Self-Verification Provably Prevents Model Collapse in Recursive Synthetic Training
Authors: Shi Fu
Yingjie Wang
Yuzhu Chen
Li Shen
Dacheng Tao
Venue: NeurIPS 2025 poster ()
Keywords: Model Collapse
Learning Theory
Synthetic Data
Self-Verification
LLMs
Primary area: theory
TL;DR: This paper proves that self-verification prevents model collapse in recursive training without relying on real data.

Abstract: Large generative models are increasingly trained on synthetic data from earlier generations, raising concerns about *model collapse*, a progressive performance decline consistently observed in empirical studies. However, theoretical understanding of recursive training dynamics and their failure modes remains limited. In this work, we theoretically show that recursive training inherently leads to exponential error growth unless mitigated by sufficient real data.  Addressing the growing scarcity of real data, we introduce a self-verification mechanism enabling models to filter their outputs based on internal confidence scores without external validation. Through rigorous analysis, we derive finite-sample error bounds demonstrating that self-verification alone can prevent collapse, even in fully synthetic training regimes. Our theoretical framework extends to large language models (LLMs), characterizing the conditions under which recursive training can maintain stability without performance degradation.

Self- Verificatiin Provably Prevents Model Collapse in Recursive Synthetic Training
Shi Ful Yingjie Wang!" Yuzhu Chen Li Shen Dacheng Tao
Coollege Computing and Data Science, Nanyang Technological University, Singapore, University of Science and Technology of China, Hefei, China,
SShnnzhen Campus of Sun Yat-sen University, Shenzhen, China shi0110e ntu. edu ag, yingjievangguuc. edu en,
cyzkrau@mail. wst.i.lusttt..mmmmm. edu. en, mathshenl 1itihennn@@@p 10gmail.....mmmmmmmm com, dacheng. tao@gmail.
Abstract
Large generative models are increasingly trained on synthetic data from earlier gen- erations, raising concerns about model collapse, progressive performance decline consistently observed in empirical studies. However, theoretical understanding of recursive training dynamics and their failure modes remains limited. In this work, we theoretically show that recursive training inherently leads to exponential error growth unless mitigated by sufficient real data. Addressing the growing scarcity of real data, we introduce a self-verification mechanism enabling models to filter their outputs based on internal confidence scores without external validation. Through rigorous analysis, we derive finite-sample error bounds demonstrating that self- seff-verifiittion alone can prevent collapse, even in fully synthetic training regimes. Our theoretical framework extends to large language models (LLMs), characterizing the conditions under which recursive training maintain stability without performance degradation.
Introduction
High-quality training data is critical foundation for the remarkable success of large generative models, such as LLMS [Fu et al. 2024a]. However, it is increasingly acknowledged that the pool of publicly available, real-world data nearing exhaustion (Villalobos et al., 2022]. As result, the training of modern LLMs increasingly relies on synthetic data produced by previous generations of the models (Briesch et al. 2023 Martinez al. 2023 Xing et al. 2025 Moreover, even unintentionally, models today are often trained on synthetic data, as many existing datasets are already polluted with synthetic content [Schuhmann al. 2022). This paradigm, known as recursive synthetic data training. has raised significant concerns regarding the risk of model collapse, phenomenon characterized by drastic and often irreversible deterioration in model performance across generations (Shumailov
et al. 2024 Alemohammad et al.) 2024a Bertrand et al. 2024 Dohmatob et al. 2024b
Numerous empirical studies have highlighted the potential risks associated with model collapse in recursive training loops, including substantial reductions in output diversity [Guo et al. 2023 Zhu et al. 2024), sharply increasing еrror rates [Dohmatob al. 2024b|c), and the amplification of biases [Wyllie et al., [2024]. To mitigate these risks, researchers have proposed various approaches, such as incorporating sufficient real data [Alemohammad al. 2024a], augmenting synthetic datasets [Gerstgrasser et al., 2024). and introducing external verification mechanisms for synthesized data [Feng et al. 2025 Firdoussi al. 2025). Nevertheless, despite these empirical advances, there remains a notable gap in rigorous theoretical understanding of both the observed phenomena and the underlying dynamics of recursive synthetic training loops.
"Corresponding authors
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Initial theoretical explorations have started to address these gaps by analyzing population risk dynam-
ies under specific assumptions [Bertrand et al. [024 Dohmatob al., 2024a Seddik al. 2024]. For
instance, [Bertrand et al. 2024] derived upper bounds on parameter deviations for likelihood-basee models when real data is incorporated, establishing convergence under strict statistical and optimiza- tion assumptions. Further work by [Fu et al. [2024b 2025] relaxed these assumptions on error bounds through more fine-grained analyses. Fu et al. [202bb] provided bounds on the Total Variation distance for simplified diffusion models, linking distributional divergence to dataset size and the proportion of real data. Complementing this, Fu et al. [2025] introduced the concept of recursive stability to establish the first generalization error bounds. They demonstrated for transformer architectures that constant proportion of real data is sufficient to prevent model collapse.
More recently, theoretical studies focusing on verification strategies have gained particular atten- tion [Feng et al. 2025] Firdoussi et al. 2025]. These works investigated Gaussian mixture models with linear classifiers in non-recursive settings, modeling external verification of synthesized data as Bernoulli random variables. In particular, /Fng et al. [2025] demonstrated, in the infinite-sample limit, that linear verifiers can significantly improve model robustness when training solely on synthetic data, while Firdoussi et al. [0255] extended these findings to scenarios involving a mixture of real
and synthetic samples.
However, existing theoretical analyses suffer from critical limitations [Feng et al. 2025 Firdoussi et [al5].022. Current frameworks lack rigorous external verification and rely on simplified proba- bilistic assumptions for data quality. These analyses remain narrowly focused on Gaussian mixture models, limiting their applicability to broader generative models, particularly vranformer-basee LLMs used in practice. Previous theoretical studies are also restricted to non-recursive training sce- narios, whereas model collapse emerges specifically within recursive settings (Schaeffer et al.) [2025]. Additionally, analyses typically rely on asymptotic regimes with infinite-sample assumptions, failing to capture realistic simite-sample conditions relevant to actual deployment. This paper addresses these gaps with the following contributions:
We propose general theoreti- cal framework for recursive training with selfiverification, where models filter their own outputs based on internally estimated confidence scores. This approach removes the need for external verification and enables model-internal quality control applicable to a broad class of generative architectures.
1. Unified Framework for Self- cuff--eiffiatiin in Recursive Training:
2. Finite-Sample Error Bounds Across Training Regimes: We provide rigorous finite-sample error bounds across three recursive training regimes. First, show that naive recursive training leads to exponential error accumulation across generations. We then demonstrate that incorporating sufficient real data mitigates this degradation. Most notably, we prove that self-verification alone suffices to prevent model collapse, even in fully synthetic settings, by ensuring provable convergence of error.
We extend our analysis to transformer- based LLMs, establishing convergence guarantees for self-verification through recursive coverage coefficient control and PAC-Bas -AC-BBaesian analysis. Our results provide the first theoretical justification for stable, fully synthetic recursive training in high-capacity transformer architectures.
3. Theoretical Guarantees for Transformer-Basee LLMs:
2 Related Work
High-quality training data is essential to the success of large generative models. However, the vast supply of human-annotated or naturally occurring data on the internet nearing exhaustion [Villalobos) et al. 2022]. As a result, LLMs are increasingly trained on data generated by earlier versions of themselves, forming recursive training loops, also known as self-consuming loops [Martínez et al., 2023 Bohacck and Farid 2023 Shumailov et al. 2024 Tao al. 2024 Schaeffer et al. 2025] Recent high-profile research has highlighted the potential for dramatic degradation in model performance, phenomenon known model collapse, which represents critical challenge for the future of AI
development and deployment /Gibney| 2024 Alemohammad et al. 2024a
Dohmatob et al. 2024b
Empirical studies have documented a number of alarming phenomena associated with model collapse in recursive training loops, including a substantial reduction in output diversity [Guo et al. 2023
Shumailov et al. 2024 Briesch et al. 2023. Zhu et al. 2024]. rapidly increasing error rates Fuu jtt al.]| 2024b), and bias amplification [Wyllie et al., 2024] To mitigate these effects, prior work has
proposed several strategies, such as incorporating real data into training (Alemohammad et al. 2024a

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Kanabar and Gastpar 2025],
enlarging synthetic datasets [Dohmatob et al. 2024a Gerstgrasser
al. 2024, Kazdan et al. 2024]. or guiding the data generation process using reward or control
signals [Gillman et al. 2024 Alemohammad et al. 2024b Feng et al. 2025 Firdoussi et al. 2025
Despite the wealth of empirical investigations, theoretical understanding of recursive synthetic training loops remains relatively limited [Bertrand et al., 2024 Dohmatob et al., 2024c. Fu ct al.
2024b 2025 Seddik al. 2024 Dey and Donoho 2024). A line of recent work has begun to explore
this direction by analyzing population risk dynamics under specific modeling assumptions, such as
linear contexts [Dohmatob et al. 2024a Gaussian models [Shumailov et al., 2024, Alemohammad
et al. 2024a Suresh et al. [2024] Jain et al. 2024], and asymptotic regimes [Marchi et al. (2024).
Furthermore, Bertrand et al. [0244] derived upper bounds on parameter drift in likelihood based
models under strong statistical assumptions, while Fuu et al. (2024b 2025 relaxed these constraints
and provided generalization bounds for simplified diffusion and attention
architectures.
Most closely related to our work are Feng et al. 2025) and Firdoussi et al. (2025), which introduce verification strategies into training loops to prevent model collapse. These studies investigated Gaussian mixture models with linear classifiers in non-recursive settings under the infinite sample
size assumption. They modeled external verification of synthesized data as Bernoulli random variables
and examined the impact on model performance. Specifically, Feng et al. 2025] demonstrated that linear verifiers can provide feedback that improves the robustness of models trained on fully synthesized data, while Firdoussi et al. [2025] extended these findings to mixed data scenarios.
Our work further advances this line of research by introducing the self-verification mechanism, extending the theoretical framework to general architectures like transformers, analyzing error accumulation in recursive settings, and deriving practical finite sample bounds.
3 Preliminary
In this section, we formalize the recursive training framework, where each model is trained on data generated by its predecessor. We then outline the self-verification mechanism, which enables models to evaluate and filter their own outputs to ensure stable training.
Recursive Training Loops. Let (G. be sequence of models trained recursively. The initial model Go is trained on a real dataset Su = {(xo.. 16,;),)5 }) S X y/, where Fo, are inputs sampled from the data distribution and yū.j are ground-trume -round-truthh labels. For each generation f = 1,..., T, model G. is trained on synthetic dataset S: (<xz,.j Dt,s)] where Ft.j X and Øt.j G (x..,) are labels predicted by the previous model. The process continues recursively until generation T.
Next, we introduce a structured framework that integrates serff-ccriiiation into each training gen- eration, consisting of two key components: (1) a confidence-guite guiidde filtering mechanism for self- verification, and (2) recursive training procedure leveraging verified data. Together, these compo- nents enable robust self-evolution through selective data curation.
Self- Velf--erification via Confidence-Guided Filtering. introduce self-verification mechanism that allows the model to evaluate the quality of its outputs. The verification score is computed using a fixed evaluator Go. trained solely on the initial real dataset, and reflects the alignment of generated output with the base model's inductive biases. Formally, the score for output given input x is defined as:
the absence of external supervision, we
Sverify (y x) log Go ()).
This design offers two main benefits: anchors verification a stable reference, mitigating distribu- tional drift across generations, and acts as an implicit regularizer that discourages semantic deviation from the base distribution.
Given an input Tt.j, we sample N candidate outputs from the previous model G.-1, denoted (ir,. N ~ 9--1(- Zt.5). We then apply a verification-bat based filtering step that retains only candidates whose scores fall within margin y >> 0 of the highest score in the sampled set:
Bo.. Sverity (Br.5.k xz.5) 2 max Sverify (De, w Iess - 7)
ISK'NN
3

