 10-omm16 ( -
n1
a² Ee [mz(a 6,, E1] o² -² Eeg (mz(a - cz) (1))+6)) +
(12)
M1
For the first term in 12, we consider the event Ei == fleil no t/00 the function m²(-.)) is Lipschitz continuous in neighborhood of (a, (), so we have
Imz(a- - 3 (1) - m2 B) = leil m2 (a - E, -6)
Mi
(13)
1/3
11
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for some E (0, c1), where define
M1
sup
m2
(1) (a
E, - 6)
K
100
and M1 is a constant independent of no as long as 7og 2 100. Event E1 hold with high probability:
1/3
1/3
150
exp
exp
1/3
(1ea1 [ < no 1///
n
> exp
VT/22-10 n2 1/6
VT/2-
1004/6
2
Then we consider then second term in 12. The Taylor expansion of the function
mi (e1) mi (a E1,B -1)
up the third-order terms is:
1 2 ma(a, 3) + mi (3) (E) for some & E (0,
m1(e1) mi (a, B) [1 - (2 3)] 61
(14)
where m (8) (5) denotes the third derivative of my evaluated at some point between 0 and Then we
can get
2
(imi (a - -I, 3-, -e) + =e1)²] = (mi(a, 3) m2(aa B)e + m B) En
+ 6 (s)e)
1 (3)
= =m B) m B) + mi (a, B m) 3)
3m3 3)
71
4n
(3)
+ E
mi (a, (3) + m2 3)& m,
m
(E)
3
m (E)
+ E
(15)
36
First, using the fact that there exists constant M that only depends and 8, such that |m, (3) (x)( M for any x, we have:
<
E
mi B) m2(a, B)c1 + mms(a,
m
(E)
c
3
M
E
(cx(3,))) mz(a, 3)leil + mm 8)1e
3
1e113
M
M m2 3) + ma B)1le1
M
E
mm (a, 3)le,1)
3
K₁
3/2
11g
for some constant Ki depending only and 3. Secondly, the last term in equation is bounded by:
E
m
(3)2
36
(E)
C
<
M
36
Eie
5M
12ng
K2
700
3
for constant K2.
So the second term in 12 is bounded by
E1, E,, [m, e1) + =1)? m (a, for some constant K. Combining 12, 13, and 16 completes the proof.
mē(aa 9) mi (a,, (a, 3)
K
(16)
710
n0
3/2
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A.3 ITERATIVE RETRAINING AND LONG-TERM DYNAMICS IN ONE-DIMENSIONAL GAUSSIAN MEAN ESTIMATION
Now consider the verifier-guided synthetic retraining in the Gaussian mean estimation setting. The iterative retraining process can be described by the following algorithm.
Algorithm 1 Iterative Verifier-Guide Retraining for Gaussian Mean Estimation
1: Input: Initial estimate X from real data
2: for 0, 1, ... do
3: Draw & iid. (0, o²) and construct synthetic samples X X E..
4: Retain points with < X < 6, yiclding na verified samples {X] : = 1.2,....44.
5: X
D X.*
6: end for
Algorithm 1 defines Markov process (XO, 81 p(X* 1x (x) given by
8*
where the conditional distribution
Th Li=1
+
X
X
p 1 gh+1 8
el6+1 i.dd ~ Verune
a
(17)
0
0
The following theorem summarizes these findings:
Theorem A.2. Let X the Markov process determined by equation 17 with initial condition
~ NOO n0
and assume ny non-decreasing k. Then the following statements hold:
lflf [b| < oo, there exists a constant 0 < p < 1 such that.
E x a a+b
2h B(X a a+ b
2(k-))-1
Moreover, if iflima- 11g oc, lim E|X a+b = 0.
1f -00 = < cc, then lim imminfg-ox 8 = -00 If -00
< a
<
00, then
limsupp-00 8 co.
Proof of Theorem A.2. Define
(18)
which represents the standardized error of the estimator X It is easy to see that [a, B + X E [a, b). where a, 3 are defined in equation 7. Therefore, suffices to consider standardized process (Ex, k = 0, 2, equation 17 can be standardized as:
Ri c++1
+
clo+1 Nirune (a ck. (2):
(19)
nk
For convenience, we shift the noise terms elk-1 in equation 19 to have mean zero. Therefore, we introduce
To (t) 2 a -
va.)(x) = Var(Z a < 7 7 x).
(20)
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where Z ~ (0,, 1).
Therefore, equation 19 can be rewritten as
Ck+1 ((k) + 77441
(21)
where 7 L E (c. 1g(h+1) is the average of independent mean zero noise equa- tion 19. particular, we have
Var(m Fs) (e,)(
Elm++1 F&] O,
where Fx == o(co, n...., 7k) and ny is the (post-filteriig batch size at round k. It is easy to see that
To,a (x) = x + mi(a - x, - x),
va, (x: m²((x x. B x)),
To (a: (a,))
We first consider lal, |4| equation 21 a global contraction. Since -o < a < B < xo, we have
oo. In this case, first show that the derterministic part ( in
a + 3
sup SupT2 (x)) sup Var a *ER
aaaa-ssss---0 a Var |2|
p <.
JER
2
Therefore, Ta (z) is global contraction. By the contractive mapping theorem that To . (z) has unique fixed point x", which solves x² =Ta (x*). It is easy to see that
(x,)(x') x² x² + x² sZsB xz) = x' x' a
(22)
By the mean-value theorem,
a B
a+
(e,p((c))
S plex
2
2
Let Vg (Cx a+082 have
E Vo+1 [ex] (Tx))(E))
12+g8
< (Ex
a+
P
2
n1
Taking expectations yields
EV
p² EV
p
(23)
n4k
Unrolling equation 23,
k-1
p ((k---1)
EV S p 2V, + p
(24)
n2
1=0
It is easy to see that
k-1 p 1
EV < EV +
< 2K + no p²)
Therefore, by the Cauchy- Schwarz inequality. lim Ec 30 easily follows. Moreover, when nk 20, let gi = 22 and 0g = / - A standard e convoluuii argument shows (g * a)
k-- 9k-1-ja, Σn=1 ((--11))
0. Therefore lim EV limg E(Es a+ 0.
n5
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Now we consider the -00 < b (equivalently -00 oo). We will show that lim Infk-40e Eg = -00 a.s..
Let tg = 3 Eg and the recursion equation 21 can be rewritten for ty:
tk+11 t/ + X(tx) 4]+11
where A(tx) = E(ZIZ < 3 (.) = Z Z 2 -tx).
Consider the hitting time TM inf k: t/ 2 M] } for any M 0. Fix M and define
m (M) s
min ((t)
2 M] > 0,
which strictly positive the fact that ((t) > 0 and A(t) is a decreasing function. On the event
(TM K3 we have i, < M for j .,... K 1, hence X(t,) 2 m(M). Summing the recursion
yields
K-1
K-1
tk = tu +
Σ
M(tz) Σ
7jt1 2 to K m(MM) - SK:
1=0
1=0
where SK = 2i=0 75t1 and to = 3 - Eo is Fo measurable (hence random). Therefore,
K-1
{TM > K} c Sk 2 to + K m(M) M 3
(25)
Define the (random) burn-in index
Ko 2 (2(M-(0) 10) -00)
m(M)
Then for all K 2 Ky,
to + K m(M) M 2 m(M)
K.
2
and equation 25 gives, conditionally on Fo.
m(M)
(TM K} S { Sx 2
K
for all K > Ko.
(26)
Next, we will show that Sx is a sub-exponential random variable in event {TaS > K}. Since SK = ΣØ Miti = Σ5=
1 n, Em (gri Be/i+!). we will first show that < g Egli+1
sub-exponential.
Since t Naune (-00, , k2) Nunue (- (o,, ty). the event {TM K} we have
g - - <t, EiZ Z tyl < M EiZ Z M] b(M) co.
The above inequality follows from the fact that E Z
an increasing function of and
t, < M for j O.... K-
on the event (TM > K}. }. In addition, Var(s. War(Z Z < t2) S
Therefore, ei+L_ - mean zero, bounded above by b(M) with Var - - RC((tt By Bennet/Bernstein MGF inequality, we have
(
log ME? -ZE7+1) < 2(1 - b(M)\/3)
X2
for 0 < A < b(M))- This shows that go+t Eg sub-exponential with pa- rameters SE(1, 2b(M) By standard properties of sub-exponential random variables, 1 Σn (c/t2 - -Be/+1) is SE(//n; (3m)) and Sx = EØ= 1it1 is SE K-1 1/n,, 26(M)/ (331)) since n, is non-decreasing. Therefore, for any t > we have tail bound
1
nul eexp (-2 min 11 ml 1)
IP (Sx 2 1) exp
exp
- min
K- 1/nj Σj=0
26(
K 2
(27)
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Use the tail bound equation 27 in equation 26, have m(M)
F > K Fo) S P
SK
K S exp (c((M)nLk)
(28)
for all K 2 Ko with c(M) min
m2 8
ní (M) 1
8b(32)
P (TM > K) EE [P (TM > K Fo)]
S EF exp (- - K) [<>>Ko]] + PP(K S Ko)
(29)
Let + oc in equation 29, we get PP(TM < 0o) Since M is arbitrary, this implies lim infx Eg -O a.s..