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A verification-wiggiee distribution over Vi is then defined via softmax over the scores:
exp (Sverity x(,1))
(
Egevis exp (Everity (i/ [i.5)) where i E yiga.
Then, verified output Ut.j is sampled from this distribution. This mechanism allows the model to act as both generator and verifier, enforcing quality control in recursive training.
The verified outputs from confidence-guided filtering are used construct the synthetic training set for the next generation. For each input æt.jj. we sample verified target Dt.j (- (t.j), forming the dataset S. = ((tt.)) j=)r trained by maximizing the log-likelihood over St:
Recursive Training with Verified Data.
The next model G; is
G: arg max GER
Σ log ( [xt,), j=1
where H. is the model hypothesis class. This process facilitates iterative self-improvement by leveraging verified high-quality outputs for training.
4 Main Results: Understanding and Preventing Model Collapse
In this section, we present our theoretical analysis of model collapse in recursive synthetic training to understand when and how this phenomenon be prevented. We first demonstrate that, without intervention, error accumulates exponentially across generations, leading to inevitable model collapse. We then demonstrate that introducing real data mitigates this collapse. Finally, prove that self-verification mechanism alone can prevent collapse, even under fully synthetic training.
4.1 Assumptions and Definitions
We begin by stating the core assumptions and definitions that underpin our analysis. Assumption 1 (Confidence- Callbraatd Agreement). Let Go denote the base model trained on ground-truth real dataset So = {(z,, ((1),,5))3 \ー}} For any input x E X, define the verified high-
confidence prediction as: y = {ul Sverify x) 2 maxy Sverify (v/ x)) ->} where Sverify is chosen log Go- Let yi (x:) arg ma Gr x)) denote the prediction G. Then, we assume
the existence of a constant 7 E (0, such that:
Pa~x yyt (x) (x) (x)) 2 1- <(7):
where <(>) is function that controls the residual error, T is a fixed constant representing the confidence threshold, and y" (z) denotes the ground-truth label.
As the filtering becomes more selective (7 - 0), we expect <(>) + 0 if the base model Go. trained on real dataset, well-calibrated. This expectation is reasonable, extensive prior work has explored confidence calibration techniques that enable models to achieve well-calibrated predictions [Mehrtash al. 2020 Wang al. 2021 Zhu al. 2022 Liu et al. 2025). Furthermore, similar assumptions have been extensively employed in the literature on self- paradigms (Huang et al. 2025a b).
Definition (Minimum Confidence). Let y (z) = arg max Gi 2)) denote the model's top prediction at generation t. We define the minimum confidence level of these predictions as: Co min tE(o,r] G. (v (z) 1z).
The quantity Co denotes the minimum confidence assigned by any model in the sequence to its top prediction over the input distribution. Ensuring Co guarantees that a non-trivial level of certainty is maintained throughout the recursive training process. This assumption is standard in prior
works (Zhang et al. 2023 Huang al. 2025a
Definition (Test Error), We define the test error of the final model Gr after T recursive generations
as: Err(gr) I F [ (x) # y" (x)). where y (z) arg max Gr(y x)) is the model's prediction, and y" (x:) is the ground-truth label.
This metric provides clear measure model performance and serves as the foundation for analyzing error propagation in recursive training.

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4.2 Model Collapse in Naive Recursive Training
We begin by analyzing the failure mode of naive recursive training. where models are trained solely on synthetic data generated by their predecessors without any verification. In this setting, performance inevitably degrades over generations, leading to eventual collapse. Our first theorem shows that prediction errors propagate and amplify exponentially across recursive steps.
Theorem 1 (Error Propagation in Naive Recursive Training), Let {GelT=o denote a sequence of models trained via naive recursive training, where each 91 is trained solely synthetic data generated by its predecessor GL-1. without any verification. Assume that for all E [1.T], the Total Variation (TV) distance between successive generations satisfies Dry (G.(- x),9,-1(- x)) Drv. where Drv denotes the characteristic magnitude of distributional shift across generations. Define the coverage coefficient C. E [ (y., (a: =)]. Suppose the final model Gr satisfies Assumption 1
Then the test error after T recursive steps satisfies:
) 2
Err(Gr) S (() 1 Drv + (c,+ log Drv)
( (
1
1
log 1 Co )
+
(I)
2Cu
Remark 1. Exponential Error Amplification in Naive Recursive Training. This theorem un- derscores the core limitation of naive recursive training: the prediction error grows exponentially with the number generations T when the condition 20% + log čo > met. Here, Co. defined in Definition]] T denotes the minimum confidence assigned to the model's top prediction across all generations. In practice, Cg tends to be small, making the condition readily satisfied. Notably, when Co S 0.5, the error term in Equation Srriiooo@eexiibeee exhibits exponential growth as T increases. This growth stems from the fact that each model not only inherits information from its predecessor but also compounds its biases and errors, leading to eventual model collapse.
This result aligns with prior theoretical works [Bertrand et al. 2024 Gillman et al. 2024 which shows that recursive training without control leads to exponential error amplification, However, these analyses directly assume upper bounds on optimization and statistical errors from finite sampling, and typically posit unrealistic iterative retraining where each model inherits its predecessor's parameters and optimizer state (Schaeffer et al. 2025)_ In contrast, we analyze training dynamics for specific generative models, including Transformer-bant based LLMs (Appendix [[] Theorem 77). under a more practical setting where each model is trained from scratch on synthetic data.
The phenomenon of model collapse observed here is also consistent with empirical observations in recursive training setups, where the absence of quality control mechanisms results in severe performance degradation over successive generations (Shumailov al., 2024 Alemohammad et al., 2024a]. This theoretical result formalizes these observations, providing a rigorous explanation for the inevitable instability in naive recursive training.
4.3 Mitigating Collapse with Real Data
common approach to prevent model collapse is to inject real data during training. While this strategy has been empirically validated in previous work (Alemohammad et al.) [2024a]. we provide theoretical characterization of how real data injection affects error propagation.
Theorem 2 (Error Propagation with Real Data). Let { be a sequence of models trained via recursive training that incorporates a proportion generation. The remaining 1 a proportion of the training data comes from synthetic data generated by the previous model Gr-1. Suppose that the TV distance satisfies Drv (Gr(- x), 9- |x)) x
E [0, 1] of real data Dreat = {(, {xxxx")] y²"}} at each
DTv for all and that the final model Gr satisfies Assumption Assume > 21/Co-13 Then as T + +00, the final error after recursive steps satisfies:
1 Dry log Drv Co
Eir(Gr) S <(>) + Drv +
Remark 2. Error Stabilization through Real Data Injection. This theorem demonstrates the stabilizing effect of incorporating real data into recursive training loops. Specifically, when the proportion of real data a satisfies the condition a > 1 2T the exponential error growth observed in naive recursive training (Theorem T is mitigated. The resulting error bound becomes stable as + +00, with the dominant terms depending on <(7). the total variation distance Drv, and the proportion of synthetic data a).
5

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Notably, as discussed in Assumption I <(7) can be made arbitrarily small through confidence calibration techniques [Liu al. 2025 Meanwhile, the TV distance Drv can diminish under
certain theoretical conditions. Prior studies have shown that (Tv(n) converges at rate of O(n O(n-1/4) 1/4) in diffusion models [Fu et al. 2024b), with similar rates established in GANs [Liang] 2021]. These results suggest that as the number of training samples n increases, the distributional shift across generations diminishes, further stabilizing the overall error bound.
The key insight from this result is that introducing even modest fraction of real data, > 1 TTTC fundamentally changes the error dynamics and ensures that the exponential amplification factor from Theorem D is suppressed. From a practical perspective, this theorem underscores the importance of incorporating real data in recursive training loops to ensure the long-term stability of model performance. Even a small fraction of real data acts as a corrective mechanism, counteracting the biases, distributional shifts, and errors introduced by synthetic data. This result also aligns with prior empirical and theoretical findings [Briesch et al. 2023 [Bertrand et al., [2024] Fu et al. 2024b 2025 Seddik et al. 2024) that emphasize the critical role of real data in mitigating error propagation.
Remark 3. Limitations of Real Data Dependency and Motivation for Self- Verification. While this result confirms the effectiveness of incorporating real data in preventing model collapse, also highlights significant limitation: the approach relies on access to an ongoing stream of high-quality real data. In many practical scenarios, such data may be scarce, expensive to obtain, or entirely unavailable VVillaoboss et al. 2022 Alemohammad et al. (2024a]. Furthermore, in publicly available datasets, it may be challenging to accurately distinguish between real and synthetic data [Sadasivan et al. 2023]. which can undermine the stability of the training process.
This limitation motivates the need for alternative strategies like ceff-verification. By internally validating and correcting the quality of synthetic data, a self-verification mechanism can prevent error propagation across generations, offering a more resource-efficien and generalizable solution to model collapse without depending on external data.
4.4 Mitigating Collapse through Self- Verification
Our next main contribution is to demonstrate that stable recursive training is possible without real data or external supervision, by leveraging a self-verification mechanism. This builds on the insight that a model's verification ability often exceeds its generation capability (Weng et al.) 2023 Huang et al. 2025a), allowing it to effectively filter its own outputs. We then present our main theoretical result, showing that self-verification can prevent model collapse:
Theorem 3 (Error Bound with velff-eriiicattion). Let [G.) be sequence of models obtained via recursive training with the self-verification mechanism described in Section 3 Assume that for all t E [1, the TV distance satisfies Drv (Gel- æ), 9--1 æ)) x Dry. Suppose the final model Gr also satisfies Assumption 7 Then, after T recursive generations, the final model Gr satisfies:
Err(GT) S <(()) + DTV+ +
(1L, log
2 Co
T C
log 1 1og T
(No DTV log Drv
1 NNO
-DV
1
1 log
(Noo
N
T
Furthermore, if N
(1) Co Co
1 log
the expomential term vanishes as T + co, yielding:
1 (1 1 DIv + N
1 (1+ (1+co Drv log Drv
1 log T
EIr(Gr) S c(7)
(2)
Co
3 demonstrates how self- verification prevents the exponential error growth shown in Theorem during recursive training loops. Specifically, the method enforces quality control using a verification score Sventy (y x) = log Go x) which quantifies how well a generated output aligns with the distribution of the initial model. This reference model, trained on real data, remains fixed across generations and serves as a stable anchor for self- verificciii.. By selecting outputs within margin 7 of the maximum score, the mechanism creates a filtered subset of high-quality candidates for training. This anchoring to the initial model Go prevents distributional drift from compounding, ensuring each recursive generation remains tethered to a consistent inductive prior. As generations progress, outputs deviating from the original data manifold are excluded, effectively suppressing error amplification even without external supervision.
Remark 4.
The Role of Self- Verificaiii in Error Control. Theorem
Specifically, Theorem establishes that the exponential decay term vanishes asymptotically when N > log(l/Co)/Co, where N, as described in Section 3 represents the number of data samples