The case 00 oo can be proved in the same way, therefore omitted.
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B COMPARISON WITH REWARD MAXIMIZATION FRAMEWORKS
While sharing the conceptual similarity of evaluating generated data via an external feedback mecha- nism, our modeling approach substantially differs from reward maximization frameworks in various aspects, including preference matching (Ferbach et al., 2024; Wei & Zhang) and recent Reinforcement Learning with Verified Rewards (RLVR)(Guo et al., 2024; Yu et al., 2025). Theoretically, these meth- ods frame the problem as policy optimization, where the objective is to maximize a provided reward signal. While highly effective for alignment, the definition of "good model" is tied to the specific reward formulation, which may not correspond to recovering the true Lata-generating distribution. Practically, reward optimization relies assigning scalar rewards pairwise comparison signals (Ouyang et al., 2022), which are often difficult and noisy to define, For instance, evaluating image quality open-ended language generation with single numerical reward is inherently subjective. While recent methods like RLVR avoid this issue by restricting themselves to domains with clearly verifiable rewards (Guo et al., 2025; Wu & Choi; Yu al., 2025), many important training settings lack such reliable reward functions. In contrast, we study the widely used "generate-verify-rerne paradigm, which utilizes binary accept/reject filtering. Theoretically, our framework defines a "good model" at the parameter level, explicitly modeling the relationship between the verifier's filtering rule and the ground truth. By formalizing this link, we can directly analyze model performance during iterative retraining, even for an imperfect or biased verifier. Practically, this filtering mechanism less noisy, highly stable, and serves as a core scalable primitive in modern LLM pipelines like DeepSeek-Coder (Guo et al., 2025).
C PROOFS OF ALL THEOREMS IN SECTION 2
Scheme 2 Iterative Retraining with Verified Synthetic Data
Input: Initial estimator 8 from no real samples
2: for k == 0, 1, do
3: Generate: Synthetic covariates X are constructed (details below), and responses are
generated as:
y++ = x + E*+1, &* N0000 02))
4: Verify: Each synthetic sample (a
(+1 ./ 1 of (x++1, yk+1) is filtered by the verifier Condition equation 1, retaining only the verified subset (X yk+1/). Retrain: new OLS estimator is computed using only the verified data:
5:
@++1 = (X++11 x -1x--1 1T y
6: end for
Given the orthogonality of {vi} in the block design equation 2, the OLS estimator decomposes into a of one -iedimensional problems, each estimating the coordinate of along direction uy. In particular, choosing {ry} as the right singular vectors of the real data matrix X yields the cleanest interpretation. making explicit how verifier bias, synthetic sample size, and noise variance interact. Accordingly, the retraining procedure can be formalized as follows:
'Since only the most recent round of synthetic data is retained for training, this scheme is sometimes referred discard workflow in the literature (c.g., (Dey & Donoho, 2024)).
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Algorithm 3 Iterative Verifier-Gudde Retraining in Linear Regression
Input: Real data (X Y
2: Compute initial estimator 80 = (xx] x²)-1xo yoo
3: Let X UEVT be the SVD of X with right singular vectors V (E1,...: Up)
4: for k 0.1.2.... do
5: for 1 do
6: Construct synthetic design matrix X with all rows equal to
7:
Generate synthetic responses yk+lJ = X + gh+l,,, where gh+iJ ~ NNO
Apply verifier to each (I; 1.. U/ k+1,) and retain valid samples satisfying
8:
k+J M/
(x; ( (.j) 0el S <r|10
R+ + oc.
(30)
yielding ny verified samples (x 1/s Compute one-dimensional estimator
1 sk+ 1.))
9:
10:
gh+l.praj. k
(31)
11:
end for
12:
Update overall estimator:
gu+1_ Σ /=1
8j g pktl.projj
(32)
13:
end for
Proof of Theorem 3.1. We consider the one dimensional projection estimator of (1.proj5 defined in equation 31. The filter condition equation 30 is equivalent to:
|o + v,J (00 0c)l S r de
- v = o + +03 E (-r- oc o 0c, + + +00 0z)
(33)
Note that @ N'00, (X0 x and vi the j-th right singular vector of X therefore of (0 ~ NNu 0-, oo²u² Therefore, (2.projá J correspond to the verifier-filtered mean estimator of one-dimensional Gaussian mean estimation problem with true mean v/ 0, variance o²p and filtering interval (-r de + 0. cc. r de + 0. 0c). Let aj = de 1z (0. 0*) 0
8g t + nz (0c 0*)
(34)
Under the assumption M4 w( (V))), there exists a constant L > O, such that 33 > Lmp for all j 1, ,.. Therefore, by Theorem A.1, there exists constant K2 depending only on aj. 3g such that if n1 > no 2 100,
prooss 1/ 002 me(ay. 3,)
(mi(a,, 3,) m 8,) ms(aj, 3, ( Br)
12
T1
1
Kj
(35)
1/3
3/2
Th0
will hold with probability at least exp(-Imn) 1/3 mı. m2, my are defined in equation 8 and
equation 9. By equation 32, we have g) gl.ppoj.j = v @1. In addition, since V =(11, U2, up) is an orthonormal matrice, have
E(A (Qllpoo - 1z 00)2
E(v $1
00 V (0) -1 E - 12 (36)
1=1
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Therefore, by summing over on both sides of equation 35 and using simple union bound, have
_/2 B10 0²112
M2.j
+ +mi.j + m1,m m2g
< K
1/3
3/2
Synthetic Variance
Verification Frror
(37)
with K = maxx K2 and
mi, == m1 m1(a,, 8y),
ma. m2 8j),
ma.j = ma 8,).
Proof of Theorem 4.1. We consider the transition dynamics of 0 in Algorithm 3. Since we designed Xk+11. to be the rank one matrix correspond to singular vector újs therefore equation can be rewritten as
G++ I.rood =1 0t
Nk
1
(38)
Tk
where 5 *++ld is the truncated noise term after verification. By equation 30, we have
< M++ iid
Nerune
(-
Oc
gh
0
de
1
gh
0g
(39)
0
0
0
We consider the rotated standardized estimator
c := 6 0
c = V 0 O
equivalently
Since gi+lpp v g by equation 32, equation 38 can be standardized as
k+1 e
kk11 i.id ~ Nerune (-B-ct,0 ct)
(40)
7
where 3 = + Co We note that equation 40 is exactly the same dynamics we consider in the proof of Theorem A.2 with 3 -a < cc. In other words, the evolution of the iterative estimator ef is diagonal and each cordinates follows the same dynamics the one dimensional gaussian iterative mean estimator. From Theorrm A.2, we known that there exists a constant p < 1 such that
1-1 2(k-5))-
+
Σ
== 1=
nj
1=0
This implies that
2(k-
El
E
0,112
12l12
po
n2
j
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D ADDITIONAL DETAILS ON CVAE EXPERIMENTS
Data preprocessing. We use MNIST (28 x 28 grayscale) and normalize pixel intensities to [0, 1]. Class labels are represented as one-hot vectors y E {0, 13K (K=10),
Experiment Details. We use convolutional CVAE model consisting of an Encoder with two convolutional layers (1-+3 and 32->66 channels, x 4 kernels, stride 2, with GELU activations), followed by linear projection that outputs the mean and log-variance of dN 20-dimensional Gaussian latent space. The Decoder mirrors this structure: a linear layer maps the latent code to a 64 X 7 tensor, which is upsampled by two transposed convolutional layers (64-32 and 32-+1 channels, 4 x 4 kernels, stride 2, with GELU activations) to reconstruct 28 x 28 images. We train the CVAE with the standard objective, i.e., binary cross-entropy reconstruction loss plus KI. divergence regularization.
Discriminator for filtering. We additionally train a discriminator D to distinguish real from synthetic samples. D is implemented as a multi-layer perceptron: five fully connected layers with hidden sizes 512, 256, 128, and 64, each followed by a LeakyReLU activation, and final linear layer mapping to a single logit. The output is passed through a sigmoid to yield the probability of the input being real. The discriminator is trained with binary cross- cross-entropy, labeling real MNIST digits as positive and CVAF-generated digits as negative.
Synthetic generation and filtering. After each training round, we generate conditioned samples by drawing z ~ (0, I), choosing labels y (uniform over classes unless specified), and decoding ã == ==ge(z, v). To control sample quality, we score each (ī, y) with the discriminator (ī,, y). For each class, we retain only the top 10% of generated samples with the highest discriminator scores. These filtered synthetic samples are then combined with the real dataset to form the training data for the next round.
Supplementary Results on ELBO We also evaluate generative performance using the test negative ELBO, a standard likelihood- based loss metric for VAEs. To prevent overfitting the discriminator (i.e., the verifier) and ensure stability during the retraining cycles, we incorporate standard regularization techniques, specifically applying a dropout rate of 0. and label smoothing with a parameter of 0.05 when training the discriminator. To investigate the effect of synthetic data size ni., we employ three linearly increasing sample size schedules. Starting with an initial CVAE trained on only 500 samples, we scale up the retraining size by adding 5K, 30K, 50K synthetic samples per iteration, respectively. The models are retrained for 50 iterations until the test negative ELBO stabilizes.