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used by the self- velffiiriiination mechanism to select subset of high-quality samples for training. This condition highlights the importance of both the size of the data pool and the quality of the selected samples in stabilizing the recursive training process. When this condition is met, the error bound converges to an asymptotic form, ensuring that the error remains bounded even as -> Dc.
This result is particularly significant because it demonstrates that, through proper quality control mechanisms performed entirely within the model itself, recursive training can achieve sustained improvement without relying on real data or external verification. By leveraging self-verification to prioritize reliable outputs and suppress low-quality generations, the training process becomes robust to error amplification, enabling scalable and stable model development in shource-constrained or fully synthetic environments.
Remark 5. Comparison with previous works. The study of model collapse in recursive training loops was initiated by theoretical insights from (Shumallov al. (2024), [Alemohammad et al. [202aa]
which analyzed simplified Gaussian models under fully synthetic data settings. Most closely related to our work is the emerging line of research that introduces verification strategies into training loops to
prevent model collapse (Feng et al.] [025 Firdoussi et al.) 2025). Specifically. [Feng et al. (2255 and
Firdoussi al. (2025) investigated Gaussian mixture models with linear classifiers in non-recursive settings. Their analysis examined how external verification of synthesized data influences model
performance as training dataset size approaches infinity. [eng al. [2025] primarily demonstrated that appropriate feedback mechanisms can significantly improve the robustness of models trained on
fully synthetic data, while Firdoussi et al. (2025) extended these findings to mixed data scenarios.
Our work makes significant theoretical advancements over Feng et al. 2025 and Firdoussi et al. (2025) in several key aspects:
Self- Verificcaitin vs. External Verification: While prior work often relies on external verifiers to guide data quality (Feng et al. 2025 Firdoussi et al. 2025), we propose self-verification framework in which models assess their own outputs. This intrinsic approach removes the reliance on external quality signals, enables autonomous self-improvement. For advanced or superhuman models, internal verification capabilities may surpass what weak external mechanisms can offer [Wang et al.) 2025
2. A General and Practical Theoretical Framework: Our work considers more general and practical theoretical framework compared to Feng et al. [2025] and Firdoussi et al. [2025]. which focused on Gaussian mixture models and linear classifiers. Importantly, we do not impose any assumptions about data distribution, making our findings applicable to wide range of real-world scenarios. Moreover, our framework addresses more realistic language model settings, and in Section [5] we extend our theoretical results to Transformer-bast based LLMss, demonstrating their applicability to modern and widely-used architectures.
3. Recursive Training vs. Mon-Recursive Settings: While WhileFeeng et al. [0255] and Firdoussi et al. (2025) investigated non-recursive training settings, our work tackles the more complex and challeng- ing recursive training setup, which the primary scenario in which model collapse occurs (Shumailov et al. 2024 Alemohammad et al. [2024a] /Schaeffer et al. 2025]. Recursive training involves com-- pounding errors and distributional shifts across multiple generations, making it significantly harder to analyze theoretically. By addressing this setting, our work provides deeper insights into the dynamics of recursive training loops and offers solutions that are more meaningful and impactful for preventing model collapse in practical scenarios.
4. Finite-Sample Guarantees vs. Asymptotic Analyses:
Prior work [Feng al. 2025 Firdoussi
jet al. 20225 primarily established asymptotic results under infinite-sample assumptions, limiting their applicability to real-world training regimes. In contrast, we derive finite-sample generalization bounds that remain valid under practical recursive training scenarios.
In summary, we advance the field by introducing self-verification, developing general framework applicable to transformer- L.Mss, analyzing recursive training dynamics, and deriving finite- sample guarantees. These contributions mark significant step toward understanding and mitigating model collapse in generative models.
Remark 6. Proof Sketch of Theorem Theorem 3 provides a recursive generalization bound for self-verifying models under synthetic training, and shows how error amplification can be effectively controlled through reffidence-guided filtering. The key innovation lies in leveraging recursive coverage coefficient control to quantify the reliability of verified data and to suppress error propagation across recursive generations.
7

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Step 1: Verification-Indueed Quality Control and Error Decomposition. We leverage the verification score Sverify (y
x)) log Goly | a) to filter model outputs and restrict supervision to high-quality generations. Let y (x) denote the verified candidate set, and define the confident support region:
Di { X Go (x) a)) 2 log(2/7)/NV)
Using standard decomposition, the failure probability of the first-generation model can be split as:
Pr~x (v, (x) x) < - P (9 (a) ( <
x E DT] + + [x $ D
The first term is controlled via a Monte Carlo coverage argument and a TV distance bound between G1 and the empirical verification distribution Gi.N. while the second bounded using the expected inverse coverage. This yields:
C7 log 2 N -
P [G (x) x)) & - 111-77 = Div(GI.9i,N)
with coverage coefficient C, == Eg [1/90 (a:) x)] This completes the base case.
Step 2: Recursive Coverage Propagation and Error Control. To extend the analysis across generations, study how coverage evolves recursively. Specifically, the coverage coefficient generation satisfies the recurrence:
C--1 S + aCr, with
(2)) 1 log NCo
DIV NCo
log
- DTV
1
N
where Co is the minimum confidence. Solving this recurrence gives:
1 Co
C
+ Co log (1) N
1
C. S
log
NCo
N
log
Co
Co
Substituting this into the error bound completes the proof.
Key Insight: Recursive Coverage Coefficient Control. Unlike classical generalization bounds, which depend solely on data size and model capacity, recursive synthetic training introduces new challenge: maintaining high-quality supervision under the evolution of coverage across generations: C, == E, // (y (x) x)]. which quantifies the expected inverse confidence over the verified candidate region. Intuitively, a smaller C,,s implies that the model assigns higher probability mass to own verified predictions, indicating better self-consistency and internal alignment.
data. This requires bounding
5 Application to Transformer-based LLMs
This section instantiates our theoretical framework on Transformer- based LLMs, which form the
foundation of modern commercial systems such as GPT-4 [Achiam et al. 2023), Claude [Anthropic 2024), LLaMA [Touvron al., 2023], and Gemini [Team al.) [023]. Our formalization considers token sequences S, = ( 2e,) within a structured space S S xest where X' represents the
token vocabulary and denotes maximum sequence length. The family of distributions induced by the model is denoted as (G 0 e (LLM), where represents the model parameters. Then,
the initial model G.S. is trained on a corpus of real token sequences {(x ((l,),.... )) using "3
maximum likelihood estimation:
log G (211) s(0))
Gu = arg max BELLIM
j=1
where 5(1) = (z) () E S represents each prefix sequence, and n = ml is the total sample
count. This initial model then participates in recursive training described in Section 3 using
the verification score function defined as Svertyy (r S) = log GUL (r S). Then, We adopt the
standard Transformer architecture (Vaswani et al. 2017 Brown 2020 Zhang et al., 2025 Hu et al.
2024), consisting of D layers with multi-head self self-attention (MHA) and feed-forward network (FFN)
modules. For input matrix (0) e 4xd computation in layer t e [D] follows:
(0) (() IIamm (L-1 ,w()) + (0) x ((1) Imam (F) ( 72 [0y10)]
8

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where Vi 72 [-1, are residual weights, and nom normalizes rows to the unit b2-ball. The FEN module is defined as FFN(Y A = ReLU(Y()) ((00)((0){(())) A with A E Rddddr and A!
(0)
Rdxxd Token prediction logits are produced via y(D+1) = softmax ( (LB)),
where I E R is the all-ones vector, A(D+)) E Rdxd, is the output projection, B E (0, 1] controls temperature, and dy is the output dimension.
For theoretical tractability, we assume uniformly bounded model parameters, as detailed in Ap- pendix [ This standard assumption in Transformer analysis aligns with prior work (Li et al. 2023
Hu et al. 2024| Zhang et al., 2025 Fu et al., (2255. In Appendix [[] Theorem 7 demonstrates how naive recursive LLM training leads to exponential error growth, while Theorem g in Appendix [ shows how incorporating real data prevents this collapse. We now present our key contribution: emror bounds for LLMs with self eeff-veriicatioon that maintain training stability without externll real data.
Theorem 4 (Error Bound for Transformer-bast based LLMs with Self- Verification). Let 9o S) denote the conditional generation distribution of Transformerrbasee LLM at the t-th recursive generation, where S E S X is a token prefix sequence and x E X is the next token. Then, after T recursive updates, the final prediction error is bounded with probability at least 1 - 8:
Err(ght)) ) <c(7)+ + (+
1
log =)
log(nB)D Vni
(N
1 Co.
T
log 3 +
log
log -
NCo
NCO
N
1 log (c)). ). the exponential term vanishes as T - co, yielding:
Furthermore, if N Co
1 N log ( NCo NCo
+)
log(nB)D Vn
EIr(ger sc(7)
log + N
log
where = mm is the total number of training samples, and D, B reflect the model capacity and prior radius respectively. A complete version of the theorem is presented as Theorem 5 Appendix H
Remark 7. Stabilizing Recursive Training of LLMs via Verification Mechanisms. This theorem LLMs,
establishes that self-verification mechanisms, when incorporated into Transformer- effectively eliminate the exponential error accumulation typically observed in recursive synthetic training loops. Through rigorous analysis of the error bound, we identify three principal components governing model performance: (i) the confidence-calibrate
agreement error (7)), which can be minimized when the verification slack parameter y is sufficiently small, leveraging established confidence calibration methods [Liu et al. 2025); (ii) the statistical generalization gap, characterized by log(nB) D/ vn. where / and D represent the prior radius and effective dimensionality of the model sapacity-parameet
that directly scale with the architectural complexity of Transformer networks; and (iii) a recursive decay term determined by verification sampling intensity.
Notably, when the verification sample size exceeds the critical threshold N log 1/Co)/CCo the decay term asymptotically vanishes, enabling stable convergence of the training process regardless of recursion depth. This result not only provides theoretical guarantees for the stability of recursive training language models but also offers practical guidance for calibrating verification intensity based on architectural complexity and confidence characteristics of specific Transformer implementations.
6 Conslusion
This paper establishes a theoretical foundation for understanding and mitigating model collapse in recursive synthetic training. We demonstrate that naive recursive training leads to exponential error growth, whereas incorporating a sufficient fraction of real data can stabilize performance. To address scenarios where real data is scarce or unavailable, we propose a self- verificatin mechanism that enables models to filter their own outputs based on internal confidence scores. Our analysis provides finite-sample error bounds showing that self verfficiiiin alone is sufficient to prevent collapse. We further extend these guarantees to Transformer-based LLMs, offering the first formal justification for stable, fully synthetic training in high-capacity generative models. Overall, our framework paves the way for principled and scalable self-evolution future generations of L.MMs without reliance on external supervision.

---

Acknowledgement
This project is supported by the National Research Foundation, Singapore, under its NRF Professor- ship Award No. NRF-P2024-001.
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12

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Appendix
A Overview
In this supplementary material, provide additional details and complete proofs support the theoretical developments presented in the main paper. The appendix is organized as follows:
Appendix B Related Work: velifieriiication in LLMs.
We situate work within the emerging literature self-verification for LLMss, highlighting key differences between our theoretically grounded, training-time framework and prior prompt- based or post hoc empirical approaches.
Appendix C Limitations.
We discuss primary limitation of our framework-its reliance on a single verification strategy based on internal confidence scores-and and outline possible extensions using richer or more adaptive verification methods.
Appendix(] D. Broader Impacts.
We outline the potential positive societal implications of our work, such as reducing depen- dence on human-labeled data through self-verification, and note that our theoretical analysis poses no immediate negative societal risks.
Appendix Proof of Theorem 3 Error Bound with Self- Seff-eeiffiation..
We present detailed imite-sampl analysis of recursive training with self verificati... The proof introduces recursive coverage coefficients to quantify verification quality and demon- strates how confidence-guie guided filtering prevents error amplification across generations.
Appendix F Proof of Theorem I Error Amplification in Naive Recursive Training. We formalize the failure mode of naive recursive training without any verification and prove that, under mild conditions, prediction error grows exponentially with the number of generations.
Appendix G. Proof of Theorem 2 Stabilization via Real Data Injection.
We analyze how introducing fixed proportion of real data each generation mitigates exponential error growth, and derive sufficient conditions under which training remains stable.
Appendix H Proof of Theorem Theoretical Guarantees for Transformer-Based LLMs with Self- Verification.
We extend our theoretical framework Transformer-based language models, providing generalization bounds under recursive training with self-verification, accounting for model capacity and confidence calibration.
Appendix I. Exponential Error Growth in Naive Recursive Training of Transformer- Based LLMs.
We instantiate Theorem I in the Transformer setting and analyze how parameter confidence bounds affect the rate of error amplification during unverified recursive training.
Appendix J Mitigating Collapse via Real Data in Recursive Training of Transformer- Based LLMss
We apply Theorem 2 to the Transformer case, showing that even limited real data injection suffices to stabilize recursive training and suppress error accumulation in practical scenarios.
Appendix K. Additional Experimental Analysis and Future Work.
We provide further empirical validation for our assumptions, present experimental results demonstrating how self-verification prevents model collapse, and discuss potential directions for future work.
This appendix provides the full mathematical foundations of our results and demonstrates their applicability to modern high-capacity models used in practice.
13