Figure 6 reports the test negative ELBO over these 50 rounds. Consistent with our analysis, we observe a clear improvement (a decrease in loss) in the early stages (up to roughly iterations 10-15). Furthermore, the trajectories reveal critical dynamic: while larger synthetic size schedules significantly accelerate this early convergence, all three schedules ultimately plateau and converge to similar negative ELBO value by iteration 50. This observation validates our theoretical framework: drawing more synthetic samples expedites the initial variance reduction phase, but the asymptotic performance limit is dictated by the verifier, not the volume of synthetic data. After the initial variance- variance-reduction gains, the negative ELBO eventually reverses its trend and increases (deteriorates) as the model inevitably converges toward the verifier's knowledge center.
As discussed in the main text regarding our verifier's ' limitations, this knowledge center is demonstra- bly biased. For reference, across all three size schedules, the final retrained CVAES at iteration 50 converge to test negative ELBO of approximately 111. In contrast, baseline CVAE trained on the entire 60K real image dataset attains a test negative ELBO of 92.12 (lower is better). Because our verifier emphasizes perceptual quality over likelihood-ban based reconstruction, the negative ELBO proves harder to improve than FID. As a result, even as the negative ELBO stagnates or worsens in later iterations, retrained models continue to improve FID, achieving sharper, cleaner digits. We believe that deploying stronger verifiers with, e.g., diversity preservation capabilities could enable iterative retraining to further improve the negative EL BLO.
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aegative ELBO
Sample sizc sch.dulc
124
651m SK
ASize - 30X
316
20
-0
50
Derrtion
Figure Test negative ELBO across retraining iterations.
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E ADDITIONAL EXPERIMENTAL RESULTS
E. RANDOM SYNTHETIC DATA IN LINEAR REGRESSION
In the main text, the synthetic covariates were aligned with fixed orthonormal basis simplify analysis and make the retraining dynamics easier to interpret. To show that the observed behavior tied this structured design, we repeat the same iterative retraining experiment using fully random synthetic covariates sampled i.i.d from a standard Gaussian distribution.
Figure 7 presents the results, corresponding directly to the lower two panels in Figure of the main text, but under the random-design setting. The qualitative behavior remains the same: with a well-specified verifier, retraining contracts toward the verifier's knowledge center and avoids collapse, whereas unfiltered retraining diverges. This confirms that the verifier-induced stability and improvement patterns hold beyond the aathonormal-essggee assumption.
(6 A.l Iteration
100 0. Iteration
Rack
0.5
1.25
1.5
1.00
0.5
0.75
0.4
0.50
0.25
0.2
0.00
0.0
10
100
20
30
40
50
Iteration
Iteration
(a) Verifier with bias (random design)
(b) Verifier without bias (random design)
Figure 7: Iterative synthetic retraining under random synthetic covariates, corresponding to the structured-design results in Figure
E.2 DIFFERENT VERIFIER SHAPES
We further analyze how different geometric choices of the verifier region affect the acceptance rule and the resulting retraining dynamics. For any region RS around center 6., a synthetic point (æ, y) is accepted whenever there exists parameter perturbation A in the region that can explain i.e.
x T (0, + A)
A E Rg.
This leads to the general acceptance requirement
ly x 00l S sup
xx T T A|+00
+0.
Ac
Different verifier shapes correspond to different support functions supaeRo |x A).
(1) Ellipsoidal verifier. Consider the anisotropic ellipsoid
Ro = {0: (0 - 0z) A - 0.) x 1²²
A 0.
Let A 0 - 0.. Changing variables A = 11122 with ||u||2 yields
A = r|| 1/2 2 = .rve A-1x.
sup AT 2
Thus the acceptance condition becomes
ly x 5 A-1x oe.
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(2) Polyhedral 4 verifier.
For the & knowledge region
Rs {1|0- Bella S rr
the perturbation satisfies ||
|1
Using Hölder duality,
|x A = r||2-100-
sup 1A
The corresponding acceptance rule is
ly - x 0el S 11|/2|1000 + de-
Although ellipsoidal and & (polyhedral) regions induce different forms of acceptance sets, both yield the same qualitative retraining behavior: ((x)) consistently move toward the verifier center Og. The empirical trajectories under both shapes are shown in Figure 8
|100
Rell Iteration
1
Iteration
1.2
1.2
0.8
1.0
0.8
0.8
0.4
0.4
0.0
0.0
Iteration
Iteration
(a) Ellipsoidal verifier
(b) Polyhedral & verifier
Figure 8: Retraining trajectories under two different verifier shapes. In both cases, ((k) empirically converges toward the verifier center O,.