---

B Related Work: Self-Verification in LLMs
Recent research has explored celf-verification as a means of improving LLM reasoning performance
during inference [Weng et al. 22023 Stechly et al. 2024 Ma et al. 2025 Lightman et al. 2023 Xu
ct al.
2024 Kamoi al. 2024 Tao et al.
2024]. Most existing approaches adopt a prompt- bbssd
paradigm, where LLMs are instructed at in inference time to critique or revise their own outputs Huang ct al. 2023 Tyen et al., 2023 Zhang et al. 2024a]. or are trained post hoc to mimic such behavior via
curated demonstrations or reiffrreemnt learning (Saunders et al. [2022] Jiang al. 20244] Kumar cet al. 2024 Zhang et al., 2024b Ma et al. 2025 While effective in certain settings, these methods
largely focus empirical improvements, and rely externally crafted prompts or behavior cloning pipelines.
In contrast to these empirical efforts, our work develops formal theoretical framework for self- verification in recursive training settings. Rather than prompting models to verify inference time, we integrate self-verification directly into the training process via internal confidence-based filtering. This allows us to establish provable error bounds, showing that self-verification alone can prevent collapse under fully synthetic training. Furthermore, extend these guarantees to Transformer- based LLMs, providing the first formal justification stable self-evolution in high-capen sighccapcctyy models without external supervision. Our contribution bridges gap in the literature by grounding the concept of self-verification statistical learning theory, rather than relying on prompt design or behavioral imitation.
C Limitations
A primary limitation of our framework lies in its reliance single verification canttegy-ooffidence based filtering using a fixed base model. While effective in our theoretical setting, this approach may be insufficient capture semantic correctness in more complex tasks. Future work should explore richer and more adaptive verification methods, such as ensemble-based voting or task-specific evaluators, enhance robustness.
D Broader Impacts
This work provides theoretical foundation for stable and data-efficient training of language models through verificiiii,,, potentially reducing the reliance on large-scale human-labelle datasets and enabling broader access capable models. As theoretical contribution, does not pose any immediate negative societal impacts.
E Proof of Theorem 3: Error Bound with Self- Verificaiii
In this section, we present the proof of Theorem which provides finite-sample eπrоr bound for recursive training with self-verification. Our analysis draws inspiration from the confidence framework introduced in Huang et al. (2025a), but extends it to more general recursive setting that captures the dynamics of synthetic supervision over multiple generations.
The key insight lies in controlling the evolution of the recursive coverage coefficient, which measures how well the model concentrates probability mass on its own verified predictions across generations. While classical generalization bounds typically depend only on data size and model capacity, recursive training introduces new challenge: maintaining reliable supervision when the training data is generated by the model itself. To formalize this, we define the recursive coverage coefficient as
C : Exx
Gilyt (x) x))
which captures the expected inverse confidence the high-quality self-verified region. A smaller value of C,,, implies that the model assigns higher likelihood to its verified outputs, indicating better internal consistency and semantic alignment.
By bounding the growth of this coefficient over generations, we are able to characterize how self- verification mitigates error amplification and ensures stability, even under fully synthetic training.
14

---

The proof proceeds by decomposing failure events and recursively controlling the probability mass over verified subsets.
We begin by recalling that the initial model Go is trained via maximum likeli- hood estimation on ground-truth dataset So {( M7 1 that is,
Proof of Theorem 3
Go arg max 9EM
log G(y xz));
where H denotes the model hypothesis class. The resulting model Go encodes a set of inductive biases aligned with annatttdetttted supervision, and serves as fixed and calibrated reference model the recursive training pipeline.
Defining Confident Support Set DI
We define the subset of inputs with sufficient sample coverage
log N
Dt
x
Go (v (x)
where the high-quality candidate y (ar) is defined based the internal self-verification score function:
y =
Sverity (y x) 2 max Sverity (3/ (x) 7).
y'El
with self-verification score function defined as Sverity (y x)) = log Go (x). This gives:
y
y/ log Go (x) 2 max log Go (x) -7
y'EY
y Go(y x) 2 max / /o (v/ x)
(3)
y'E)
where 7 = ?
Decomposing the Probability Low Confidence Generation
We aim to bound:
PrIx (1 (x) x)) < -]] < Panx (1 (x) x)) S - T, E DF] + PANx [i $ D.].
(4)
Bounding the First Term: Using Monte Carlo Estimate
For DJ', we aim show
(i (z) 7/2 Indeed, observe that y ~ (i 2) $ y (x) and only if g1.1. y (z: for all since E DJ which happens with probability
VI,N ~ Go(.) have Vis $
log N
(1 - Go (x) x)) $ (1
f N
(5)
Then we show that:
(1 log N
N <
2
To do so, we analyze its asymptotic behavior, and verify that it satisfies the above inequality. This is now typical exponential approximation problem, where the base (1 + plays key role. We notice that:
N
lim
No+00
N
15

---

Thus, we get
log
log:: 2
Nv.
(1
log 2
loc
log
1
N
N
Then, obtain
(1 - Go (x) |x))x² - 7
2
Thus, we get
(v. 2 1 5/2.
Bounding Diserepancy via Total Variation Distance
The total variation (TV) distance between two probability measures P and Q is defined as
1 Σ -(V)) 2
Drv (
Importantly, the TV distance provides a direct measure of the pointwise probability differences between P and Q. In particular, for any measurable event A, the following inequality holds:
[P(A) - Q(A) - (D)v(,,Q).
Specializing singleton events {y}. follows that GI z) - gī,v (v) =) s 2D1v (91(- | "i, |z)) Equivalently, we can lower bound the TV distance by
Drv (9) 1), ( x)) 2 2 G (z) x) - -i.N (I) x) From the definition of the set Dj, we know that for any x E D
(x)(())()) 2 2
Suppose that G1 (1 (x) x) < - T. Then it follows that
| (v |z) - Ii.N (x+ =) 2 7
2 2
Substituting back into the previous inequality. we conclude that for all E D
Drv (9 ()) 2 (T < - -r..
Thus, get:
P PaNX (9 (x) 2) < 1- T, D1j < 4
- Div Div 2))1) 1x))
Bounding the Second Term: Using Markov's Inequality
Let C.yy = Exex
Then:
9 (x))()))
log 2 g
Pa~x & Dt1 = Panx
Go (y (x) xx) <
N
log
Pa~xx
NGo (yt (x)) (x)
log r
E
N
Go Guytt (x) x))
C₂
2
log
N
-
16

---

Final Bound for G1
Substituting equalities
and into
we obtain
C 2 N log
Pa-x (y (z) x) S 1 - r| - - ( (. +
(9)
Recursive Bound for the Second Generation Model 92
Building upon the prior analysis, we now investigate how a self-verification mechanism can be systematically employed refine model training across generations. Specifically, we focus filtering the outputs generated by Gi through confidence-aware criteria, and subsequently utilizing only the verified outputs train the next-generation model G2. This recursive training framework ensures that each successive model benefits from high-quality and semantically coherent supervision.
Definition of Verified Input Set for 92
We define the subset of inputs with sufficient verified coverage as:
DJ = or E 9i (
log N
(x:
x))
where the high-quality candidate set yt (x)) is defined based on the internal verification score function:
y { Sverity (y/ x)) 2 max Sverity (y' 1x) -7
-
V'E)
Bounding the Cow-Confidence Region for G2
Following the same logic as in Eq. 77, derive the following bound the probability that G2 assigns low confidence to the verified candidate region:
4 Drv (9a() x)) F2 (
Pa~x (2 (x) x) < v, DJ1
(10)
Bounding the Failure Rate of Verified Samples
Similarly, we follow the derivation of Eq. 8 obtain upper bound on the probability that an input falls outside the region D
log 2 N
P Pa~xx1 & D
PA~x
91 (y (x) (z)
log 2
P
N91 (y (x) 23)
log 2 N
1
E
(11)
Ir (yt (x) x²)
The quantity Ex>x across all imputs, and plays central role in the recursion. To proceed, we introduce the variahle:
captures the expected inverse confidence in the high-quality region
G (v: (=)]
(
91 (v (x)
Its expectation is computed via the integinteaaal identity for nonnegative random variables:
Ex [2(x)]
P(Z(x) 2 t) dt.
(12)
17

---

Since the posterior satisfies Co S 91 (y, (x) x) S 1, it follows that
1
1
1 ((x)
91 (07 (x) 2)
Co
Hence,
1,
0 0st < 1,
P P(Z(x)2
0,
t> +.
Thus the integral in reduces to
1dt f00 do P(Z(x) 2 t)
E [[Z(x)]
1 P(Z(x) 2 t) dt.
Recall that Eq 9 for t 2 1, then we can deduce that:
4f
Drv (91 x)), 9in( x)) + C N
P (xx)
log
t
t-1
The rest of the proof proceeds by upper bounding this integral through sharp relaxation using B(t), characterizing tail probabilities of inverse confidence values, and leading to a recursive generalization bound.
Explicit Upper Bound on Recursive Confidence
We define:
4t B(I) - - Drv (Gi(: (1), x)) + C2 N
log 2
1-1
Then we Upper bound for the tail probability via min(1, B()). However, probabiiitts cannot exceed Hence tighter statement
P(Z(x) 2) S min (1,
and therefore
Z(x()) 11 1+ f do
P (((()) 2 t) $ $1+ f do min(1, B(t) dt.
Since B() is monotonically decreasing for E (1+, +oo), we focus on the non-trivial case where B 1. In this setting, there exists a unique point to
(1, co
such that
2 1, < 1 S lo:
B(to) =1 and B(t)
1 1, to t
Consequently,
1, 1 [1, to).
minf1, 1, B())
B(t), te [to. 1/Cu).
Hence,
min B()) dt 1" di f do B(Idt (lo (6-1) + f do
B(I)dt. (13)
Then, we compute / B(I) di explicitly. Let us denote
t B(t) A 1-1 + C Clog(
( 2t
--11)
18

---

where
Cg
A 4Drv 1x)
=
N
Then we decompose and integrate each part:
I t-1 dt
[(1 t -- )atat= dt + ++ 1| constant,
flog log
1 1-1 dt / log(2)) dl - / log(t - 1)
( log(2t)
$) - [(t- 1) log(t 1) (t 1)}
constant
t log(2t)
(t-1) 1) log(t - 1) + constant.
Therefore,
/ [(t)dt= A | In|t 11] + C c log(2t) (t 1) log(t - 1)] constant.
We shall denote a particular antiderivative by F(t). Then
CD
B(I) dt
F( - Co
F(to), -F(t),
where lo is the unique solution B(la) Then conclude the final upper bound. Combining with the preceding results, we see that whenever B 2t) < 1, we obtain
13
J min 1, B(I) de (to 1) + |r( -(tt))
and thus
((Z(x) = +1+ , t P((()) 2 1) s + (1o-1)
1
[F(
|F
F((0)]
Co
Recalling that Z(x) 1/91 (y (x) 1). we finally arrive
s 1 (to - + F J F((t): Co
E E 9 y (x) x
which provides an explicit upper bound on the coverage coefficient in the regime B C, < 1. Here, E (1, ) is the unique point where B(4g) 1, and F(-) is any antiderivative of B(-).
Then we further derive the explicit coverage coefficient upper bound. First evaluate ( F(l)). From the explicit antiderivative, ignoring any additive constant (which cancels in the difference), we get
F( 11) Co
-F(t)) = =A to) In
Co 1) log
to
+ c c((22/ log( 2) - to log(2to))
( C
1 - 1) - (to - 1) log(to - 1))]
Co Co
Co
Thus,
(6 F 1
F(to) = (to 1) + A A to + In Co
- + c(..].
Co
to
where the bracket [...] in the C-term is the obvious difference from above.
Then we incorporate the condition B(to) = 1. Recall
21 to + C log
to - ) = 1.
B(ta) = A to 1
19

---

This equation confines to to the interval (1, c₂ Although to generally cannot be expressed in an elementary closed form, we can appropriately scale the equation to obtain an approximate range for to-
Moreover, elementary inequalities for log imply
t-
S logt $ t-1
for all t 0,
and setting ( = 24g yields
24o
log
to -1
S 1 + to 1
(14)
2tg
Hence, from (too) 1 deduce bracketing inequalities:
A to c(1+ -1) to
2
-1)
1,
(15)
to
to + +2+ 2to
1)
(16)
to
Next, we consider the non-trivial case that aligns more closely with practical scenarios, A and C are small enough that A + 1. Under that assumption, 15) and (T)) is valid and yields an upper and lower bound on 40. Thus
1 + C 2AC
1+ C
5
2 (2A+
CA+
Thus,
to - 12 1 22 +2AC
2 C)
V1 - 2C -2 2AC - (1 C) + 2A
2 12AA+
2A
(17)
2 (2A+
Our goal is to convert these into simpler numerical bounds on tg. which can then be substituted into the final coverage coefficient bound
C2,1 = Exaxx
2 + (lo 1) 1) + |F To - -F((o]]
G1 (1(
x))
Thus, get
[F( 1
C₂21 = Exnx
$ to +
-F(to))
Gi(v1 (x) x)
Co
$ A [(6] -to) + ln
C5 -1
-1 )]
(18)
to -
+c[(2 log(
2 2) to log log(2ta)
((2)
1
1) logl 1 -1) (to 1) log(to 1))]
Co
Co
1
to + 1
log(
t0 ) + +
log(
Co
Co
Co
1+C
2- (2A + C)
C 1 log( 1
$ 11-4+
+
log(
Co
2A
Co
Co
$1+C+
log
C
log
Co
A
Co
Co
$1 1 1
Co
log( / +
Co
log(
Co
)
(19)
20