E.3 MNIST-SPECIHIC FID EVALUATION
The standard Fréchet Inception Distance (FID) is widely used in generative modeling, including on MNIST, following prior work such as Dai & Wipf (2019); Leontev et al. (2020): Chan & Sithungu (2024). Nonetheless, we agree that Inception embeddings are not tailored handwritten digits and may not fully capture perceptual similarity on MNIST.
To address this point, we introduce a MNIST-specific FID variant. We train lightweight convo- lutional network directly on MNIST classification, and compute FID using the penultimate- layer activations as the embedding space. This produces a domain appropriat FID measure while preserv- ing the same statistical structure as the original metric. These results confirm that our conclusions are robust the choice of embedding and do not depend on the use of vanilla FID.
Results. Figures 9a and 9b report the new FID scores under our retraining framework for all verifier sizes. Consistent with the standard FID curves in the main paper.
PID va Iteratio
FID vs Stertion
144
320
0
3300
(a) MNIST-specffic FID over retraining iterations.
(b) Zoom-in view for filtered runs.
Figure 9: MNIST-Specific FID using our MNIST-trained feature embedding, Both results confirm that our conclusions remain unchanged when replacing standard FID with domain-specifie metric.
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E.4 DIFFERENT INITIAL SAMPLE SIZES
We assess the robustness of verifier-guided retraining by varying the number of real MNIST images used to train the initial CVAE (1k, 2k, 3k, 4k, 60k). For small and medium initial sample sizes, verifier filtering yields clear early FID improvements and then stabilizes performance, whereas unfiltered retraining quickly degrades. When the generator initialized on all 60k real images, verifier filtering no longer improves FID over the initial model, but it still effectively prevents the severe collapse observed under unfiltered retraining.
We perform our main experiments in a low-real-dar regime (e.g.. with 500 initial images), where the verifier, having been trained on much larger subset of MNIST, holds strictly more external knowledge than the generator. According our theory, this is exactly the regime in which verifier- guided retraining should provide true improvement rather than simple stabilization, because the verifier contributes additional information. In contrast, when the generator is initialized on the full 60k training images, the verifier would need access to an even stronger source external information to achieve improvement; otherwise it can only prevent collapse. For this reason, the small initial sample size serves as the most informative regime for highlighting the verifier's 's snowledge-injectinn effect and demonstrating the improvement phenomena predicted by theoretical framework.
(Filtered
Fratial Campri Goe
120
100
Unditeeed
10
15
25
3.
35
Figure 10: FID across retraining iterations under different initial sample sizes, comparing verifier- filtered and unfiltered synthetic retraining.
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F USE OF LARGE LANGUAGE MODELS
The authors acknowledge the use of ChatGPT for assistance in improving plot figures, as well as for checking grammar and spelling. All scientific contributions, analyses, and interpretations are solely the work of the authors.
30


Reviews:
[{"review_id": "Pvz62EC0BZ", "reviewer": "Reviewer_Lw2s", "summary": "Authors analyze the learning dynamic of generative models iteratively retrained on their own data. The novelty comes from the fact that each step, (potentially bad) data are discarded from the retraining loop, using an external verifier.\nAuthors validate their theory on synthetic experiments and MNIST.", "questions": "- \"Suppose each eigenvalue of the design matrix X 0 is ω( n0 )\", what is w? do you assume that the eigenvalues are strictly a function w of n0?\n- I do not understand the takeaway of Theorem 3.1, especially I do not understand the discussion around Equation 9.\n- \"This contribution also clarifies a common misconception: even with a perfect verifier (θc = θ ⋆ ) and infinitely many synthetic samples in one iteration, convergence cannot occur in a single step. As\nshown in Theorem 3.1, while infinite samples remove the synthetic variance term, the verification bias+variance term persists.\" I do not understand this comment: if the verifier is perfect, why would the verification bias persist?\n- Could authors comment on the conclusion line 299, with the \"3 phases\" depending on the verifier, isn't it obvious that if the verifier is unbiased, it will help, and it is not a well-suited verifier, then the filtering procedure will not be helpful?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "The topic is very timely: quantifying how much can be leverage from synthetic data is a central question for generative modelling.", "weaknesses": "- Disconnection between the analysis and motivation. As [1, 2, 3], authors analyze a linear regression setting setting, that does significantly differ from the generative model setting (as started by the authors in the conclusion). Did author try to make the analysis in the true generative model setting? Like [4]? Can authors provide justification of why to study this setting? and it is relevant to generative modelling?\n- In particular, the proposed setting looks like well-studied semi-supervised learning results [4, 5]?