---

Final Bound for G2
Combining the bounds from
10 and 11 we obtain:
Pr~x (22 (x)
x) S 1 --r] r]
4 Div Drv (92(- x)) 1), G2, (. |x)) +
log 2
1
BENx
91
(3r (x))
x)
< - (9al- x), G2 x))
Cr.1 2
log
(20)
N
where Cy,1 is the bound derived above. This completes the recursive enror bound analysis for the second generation model.
Recursive Generalization Bounds for Arbitrary Generation
We now extend recursive analysis to arbitrary generation steps beyond t 2. To simplify the discussion and maintain consistency, we make the following assumption:
Assumption: Uniform Total Variation Distance
Without loss of generality, we assume that the total variation distance across all generations S S remains approximately the same, that is,
Drv (Ge(- (x), xx) = Drv.
This allows us to write for the second generation:
A₂ = Drv (9e(- x), 9 G2 x)) x DrV,
C₁
1
C
1
C N
- N
Co log
log
(21)
Co
A
N
Co
Co
Continuing this recursion, arrive at:
C2 = Exn.
S1
A2
log(
+
C2
log
(22)
22(y (x)
Co
A2
Co
Co
leading
Ag x A2
(23)
C3 C222
1
C2
N
Co
Co
log
Ao
N
Co
log
Co
(24)
This recurrence generalizes to all future generations:
A x A,
(25)
1 G S N
1
C.
1
log
log
(26)
N
Cp
A
N
Co
Co
Solving the Recurrence Relation
Given the following definitions:
1
1 N
A
(
NC.
log
C
B
+
NCo
log
7
have the recurrence relation:
C S 3 + aCC,
which can be solved as:
G C
1-a (1 (l((a)))
B
21

---

Thus, we have:
C S (No log
)
N +
NcO log (1)
1
1
C
1
log
NCo
Co
1
log
NCo
Co
NCo
Now recall C.
Under the assumption N
log
1 ( we conclude:
N
Co
1
C2
1
4 log (1)
C2 $
log
+
(27)
NCO
Co
N
N
log
Ca
Co
Substituting this into the bound on Cy we get:
1
Cz S 1 At log
+
C.
log
Co
A
Co
Cu
1 Co
1
< 1 Drv log
(28)
Drv
((1C)
C
+
log
(7)
(NCo
log
(o)
Co
log
(29)
Co
N
N
log
1
Co
Cu
Co
Final Error Bound After ' Recursive Generations
Thus, after T generations of the recursive training loop, obtain
P FaNx [Gr (u
(x) (x))
x²) I -
<1-1]= DIv +
Crit
log( )
N
S Drv + N log( 1 )
log(
Drv log
NCo
DIv
(No log
Cr.y
1
Co
log (1))
1
C/
1
(
log(
NCo
Co
N
Co
Co
N
N
log Co
Co
1
1
DTV + N
log(
})
+
log(
Drv log
NCo
Drv
(NCO log
1
C
1
1
+
log(
log
Co
N
Co
Co
N2
1
1
1
Drv +
Drv log
+
log
log(
N
NCo
Drv
N2
Co
Co
1 log(
))
1
Cr
log( b
(30)
NC Co
N
Furthermore, we obtain:
Pawx (Gr (U+ (x) xx) 21-1 1-
2 1 1
Div (1
1
1
1 1
log( (1)) log(
T
N NCO
Drv log
Div
N2 Co
00
-(
log(
11
1
Cr
log( 1
).
(31)
NCo
Co
N
22

---

Final Accuracy and Error Decomposition
Define (x) = arg max, Gr x). Then:
ACC(GT)
Pantx [ (a) (x)]
-Paxx 1 (x) = (x) GT
| x0)
1-
Panw (9r (2 (2) ()) 2 1 -
2 1-())) 1 1-
c(1) 1-e(1)
(NCO
1
Cy(1 c(>)) log( - =
DTv
log(
5
NCo
Co
(1 c7)) ( N NCo
1
1
1
log( (
log(
Dry log
+ N2 Co
(32)
DTv
Co
Furthermore, we get
Emr(Gr)
PPaxlyy((a + (x)]
1 1 -c(())
(
1
)
C1111 c(7))
S c(7)
Drv +
log(
log -
NCO
Co
N
(N)
1
1
log( -
+ (1 <(>))
+
Drv log
+
log(
N
NCo
Drv
N Co
Co
$ 1
1
1
1
1
Drv +
+
Drv log
DTv +
log(
log(
N
NCo
N2
Co
Co
Oz
1 + +c()).
log
log(
(33)
NC
Co
The proof is completed.
F Proof of Theorem I: Error Amplification in Naive Recursive Training
In this section, we prove Theorem I which characterizes the failure mode of naive recursive training Seening-whher each model is trained solely on synthetic data generated by its predecessor, without any verification or quality control. The key result shows that prediction errors amplify exponentially across generations, ultimately leading to model collapse.
This phenomenon stems from the accumulation of distributional shifts each generation, which progressively degrade supervision quality. Our analysis formalizes this effect by bounding the total error in terms of the cumulative divergence between successive models. Even when per-step errors are small, their unregulated propagation drives the model away from the original data distribution.
This negative result underscores the need for intervention whetteh whether via real data, external feedback. or self verification to stabilize recursive training. We now present the detailed proof.
We establish results by carefully analyzing the propagation of errors in recursive training paradigm without verification mechanisms. Our analytical approach parallels the methodology used in Theorem measures.
Proof of Theorem /
3 with critical distinctions reflecting the absence of quality control
Foundational Definitions and Decomposition
We begin by defining the subset of inputs with sufficient confidence under the base model:
D = {or E X Go (3, (x)
x) 2 1 21-1]
where the high-quality candidate set yit (x)) is defined via the verification score function:
y = Sverify (y x) 2 may svertyy (3/ xx) - 7).
23

---

with verification score function defined as Svenit (y x)) := log Go a)) To establish our primary results, decompose the probability of low-confidence generation:
Fanix [9 (r) x) S 11-
< X (I (x) x) S 11-T,EE Panw [x $ Dj]
(34)
Analysis of Model Divergence via Total Variation Distance
To quantify the disagreement between consecutive model generations, we leverage the total variation distance. For any input æ and output y:
C .) Go x)] < 2Dvv (91
2 ( a))
This inequality can be reformulated to establish lower bound on the TV distance:
1 Drv (- x)) Go x)) 2 2
9 (2,, (z) ()) Go(u; (1) x)
For inputs in the high-confidence region DJ, we have:
Go ( (z)
2
When 91 assigns insufficient confidence to the optimal region, i.e., Gi (x) x) 1 T, the
discrepancy between models is bounded below:
( (z) x) Go (v (x) x)
7 2
2
This allows us to express the probability of low confidence for inputs in D in terms of the TV distance:
P Pa-x (9 (x) x2) x) S 1 T, E < Div (9i(- 2)) 20. x)).
4
(35)
Characterizing the Complement Set via Inverse Confidence
Let C., := Ex>x Co( (z)]a)
denote the expected inverse confidence under the base model.
Applying Markov's inequality, we derive:
Paax (or & Di1 Panx Go (y: (x) x))
2
P xm2
2go (v (x) (t)
1 2.) Exwwx
Go (v+(x) (x) (x)
<
Cy.
(36)
Integrating equations
35 and
36 into 34 we establish the comprehensive bound for the first
generation:
P Pawx (1, (x) 2) < r]
Drv (Gr(- x). Gol x)) (1 (1-37) 7) C..
(37)
Recursive Analysis for the Second Generation Model G2
Following an analogous approach, we define the high-confidence region for the second generation:
D = { x E X 9 (1)())))211-3
24

---

By symmetric reasoning, we obtain:
4 Drv (92(- 1x) Gi [x)).
Pa~x (22( x)) s 111-1,a0 11-1, E
Gi
(38)
5
For the complement set:
P l2 D = -Panx
91 (y (x) x)) <1 т
2
$ (1-22
T Bxnxx
(39)
91 (y (x) (z)
Advanced Analysis of Expected Inverse Confidence
The expected inverse confidence Egex (vt [z)] represents a critical quantity for understanding
emror propagation across generations. We analyze this quantity by introducing the random variable:
(
91 (y (x) z)
Utilizing the tail-integral identity for non non-negative random variables:
[x)) = P(Z(x) 2 1) di
1dt / co P(Z(x) 2) dt
To
++
P ((((x) dt. dt.
From equation 37 for 2 1, we derive:
4t
( -
P P [(xz)
t
Drv (91( xx) Gol- (-)] +
2t Cr.
sareshold-basede Analysis of Inverse Confidence Distribution
We define the bounding function:
4t B(I)
1
1-1 Drv (9 (-
Go x)) +
2 21
Cn.
Since (t) is monotonically decreasing for >> 1, focus on the non-trivial case where (t))
then there exists a unique threshold to (1, टτ such that:
2 1, 1 < to.
B(to)= and B(t)
< 1, to t
This threshold partitioning enables us to refine our expectation bound:
d/ B(t) dt.
[Z(z)) < + (to 1)+
t0
25

---

Explicit Evaluation of the Integral Term
For analytical tractability, we represent (t) in the form:
t 1 C2
B(t) A -- C +
2'
where:
C= C
A 4Dv (91 x)) Gol. xx),
This decomposition yields:
E [x)) to + C 1
to
t6 (1--
/ + +c,) dt.
Co
Through careful integration and algebraic manipulation:
C0 A
1 dt = A
Co 1
+ C.
- to + log
+ C'log
t0 t
t-1
t
Cu
tu
Coto
(
S A
+ log
+ Cloo
Co
to
1
Co
Determining the Threshold Value 4g
The threshold lo characterized by the equation B((o) Under the reasonable assumption that C + < 2, we can express la explicitly:
VlC, - + +1
10
(40)
2 - (C + 2A)
This allows to derive the lower bound:
2A
1
(41)
2 (2A + CT)
Analytical Expression for Expected Inverse Confidence
After substituting the threshold expressions and simplifying, we arrive at:
2 C, + A
C log Co
1
E
log
+
(42)
Gn (x)
Co
Consolidated Bound for G2
Denoting C,.1 = Egex
and combining equations 38 and 39 we establish:
Gi (1TT()))))
Pawxx [92 (x) x)) & - 7] < - Drv (Ga( x)) x)) + (1 ) C..1-
(43)
Generalization to Arbitrary Generations
To extend our analysis beyond the second generation, assume uniform distributional shift across generations:
Drv (Gi(- x)) G.-1 x)) DTv,
This assumption enables derive the recursive relation:
Cz.t + A
1 A log
1
Ct,t
+ Cr log
Co
A
Co
C
<
+ log
+ log
(44)
Co
Co
2
26

---

Solution to the Recursive Relation
We reframe this as a standard first-order linear recurrence relation:
Cz,t S aC + 3
1 where := ( (d. + log Co > 0 and 3 == Alog A. Crucially, > 1, indicating exponential growth.
Unrolling this recurrence yields:
Cy.t S a
S a (a + - aB +
< a a
S aa C2 + 8
k=
Applying the formula for finite geometric series:
t-1
1 - a
Σ k=
This gives:
Cg S a' Cy
-
-1-
B
1-a
With simplification and substitution of our specific parameters:
C..tt S (C, + A log 7
log
(45)
2Co
2
Co
Asymptotic Error Analysis After Generations
Extending our analysis to the terminal generation T, we establish:
Pawx [9r (y (x) 1z) S r] -
1 Drv + (C, + Drv log DTV
1
1
1
T
+
log
2Co
2
Co
(46)
Correspondingly, for the complementary event:
P (GT (x) x) 2 1- - _ Drv - -Cw-(C,+ + Drv log
1
1
)
log
Div
2Co
2
Co
(47)
Quantification of Model Accuracy and Error
For the final model Gr. we define y (x) = arg max, Gr | x) as its preferred prediction. The accuracy is then:
Ace(Gr) P (vt (I) y² (x))
P [vit (x) y (x) Gely (x) 2x) - 7] PaNx [GT (ut (x) (x) 2 1-7
2 (1 ((+)) (1 (1-
1 -Dv (C, Div log
( 1
1 T
+ log
DIv
20
Co
2 1 - 11)(()) 1 1- c-c())
1
1
1
Drv (1 (C Drv
+
log
DTV
2Co
Co
(48)
27

---

The corresponding error rate is:
Err(gr) = (x) + (x))
1 <c(r) + - Drv (C, + Drv log DTv
T
20% +
log Co.
(49)
The critical observation is that if 20% + log 1 > 1, the error term grows exponentially with T. Co inexorably leading to model collapse as + oc. This establishes the theorem.
G Proof of Theorem 2: Stabilization via Real Data Injection
In this section, we prove Theorem 2 which establishes that incorporating a sufficient proportion of real data each generation can prevent model collapse in recursive training. Unlike naive training that relies exclusively on synthetic data, this hybrid strategy anchors the training distribution to the ground-truth data manifold, thereby mitigating the accumulation of distributional shift across generations.
The proof shows that even modest amounts of real data are sufficient to dominate the recursive error dynamics and ensure bounded generalization error over time. This result provides theoretical justification for widely adopted empirical practices that mix real and synthetic data to maintain model quality, We now present the formal derivation.
Proof of Theorem 2 We establish our results by analyzing the error propagation recursive training paradigm that incorporates proportion of real data at each generation. The remaining proportion of data generated synthetically by the previous model.
Foundational Definitions and Decomposition
Let the training dataset for generation be mixture of real data and synthetic data:
D. = a Dreal + (1 a.) D symibetic
where Drcal = ((I,)') x y ground-truth label] and Deynthetic {(x 0) x X, (x)1
We define the mixed distribution as G. = ag* + (1 - a) where g" represents the ground truth real data distribution.
We define the subset of inputs with sufficient real-data confidence under the base model Go:
DJ = { x X' Go
T 2(1 a)
(v"
x)) 2
where y" is the ground-truth label, and r the confidence threshold.
The probability of low-confidence generation at generation can be decomposed as:
P X (G x) & T] < Pa~x [Gr | xx) 1 7, E D
+ P P [r & D.).
(50)
Analysis of Model Divergence via Total Variation Distance
To quantify the disagreement between consecutive model generations, we leverage the total variation distance. For any input x and output y::
G1 ( 2)
g (y =)] 2Drv (GI 2)) gTi =))
This inequality can be reformulated to establish a lower bound on the TV distance:
Drv (9 gmix =)) 2 G (vt (z) x)) gri (z) (x)
28

---

For inputs in the high-confidence region Dj", we have:
Go (y, (z) x)
2(1 a))
When 91 assigns insufficient confidence to the optimal region, i.e., Gi (x) x) 7, the
discrepancy between models is bounded below:
(1 2) g (u (x) x)) a (1 (1) a) (1-1)11211-a))
(1 r) = 2
2(1 aa)
This allows us to express the probability of low confidence for inputs in Dj in terms of the TV distance:
Pa~x (Gi(y (3)) x) S -,, = Drv (gil. x), (.
(51)
Characterizing the Complement Set via Inverse Confidence
Let C. == Ex-x
denote the expected inverse confidence under the base model.
(u (=)]=)
Applying Markov's 's incquality, we derive:
Pa~xl= $ DJ] Panx
Go (y (x) (x) <
2(1 -a)
(1- 2(1 α))
T
Pane
Go (ut (Ir) xx)
7
Bxnx
2(1 a))
Gu (y (x)
Cr.
(52)
2(1 a))
Integrating equations 51 and
52
into
50
we establish the comprehensive bound for the first
generation:
4 Pa-x ()(((()) x) 1 11-]] Drv (91(- (T) |x)) + (1- 2(1 a) Cr- (53)
Recursive Analysis for the Second Generation Model Gz
Following an analogous approach, we define the high-confidence region for the second generation:
DJ= { x x G1
2(1 a)
Then, we obtain:
P P Co x)) -1-,,-- E Dt] < Drv (g2(. x), (),g (. (.1)))
(54)
For the complement set:
P & D = PaNx [ Øi (yt (x) x)) < 1
2(1 a)
T
Exax
(55)
2(1 a1)
Gi (3 (x) ②
29

---

Advanced Analysis of Expected Inverse Confidence
The expected inverse confidence E error propagation across generations.
91 (v. (x)]z) represents a critical quantity for understanding
We analyze this quantity by introducing the random variable:
(x))
9 (y (z)
Utilizing the tail-integral identity for non-negative random variables:
E [[))) = P(Z(x) 2 2()d de
[ 1dt S,¹ + P(Z(x) 2 t) de
1+ [ to ((((x)) t) dt.
From equation (53)
for 2 1, we derive:
4t
Div (91 x)) g 22)) + (1
PRN,
[(xx)
I -
2(1 ait Cz.
2(1 - a)
Threshold-basel Analysis of Inverse Confidence Distribution
We define the bounding function:
4t B(t)
t-1 Drv (Gil (x) x)) (1
2(1 a)t Cz.
2(1 a)
Since (()) monotonically decreasing for >> 1, we focus on the non-trivial case where (1)) <
then there exists a unique threshold lo (1, Co such that:
1, 1 1 S to:
Btoo) and (t)
S 1, to S S Jh
This threshold partitioning enables us to refine our expectation bound:
f d
MV [x)] S 1 (to 1)
B(t) dt.
Explicit Evaluation of the Integral Term
For analytical tractability, we represent B(t) in the form:
B(t) A 1-1 + +o+ + (1-
1
2(1 2(1-a) a) Cr.
where:
C
4=4Dry (G1 | gTix (+12)),
C =
2(1 -11-a)
This decomposition yields:
1
c
( + dt.
E [[())) $ to
1
C,(
to) +
2(1 11--0
Co
30

---

Through careful integration and algebraic manipulation:
|
c6 A
1
Co 1
+ C
dt
A
to
log
+ Colog
t t-1
t
Co
to
Coto
to
1
1
2 A
log
Colog
Co
to
Co
Determining the Threshold Value
The threshold tn is characterized by the equation B(t)) = 1. Then, we can express to explicitly:
2A + -1+
C 1+ V (L- +1) - - 2(1-a)
C2
+1)
C₂ 2(1-a)
10 -
-2A - 1.0--0
C2
+2
T-a
2A C. - 1+ v
I-a V
Cga
+ +1)² + (2A +,4+202
I-a
C_
-2)
(1-a)
C
(56)
-24 20, + 1-a
C₂
+2
This allows us to derive the lower bound:
2A 2 (2A + C.)
12
(57)
Analytical Expression for Expected Inverse Confidence
Let A = Drv (G1() x), 9Tm (. x)). After substituting the threshold expressions and simplifying. arrive at:
Co
41 t
1
1
E
1 +
A +
+
C,dt
(iyyt xx)
1
2(1 a)
2(1 (1--att a)t
A
+ C, log
1
$
C
+
+ A Alog
(58)
2(1 - a)
Co
C.
A
2(1 a)
Co
Consolidated Bound for G2
Denoting C..1 E
and combining equations 388 and 39
establish:
G. (x+((²)
P Faox G2y (x) x) <1--11 Drv (Gol- x) x)) (1 2(1 - a))
Cr.1. (59)
Generalization to Arbitrary Generations
To extend our analysis beyond the second generation, we assume uniform distributional shift across generations:
Drv (Gx(- x)) 9- x)) = Drv,
This assumption enables derive the recursive relation:
Cy A
C log
Cz.t S
log
+
Co
1 - a
Co
1
1 C Alog
log
(60)
2(1 a))
Co
2(1
σ))
Co
A"
Assume > 1
T + co, then we obtain:
2( 1)
1 -
a
Cg,r S
A A
(61)
Co
A
31

---

Asymptotic Error Analysis After T Generations
Extending our analysis to the terminal generation T. we establish:
Pa-x (9T (yt (x) 12) $1- _ Drv +
a
Drv log DTV
(62)
Co
Quantification of Model Accuracy and Error
For the final model Gr, define yF arg m GT x)) as its preferred prediction. The accuracy is then:
Acc(GT) P Panx (x) y
(x()]]
PPnx [v, (x) yz)xx) Go (2) xx) 2 1211111 1-1 - [or (x) (x)
21-1
2 (1 c(1)) (1- DIv 1
1
1
Drv Divllog log
Co
Drv
2 <(>) 1- Drv (1 (1- =(1))
1 - Drv log
1
(1-e(1))
(63)
T
Co
DTV
The corresponding error rate is:
Err(Gr) = Pawxxyt # (x)]
Se + 1
Drv + 1-a
Drv log
(64)
Co
DTV
Conclusion
The inclusion of real data at each generation mitigates error accumulation, ensuring that the model maintains stability over recursive generations. This completes the proof.
H Proof of Theorem 4 Theoretical Guarantees for Transformer-Based LLMs with Self- Verificaiinn
In this section, prove Theorem 4 which extends self- self-verificttion framework to autoregressive Transformer based LLMs. This result establishes that, even in high-capacity architectures, self- verification can effectively prevent eиror amplification across recursive training generations.
To enable this analysis, we assume that the Transformer model parameters are uniformly bounded, standard condition in prior theoretical studies (Li et al. 2023 Hu et al. 2024 Zhang et al. 2025 Fu
al. 2025 as follows:
Assumption 2 (Bounded Parameters [Zhang et al. 2025 ). The parameter space θ of the Transformer model is constrained as:
0 0
ADD+1..
< BA, max (11(01,17200)
1,, | s BAL1
|| & BA..
=
1.2
[
|
| S By for all 1 E [D), i i mm).
s Bo,
< BR.
where h denotes the number of attention heads, and BA. BA1, BQ. Bx By > are pre- specified constants.
Under this assumption, we then present the complete theorem statement characterizing the final test envor after 7 recursive steps.
Theorem 5 (Error Bound for Transformer-based LLMs with Seff-Verification). Let GLLM S) denote the conditional generation distribution of a Transformer-basedd L.M at the 1-th recursive generation, where S S X is token prefix sequence and X is the next token. Then, after
32

---

T recursive updates. the final prediction error is bounded with probability at least 1 8:
Err(Gor S <(>)
+
-+
1
log
- ())
lo
1
NCo
1
log 1 +
))
T
C N
+ N
+
(NCo
log
log
(
(65)
N²Co
NCO
Co
Furthermore, if obtain the bound:
Co log
( then as T oo, the exponential decay term vanishes, and
"rr(Gorr) <(7) (+ NCo log +)
l)g(nB)D Vn
1 N
log
log
where = mll is the total number of training samples, and D = D d. (dr + dn + d) + d dy and B = R B By By reflect the model capacity and prior radius respectively.
Our proof draws inspiration from the PAC-Bayesian analysis developed in Zhang et al. (2025] introduces new techniques to handle recursive training dynamics. In particular, we leverage recursive coverage coefficient to track how the model's confidence evolves across generations, allowing us to rigorously control generalization error under fully synthetic training. We now present the formal derivation.
Proof of Theorem 5
We present the proof as follows.
MLE Initialization and Sequence-Based Setup
We assume that all token prefix sequences S, = (.1...., ..) are elements of a structured sequence space S C 2 where X' denotes the token vocabulary and the maximum sequence length. This ensures that all context representations during both training and generation reside within bounded and well-defined domain.
The initial autoregressive language model GHL is trained via maximum likelihood estimation (MLE) on corpus of sequences {(x(1)
(i) by
The training objective is given by:
On arg max dELLLM
log M (x(1) s(O),
t=1
S and n = ml is the total number of nefix-conditioned samples. The resulting model gLiM serves as fixed reference distribution encoding inductive biases from human supervision.
where 5 = (a) (1)
(1)
Defining the Confident Support Set St
To identify reliable contexts for self- veff-verification, we define the verification score function as:
Sverity (x ;= log GLiM(a (z (S),
and the corresponding high-quality candidate set:
x (S) = x E Sverify (x: S) 2 max 8verify (S) -5)
z' Ex
We then define the confident support set S as the subset of prefixes where the reference model assigns sufficient mass to the high-reward region:
St = { {
log
SES
G CLL (* S) 2
N
33