\n- Experiments: for Figure 3, authors mentioned they used the FID metric. I di not find any additional comment, so I assume authors use standard vanilla FID. If so, I think this is a mistake for multiple reasons:\n    - FID rely on the Inception embedding, that is a standard for natural images: to the best of my knowledge, MNIST images are not considered natural images.\n    - In addition, the point of using a embedding is to lower the dimensionality of the data, in order to have a better approximation of the empirical Wasserstein distance\nTo my knowledge, the best for MNSIT, is to train a classifier from scratch, and take the last layer for the embedding.\n- Figure 3b, what is the \"verifier training size\" in the caption? Is this a way to vary the quality of the verifier?\n\n[1] Elvis Dohmatob, Yunzhen Feng, and Julia Kempe. Model collapse demystified: The case of regression\n\n[2] Elvis Dohmatob, Yunzhen Feng, Arjun Subramonian, and Julia Kempe. Strong model collapse\n\n[3] Elvis Dohmatob, Yunzhen Feng, Pu Yang, Francois Charton, and Julia Kempe. A tale of tails: Model collapse as a change of scaling laws\n\n[4] Damien Ferbach, Quentin Bertrand, Avishek Joey Bose, and Gauthier Gidel. Self-consuming generative models with curated data provably optimize human preferences\n\n[4] Ben-David, Shai, Tyler Lu, and Dávid Pál. \"Does Unlabeled Data Provably Help? Worst-case Analysis of the Sample Complexity of Semi-Supervised Learning.\" COLT. 2008.\n\n[5] Zhou, Z.-H. (2018). A brief introduction to weakly supervised learn­\ning. National Science Review", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "BYDeTYQm3B", "reviewer": "Reviewer_SkJA", "summary": "This paper studies the role of verifier-based filtering in retraining on generated data to avoid model collapse. It shows short-term variance reduction and long-term convergence toward the verifier’s knowledge center.", "questions": "1. The paper briefly cites [1] but does not clearly explain how it differs from that line of work. Could the authors clarify what the theoretical novelty is relative to [1][2], and how this work connects to recent studies [3][4]?\n2. I have some concerns about experimental setup:\n    - Does initializing on only 500 real MNIST images limit the baseline and exaggerate the apparent early gains?\n    - The paper states that \"the verifier is trained on varying amounts of real data together with an equal number of synthetic samples.\" Which round’s synthetic samples are used? Is the verifier retrained each iteration (i.e., changing over time), or fixed?\n    - The paper mentions keeping the Top-$10\\%$ of generated samples. If the verifier is binary, why is there a \"top\" ranking rather than simply using all passing samples (and reporting number)? If it is deterministic top-scoring selection, does this still match the theory's assumption?\n\n[1] Damien Ferbach, Quentin Bertrand, Joey Bose, and Gauthier Gidel. Self-consuming generative models with curated data provably optimize human preferences. In The Thirty-eighth Annual Conference on Neural Information Processing Systems, 2024.\n\n[2] Xiukun Wei and Xueru Zhang. Self-consuming generative models with adversarially curated data. In Forty-second International Conference on Machine Learning, 2025.\n\n[3] Kareem Amin, Sara Babakniya, Alex Bie, Weiwei Kong, Umar Syed, and Sergei Vassilvitskii. Escaping collapse: The strength of weak data for large language model training. In Scaling Self-Improving Foundation Models without Human Supervision, 2025.\n\n[4] Xuekai Zhu, Daixuan Cheng, Hengli Li, Kaiyan Zhang, Ermo Hua, Xingtai Lv, Ning Ding, Zhouhan Lin, Zilong Zheng, and Bowen Zhou. How to synthesize text data without model collapse? In Forty-second International Conference on Machine Learning, 2025.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 2, "strengths": "- The paper is clearly written and easy to follow.\n- Verification plays an important role in preventing collapse in self-consuming models. It’s a simple and intuitive idea.", "weaknesses": "- I am mainly concerned about where the incremental theoretical contribution lies. The connection to prior verification in self-consuming works is not discussed in depth (see Question 1).\n- The paper considers pure synthetic replacement. In practice, retraining typically mixes in real data or accumulates data.\n- Experiments are small-scale (MNIST/CVAE only) and lack validation on larger datasets or modern models.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "h0VQvOPm7n", "reviewer": "Reviewer_j2Ts", "summary": "This paper mainly studies the role of verification in preventing model collapse using a linear regression framework. Their analysis finds that, in the short term, the proper use of verification can enhance performance by reducing variance. However, in the long term, it may cause the model parameters to converge toward those used to construct the verifier. The results of this paper are supported by rigorous theoretical analysis and experimental evidence.", "questions": "Please see the weaknesses above.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper is clearly written and easy to follow.\n\nThe claims in the paper are supported by rigorous theoretical analysis and comprehensive experiments.\n\nThe results are quite intuitive and make a lot of sense.", "weaknesses": "I think this paper presents a solid theoretical work. Below, I will list some drawbacks; however, they do not constitute reasons for rejection, as we all understand that in theoretical analysis, it is often challenging to address more complex but practical models.\n\n1. During iterations, it is assumed that the synthetic covariates can only be the copies of a fixed orthonormal set. This assumption might be a little bit strong, can this be further relaxed?\n\n2. It seems that the overparametrized regime has not been considered. Is it possible to generalize the results to the overparametrized regime?\n\n3. In practice, the verifier may evolve over time. Could the authors consider whether a similar analysis can be conducted under this setting?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "sD6PEiyBny", "reviewer": "Reviewer_TG7U", "summary": "This paper investigates the effect of verifier quality on model collapse from training on self-generated synthetic data.  The authors structure their analysis from linear regression with a verifier that has controllable bias.  It is discovered that a mildly biased verifier may provide initial improvements, but will ultimately lead to performance plateaus or collapse over many iterations.  The authors evaluate their work on linear regression as well as preliminarily on a CVAE implementation on MNIST.", "questions": "Why is verifier-based filtering in particular important to study thoroughly - even for frontier generative models?  Whereas naively retraining the generative model on self-generated synthetic data may indeed lead to model collapse, the field has seen success in avoiding this through verifiable rewards (Reinforcement Learning from Verifiable Rewards, as used in DeepSeek [1], Tulu3 [2]).  Why do filtered retraining from a verifier rather than use all data but labeled with a reward from the verifier (such as labelling unsuccessful generations with a 0)?  The assumption for the components is the same in both approaches, but RLVR seems to work better than verifier-based filtered training.\n\n[1] DeepSeek-AI et al., DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning, 2025.\n\n[2] Lambert et al., Tulu 3: Pushing Frontiers in Open Language Model Post-Training, 2024.", "limitations": "", "rating": 2, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "The paper is quite clearly presented, and the math is approachable to verify, being based on linear regression.", "weaknesses": "The paper is quite toy from the perspective of model (linear regression), data, and verifier.  This may be fine for theoretical proofs, but it does limit the scope of takeaway when considering the applicability to frontier generative models.  It would be interesting to see actual experiments on language modeling and large-scale pretrained models beyond such toy settings.  Particularly as there are a variety of reinforcement learning from verifiable rewards works on frontier models at the moment, it should not be difficult to apply it for the verifier-filtered experiments proposed here.\n\nEven MNIST is a bit toy compared to available datasets of natural images.\n\nThe takeaways also make intuitive sense, rendering the findings a bit unsurprising - one does expect that a mildly biased verifier should help initially but performance should not be able to break free from the biases of the verifier when the scale of seen data is predominantly that filtered by the verifier.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "442ygfa8iS", "reviewer": "Reviewer_ENpW", "summary": "The paper studies verifier-guided synthetic retraining: a loop that (i) generates synthetic data from the current model, (ii) filters it with an external verifier via a binary accept/reject rule, and (iii) retrains only on the verified subset. In a linear-regression setting, the authors give a one-step MSE bound that separates synthetic variance reduction from verification bias/variance. They then analyze the multi-round process and prove it behaves as a noisy contraction that converges to the verifier’s knowledge center. Experiments in linear regression and a CVAE on MNIST show early, clear gains with verification; long-run plateau depended on verifier quality.", "questions": "- How sensitive are the results to the shape of the verifier region? \n\n- A probabilistic perspective might also be interesting since these are generative models. The retraining process can be thought of as to move the original training data distribution towards the distribution defined by the verifier, which actually defines a generative model itself. When the retraining iteration goes to infinite, we are distilling the generative model defined by the verifier into a parametric generative model e.g., a VAE.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- The work formalizes a widely used practice, i.e., filtering synthetic data with a verifier and shows how it changes collapse dynamics.\n \n- A clear theoretical framework and detailed analysis followed by empirical evidence in simple setup.", "weaknesses": "- Core analysis is linear regression with a spherical verifier and a special synthetic design; extensions to non-linear models are discussed but not derived. This limits direct transfer of the rates/conditions to modern LMs/vision models. \n\n- MNIST/CVAE is a clean demo but dated; no language or large-scale vision results. Also, while FID trends are solid, downstream task metrics or human evals would strengthen claims about generative quality. \n\n- In the MNIST setup, the useful verifiers are trained with much more training data than the initial 500 data points for the VAE. In the reality, for images, we will normally use all data possible to train both the generative model and the verifier, therefore, another more practical setup is to fix the training set for both the original VAE and the verifier, and iterate from that.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)