---

Decomposing the Probability of Low-Confidence Generation
We aim to bound the probability that the updated model assigns low probability mass to verified candidates:
Pa-S [ (a+(()
1 - < IP s [ (1x, (S) 11-,, S st]
+ Ps-ss [S & st]
(66)
Bounding the First Term: Monte Carlo Estimate for Verified Support
Let Ni be the empirical estimator from N samples drawn from Gb, For S E ST, the probability that none of the samples falls into x. (S) is most:
(1 A (x) ()) N - (1
log T
{s))s=
Σ
N
2
Hence,
(n (S) 2
Bounding Discrepancy via Total Variation Distance
The total variation distance between two distributions P and Q over X' is:
(rv(P)))= 1 Σ Σ P(r))
#EA
For any event A C X, it satisfies:
[P(A) - Q(A) & 2DTv(P.Q).
In particular, if Gb (ax (S) S) $ 1 - and (2, (S) 5) 2 1 3. then
Gym gi 2 2
= Drv 2 4
Taking expectation:
4 -
Psws (x: (S) S) < 1 1- T, SEST]
Essss
(67)
Bounding the Second Term via Markov's Inequality
Define the expected inverse support quantity:
C = Esns
GLLM (x (S) 8)
Then, by Markov's 's inequality:
log
Psws(S (S & ST = PP
cUI [x (S) () <
N
log
PP
> 1
N GLLM (x. (S) ())
C
2
log
(68)
T
34

---

Final Bound for the First-Step Generation Error
Combining inequalities and and substituting into the decomposition, we obtain the following upper bound:
C2
2 log N
PswS [oo,
5) S 1 -] & ESSS (Dv((S)) (11)). [S)] + x (S)
(69)
Since Gb) S) is the maximum likelihood estimator of the conditional distribution gi ( S) based on sampled data, we now focus on analyzing the expected total variation distance Es-s [Drv (.) (-1S), (
Step 1: Error Decomposition under Token Prefix Sequence Modeling
begin by establishing a framework for analyzing the error. Consider our dataset consisting m token trajectories, each of length &
D D= (
(
0
}
m
j=1
where each token prefix s(
(z) (...,,(())
To measure how closely our model approximates the reference distribution, we define the log- likelihood discrepancy function:
m
log (x s( g
L(O,D)
GLIM (+t1) s(5))
For applying the PAC-Bayes framework, we introduce a ghost sample D = ((5))) where 5(1) s( and 200)
-b0) me ,1tt1)
|=
(- (())). We then define:
( = L(,, - log ED exp ((().)) D]
PAC-Bayes Bound under Token Prefix Modeling
By the standard PAC-Bayes lemma, for any distributions Q, P A()) where P may depend on D, have:
Barr [g(()) < KT((PIIQ + log Ra-q [ex)(
Substituting the definition of g(f) and taking expectation over D:
Ep exp Eg-p L(0 D) - log Bn (e)p((() KLL(1))))) 1
Applying Chernoff bounds, with probability at least
8, we obtain:
EEg- [oog Ep ex D)) [D] < - (LL((II) + log
1
35

---

We can dower-bound the left-hand side using two-step approach involving Jensen's inequality and total variation distance:
E
log
exp(L(0, D))
D|]
1
m
9i Ni (a
()
15(0))
2
log Ep
exp
log
GHl (2 5(00)
DD
j=1
log 9 (1 ((0))
Ba-p
log En
exp
()) (((01)
D
m
(( Gb (+ (( D (9
9e Gul () log
(x (((i))
Banp
log
exp
D
2
GL (I 5((i))
j=1 t=1
Next, for any reference parameter 0 E O, we decompose the expected loss:
-Rg [L(0, ((0,,,) D)] -. [T(00r, (I((1, D) [_(0" D)) + (I(0, ())]
< L(0", D) + Eg- (00, D) L)),
where we use the fact that 0 minimizes L(0, D). In the realizable setting, there exists l such that Gy-M (- (S) = (i S).
Combining these bounds, we obtain:
(i s( G. |(
(70)
j=
1 Equp
1 m
(+) s(:))
log Eg
exp
log
D
2
G GLIM (x(t)) ((()
m
i
(1 gy (x
GHLM (1)
((
Egup
log
+ KT.(PIQ) log
(71)
j=1
The first two terms capture the fluctuation under the posterior distribution 0 P, which will address next.
Step Controlling the Fluctuation via Structured Posterior Sampling
To control the fluctuation terms, we define the posterior distribution P E A(9) as product distribution over each layer of the transformer:
p= II ((()))
(72)
1=1
Lp(002+1) Unif (B ((( (1-111)11111))
CP()))) Unif (B(?(") (9) 1.2.1-11)) Unif (B(55) 7. 11D) LP(A(0) -CP(((()
(t)
(r(A())) Unif (B(A(() A1. |l (Ir)) Unif (B(A() (1) 11-)
(p(w()))
II Unif [B -00 || -1l)) Unif (B(W,
K.(1) . -I=))) Unif (B(W ,0 -IIIE))))
(73)
36

---

for all t E [D], where (ni, r, II)) denotes the uniform distribution over ball centered at with radius under norm -
The radii are carefully calibrated as follows:
(1)
21 R (1 + BA,1BA.)) a /mm))
(0) R-- R
722
(0) R- B (m)l)
(1) R R 1,,,0 (007//(mm) (mme),
A22
rV
1 R 1-'ha11 BA,1 (A,2) -
0, a0,
Q
(1 R 'h
"1 + BA, B.,2) (Br)) Bx az /
( Rー (1 BA,1BA.2) -B BBy B a; (mL))
( = = =TB7//mm))
KK"
This carefully structured posterior allows us to prove the following key result:
Lemma 6. Zhang et al. (2025) With the posterior distribution defined (72). which is uniform
distribution over neighborhood around 01 with radius proportional to 1/(mL), ) we have:
go (2(3)
((())
2
B
log Euj
exp
2
log
Gj (1441 ((())
())
D
1=1
(x 1(((0)
Baw
log
=1)).
(74)
GLM
t+] (s(0))
(1)
Proof. A detailed proof can be found in Appendix F.2 of Zhang et al. 2025]. The key insight is that the construction of P with specific radii ensures that the model outputs remain stable under small parameter perturbations, allowing us to bound fluctuation terms by constant.
Bounding the KL Divergence Term
To complete our bound, we need to quantify the KL divergence term in (7 We define the reference prior distribution Q E (()) as:
D+1
Q II 1=1
Co(o()))
with the following component-wise structure:
Final linear layer:
Cq(0(D+1)) Unif (BB 111,)))
Transformer layers / E [D]:
Co((()) Unif (B(1/2,1/2,1 ()) Unif (B(1/211/2,| ()) Lo(Ai)) Lo(W()).
(a()( = Unif BA,, [[]) Unif (B(0, BA2, [le))
h
CQ(W(0) II Unif (B(0, Bq, || [[e)) Unif (B(0, Bk, ll [le) Unif (B(0, Bv. [le))
The KI. divergence between this prior Q and our posterior can be bounded as:
(T((P))) o (D2 d. (dr + dn + +d) + d dy) log (1 mf RhB BA, BA,2Q Bx By
O 1o log (1+ mLB))
(75)
This term represents the complexity of the posterior family relative to the prior and controls the flexibility allowed in posterior fluctuations under the PAC-Bayes objective.
37

---

Deriving the Final Bound
By combining 74 and
75 and substituting into 7 we obtain:
m
(9
(| s 0(1)+ 0 log (1+ meB)) + log
t=
Taking the square root and using Jensen's inequality, we get:
Div(gi, ( 1. ss()))
ml
m Dtv(91.v(-) 1( (. (s("))
(ml)² jmltm
D. log
[ mBB)
+ log
ml
Based on Lemma F.4 in Hu et al. 2024 we can further bound the difference between the empirical and expected total variation distances:
m
T
E g [Div(Si,N (((())
1 G G. ((((()))))
1 G. s(
ml
ml
1 D. log (1 (1+ 11+m + log
(76)
ml
Therefore, with probability least
5, bound the expected total variation distance:
D log (1 mm + log
Es-s (-1)). S))] &
(77)
ml
Final Bound for CLLM
Returning to our original objective in 669
we can now derive the final bound:
C 2
Psws [9 (zt S) 1 7
Esss (Dv((G (- 9i [S)] +
log
D log (1 + mLB)
+ log
C₂
2
2A
log
ml
7
D log (mê C
24
log
log
(78)
ml
N
Let ml denote the total number of samples. Thus, with probability at least 1 8, we have:
D n
1 +
C, N
Iss.
[GL (a: (S) < - r] S
log
log
(79)
5
This final bound characterizes the probability that our trained LLM fails to assign sufficient probability to the desired completion. The bound decays with the square root of the sample size but depends on model complexity D and scales inversely with the threshold parameter
38

---

Recursive Bound for the Second Generation Model gz
Definition of Verified Prefix Set for 9
We now extend our analysis to the second generation model by defining subset of prefix-conditioned inputs that have sufficient verified coverage from the first model:
log 2 N
D S S
Go (v (S) SS)
where the verified high-reward candidate set y. (S) contains tokens with near-optimal self-reward:
yズ (T)
y UEW x 8verify (v/ () 2 max Sverify (v/ |S) - 7).
y'EX
Bounding the Low-Confidence Region for C.L.M
Applying the same total variation-basee analysis for the first-step model, we can bound the probability that Øbz assigns low confidence to verified candidate regions:
Ps-s 990, (, (S) 1--1, SE D S - Drv (go S), 92. (S).
Bounding the Failure Rate of Verified Prefixes
To complete our analysis, we need bound the probability that prefix falls outside the verified region D Using Markov's inequality, we have:
log N r
sS $ D = PS-S
CLLM
(y + (S)
(80)
log 2
= -Pss
(81)
N GILM
(y (S) S)
log
EswS
(82)
N
gul (y (S) S)
This leads us to define the reciprocal of the model confidence as random variable:
Z(S)
GLLM (S)
Using the tail-integral identity for expectations:
Es-s Ss~s\ - / P(Z(S) 2 t) dt,
and combining with our PAC-Bayes bounds on the total variation distance between Gui and Gi we can derive:
log ( +C
Cy.1 = Es-5
1 A Co
log
GLLM
(y*(S) 5)
Co
Co
where:
I A V n D log represents the statistical error from PAC-Bayes bounds
C Cy is the approximation emorr from the confidence threshold N²
Co represents the minimum prediction confidence of GHyM across verified high-reward candidates
This bound quantifies how low-confidence predictions propagate into the next generation model.
39

---

Solving the Recurrence Relation
By extending our analysis recursively, similar to the proof of Theorem 1, we can derive recurrence relation for the expected reciprocal of model confidence after generations. The solution this recurrence is:
1
C, S 1+ A log
Co
A
- +
Co
Ct
Co
log
Co
1
< 1
Drv log
Co
DTV
(NCo
log
C.
1 1
Dry
Co
log
DTv
1
1
log
1 Co
Co
N Co
log
1
Co
Co
51+
1
V
/
I ln)
log
x log (vn)
1
Co
n
C N
NC, log
log
(83)
Co
N
Co
Co
This expression shows how the error compounds over iterations, with three main components: 1. A statistical error term that decreases with sample size n 2. An approximation error term that decreases with the number of samples N used in self-training 3. A recursively propagating error term that decays exponentially with the number of generations f
Final Low-Confidence Bound After T' Recursive Generations
After I' recursive updates, with probability at least - 6, the probability that the final model assigns low confidence to verified top candidates is bounded by:
1
(+)
1 V
1
PS~S [9 (1c, (5) (S) < -1
log
log(n (B) log
NCo
n
( N
1
1 Co
C2 log
log
+(NCo
log
N2Co
NCo
N (84)
This bound reveals that as 7' increases, the third term vanishes exponentially, demonstrating that self-training eventually eliminates the initial confidence errors, provided that NCo 1, which is typically satisfied in practice.
Final Accuracy and Error Decomposition
To translate these confidence bounds into concrete accuracy guarantees, we define:
(S) = arg max G (x 1S); x² (S) == ground-truth token following S. #
The accuracy of generation after T steps is: Aco(gH)M) == F s (x+(S) = x (S)
By decomposing this probability and using our previous bounds, we obtain:
Acc(Or) = Ps-s (S) x² (5)
(a: (S) 5) 2 1 r] Pswss [ (x: 5) 2
--
11-((7) <(7)) -
1 log
(1)
1
log n B3 log
NCO
(1-c(1)) 1 + N²Co
log ( +
1
C N
1
log
log
-
(85)
NCo
Co
40

---

The corresponding test error is therefore:
Eir(SUr)) = 1 - Aco(Sotr) = PS.S (2](5) + x {S)]
Which satisfies:
(
1
log ( ln log
1
Err(OoTT) S <(>) +
NCa
+ 1 N
(+) +(NO
C. N
+
N²C,
log
NCo
log
Co
log
(86)
This error bound consists of four components: 1. An inherent approximation error <(7) from the 7-approximation of optimal reward 2. A statistical error from finite training data that scales as O(1/vn) 3. An approximation error from finite self-training samples that scales as O(1/N) 4. A rapidly diminishing initial error term that decays as O ((I/NC.)T+I)
This analysis demonstrates how recursive self-training progressively improves model performance, with the error bound decreasing as both the pre-training dataset size and the self-training dataset size N increase. Moreover, the henefit of additional recursive generations T is most pronounced in early iterations, with diminishing returns as T' increases.
I Exponential Error Growth in Naive Recursive Training of Transformer-Based LLMs
In this section, we analyze the failure mode of naive recursive training applied to Transformer-based large language models (LL.ss), where models are trained exclusively synthetic data generated by their predecessors without any verification or external supervision. This setting, though increasingly common due to the scarcity of real data, introduces significant risk: even small inaccuracies in early generations can compound over time, causing the model to drift progressively further from the original data distribution.
We formalize this phenomenon by characterizing how prediction errors propagate across generations. Specifically, show that the absence of quality control leads to exponential error growth, ultimately resulting in model collapse. This result highlights the inherent instability of unverified recursive training and underscores the necessity of intervention strategies such as real data anchoring or self-verification. We now present the formal theorem and provide its proof.
Theorem 7 (Error Propagation for Transformer-based LLMs in Naive Recursive Training). Let GHA (x S) denote the conditional generation distribution of an autoregressive Transformer-based LLM recursive step I, where S S C X is prefix sequence. Suppose the model trained via naive recursive generation without any self- mechanism. Then, with probability at least 1 8, the prediction error of g is bounded as:
1 Err(Cht S (((T) FV log(nB) Doog log
1 (,,C,+ V Dlog bos (++=+ log 1 T Co
)
where C. := Es-s
denotes the expected inverse confidence assigned to top-reward
GLLM (8)1S) candidates under the initial model.
Note that since C..y > 1, the error bound includes exponentially growing term of the form
(c+ log((
(d. log d.) )²
T
As the number of recursive steps T + xo, this term can dominate the total error, reflecting the compounding effect of unfiltered distributional shift. Without confidence-baset filtering, the model may gradually amplify its own errors, eventually leading model collapse.
41

---

We establish the following result by analyzing how prediction errors propagate across recursive training iterations in the absence of verification mechanisms. While our proof strategy parallels the one used in Theorem it introduces key modifications to account for the lack of confidence-based filtering or quality control during training.
Proof of Theorem 7
Foundational Definitions and Error Decomposition
We begin by defining the subset of prefix-token inputs for which the base model 9 assigns sufficient probability mass to its top predictions:
S = {s { e (a: 5) 2 1 - 7
where the high-reward candidate set is defined via an internal self-reward mechanism:
xS) { x E
( () 2 max 'ar(aa |S) 7) -
ExEX
and the self-reward function given by:
Tsetr (ar S) == log D (xr (S).
We aim analyze the likelihood that the model Gg trained recursively without any verification mechanism, assigns insufficient probability mass to its own top candidates. We decompose this event as:
PswS (a)(()(() () S 1--1-7 r] s Pa (a (S) () < 1 1--1 S St]
1]
+ Ps-s [S $ st].
(87)
Analysis of Model Divergence via Total Variation Distance
To quantify the discrepancy between consecutive model generations in the absence of verification, we analyze their divergence using the total variation (TV) distance. For any input prefix E S and output token æ E X, we have:
CL (S)
CLLM (2
|S]
2D (GU ()) Gb ()).
This inequality can be inverted to obtain lower bound on the TV distance between generations:
1 gHi (x:(() S) CliM (x= (S) S)
Drv (G (). (- S)) 2 2
For prefix contexts in the high-confidence region S the base model assigns:
Gio (xt S) 2 1
T 2
If the updated model assigns insufficient mass to this set, i.e.,
(a S) < T,
then the discrepancy is bounded below by:
(x: (S) S)
( (S) S) 2 2
Consequently, the probability that the updated model fails to retain high confidence on verified inputs can be bounded by the expected TV divergence:
Ps-s [GL (o (S) 1 5) $ 1-1, Se st] < EsSS [Drv (9) 8). S))] (88)
42

---

Characterizing the Complement Set via Inverse Confidence
Let Ciyy := Es
denote the expected inverse likelihood mass assigned to high-
GLLM am (8)]8)
reward candidate sets under the base model. By applying Markov's inequality, obtain:
ss $ St]
Ps-s
gli
(at () < 1 2
7
1- 3
Psn
GIi (S) S) > 1
2 -ESSs
(axM( (S) |S)
2) Cy.
(89)
Combining equations (35) and (36) into the decomposition (34.. we arrive at the following bound for the first recursive generation step:
4 -Es-s Drv (GH S), G.MM [[[S))]] + (1- C..
Ps-s (90 (00t S) < 1 1-
(90)
Furthermore, from the proof of theorem 3, we know that:
EsnS [Drv (Go))( (.1 (). G (- (-1 [])] & v log (nB) D log 1 à
(91)
Thus, we have
1 + (1- - Cz. V Vo log (nB) D log g
PswS ( (5)
S
(92)
Recursive Analysis for the Second Generation Model g
Following analogous approach, define the high-confidence support region for the second generation model:
S = {s e gi (at(S) S) 2 1 う
For inputs S E Sz, a discrepancy in predicted probability mass between GHLD and G.I on high- reward candidates leads to the bound:
4 _ Es-s [Drv (GH 5), 9 G.L S))] (93)
PssS [ (x: (S) -,- - S
E
For the complement region S \ ( apply Markov's inequality as before:
Psws
[S $ St
Psns
(og ( (S) S) < 1
(1-2) 2
Esas
(94)
GLiM (x. (S) SS)
Generalization to Arbitrary Generations
We now generalize the inverse-confidence recurrence to arbitrary recursive steps. Let Cr, = denote the expected inverse confidence at generation Then the recurrence
Es-s
gULM (8)18)
43

---

relation is given by:
Cra S C A
1 1 + Aloog A Cy,1-1 log Co
1
Co
<(20 +og 2.) ) Cra-1 Alog 1 A'
Co
(95)
where A = / () D log & reflects the TV-induced generalization noise, and Co is the mini- mum confidence all prefix-conditioned top predictions in earlier generations.
Solving this recurrence yields the following bound:
Cy & (Co+Alo8 1)
( (d6 log _
)
(96)
Co
Asymptotic Error Analysis After I' Generations
Finally, applying this bound to the error probability after T recursive generations, we obtain:
Ps-S G (ae (S) 8) 1-
A (1 (1- - Co CG+ Alog 2) ( Co
"
T
log Co)
1 V
8 _ Cr (C, V log(n log } (d)
log Co )
T (97)
log(nB)Dlog
Co
This bound highlights exponential sensitivity to both the minimum confidence Co and the number of recursive steps T, in the absence verification mechanisms.
Final Accuracy and Error Decomposition
We define:
(S) == arg max OLLM sEX
(I S); x²(S) := ground- -ruth token following S.
Then the accuracy of generation after T steps is:
Acc(G]))) = Pa-s [+)(S) = =S)]
and satisfies:
Ac)(Or) PSns (x) (5) g0r (x (S) 1) 1- T] Psws [ (x+ (5) (S 2 1-1)
2 1 1() (1 - <{7)) _ V I log(nB)
- (11-(7)) C + log(n log 1
(d) + log Co
T
(98)
The corresponding test error is:
Err( 1 Acc(ClIM) = Psns (2+(S) # x (S))
and satisfies:
(5(7)) + S V Log(nB)D log 1 - C,,, + v (ogg(nB))Do log 1
( + log
1
-
Co
(99)
The proof is completed.
44

---

J Mitigating Collapse via Real Data in Recursive Training of Transformer-Based LLMs
In the previous section, we demonstrated that naive recursive training of transformer-based LLMs- without any quality control external supervision- exponential error growth across gener- ations. This failure arises from unfiltered distributional drift, which causes the model to gradually reinforce and amplify its own mistakes.
To mitigate this collapse, one natural strategy is to incorporate real data into each generation of the training loop. In this section, formally analyze how the inclusion of ground-truth supervised examples affects recursive training dynamics. Our key result shows that introducing even a moderate proportion of real data can effectively suppress exponential error amplification, ensuring bounded prediction error across generations.
Specifically, we consider setting where each generation G. is trained mixture of synthetic and real data. Let E (0, 1) denote the proportion of real data retained each step. We show that this mixing strategy induces recursive contraction effect, reducing the sensitivity of the model to its own erroncous generations.
Theorem 8 (Error Bound for Transformer-basee LLMs with Real Data). Let G (x) S) denote the conditional distribution of a Transformer LLM at recursive step t, trained on a mixture of real and synthetic data with real data proportion O. Then, under the same notation and assumptions as in T'heorem 7. and assuming a > 2,1/Co-13 and T + 0o, the prediction error of 9 is, with probability at least 1 - 8. bounded as follows:
Err(Gr) S <(7) +(7)+²1 Dlog(nB)
1
1 Co V log(nB)
D
(100)
Compared to the exponential growth in Theorem 7 effect, effectively anchoring the model and limiting the accumulation of distributional shift.
the presence of real data introduces damping
We now present the proof of this result.
Proof of Theorem This proof builds upon the key techniques established the proofs of Theorem _ and Theorem 7 particularly the decomposition of error into high-confidence and low-confidence regions, and the recursive control of inverse confidence via total variation distance. We present a concise yet complete version adapted Transformer-base based LLMs.
We analyze recursive training process where the model Go at generation is trained on a mixture of real and synthetic data:
a Dreal + (1 - a) D Syyothetie:
and the induced mixture distribution is:
a Gm a G + (1 - a)) I.
Decomposition of prediction error
We define the high- high-confidence region:
DJ x x X' G (y)
CLIM
|x) 2(1 a)
t
and decompose the total error:
P Pa [Gb. (y"
x) $
<
( (y* 11) 1 DJ1 + [ D.
Bounding the first term via total variation
Using the standard TV bound:
GLuM xx) - G. (y x3)= < 2Div 2DIv(CllM.. g.mix),
get:
P [9 (y (x)
x)) I 1 11-17 x D < DTv DTv(GH.. g.
45

---

Bounding the complement using inverse confidence
Let Cy = Bxnxx
By Markov's inequality:
GLIM (1x (z))
P
x D
1 2(1 -a)) Cy.
Cy.
Recursive coverage coefficient control
Assuming a 2(1/Co-1)* we have:
1 Drv log
1 DIv
Cyp x
Co
Finite-sample generalization bounds
In Transformer based LLMs, the finite nample deviation under PAC-Bayes analysis is:
P Pawx [9r (2r" (x) (x) < - Drv + 1 a
DIv log
(101)
Co
DTV
Final Results
Combining all terms, derive the final error bound:
Err(Ght S <(>) + 1 V D log(nB) -
1 1-a V log(nB))
n
Co
where D is the model's effective dimension, B its prior radius, and the total sample count. The proof is completed.
K
Further Discussion on Assumptions and Empirical Validation
K.1
On the Realism of Assumption T
This section aims to clarify both the intuition behind Assumption | and its alignment with real world LLM behavior.
At its core, Assumption formalizes confidence calibrated agreement condition: high confidence model predictions likely be correct. This is consistent with standard assumptions in self training (e.g., Assumption 2 on page 14 Huang et al. (2025b) and aligns with active research on confidence calibration in LLMs, which is well studied area (see Liu al. 2025 for survey).
Critical to realism, empirical work shows modern LLMSs often exhibit strong calibration (Achiam al. 2023]. For example, Luo al. (2025) reports small expected calibration errors for pre trained models like Llama-3-8B (3.52%). Qwen-2.5-7B (5,41%) and DeepSeek- V2-Lite (3.39%) on MMLU.
This demonstrates close alignment between confidence and accuracy.
To further validate this, conducted experiments the MATH dataset with Phi3.5-Mini, analyzing the log probabilities of correct versus incorrect responses. The results are summarized in Table [
As the data shows, correct responses consistently exhibit higher log probabilities, and their distribution stochastically dominates that of incorrect responses. This observation confirms the positive correlation between model confidence and correctness, directly supporting the validity of Assumption T
K.2 Empirical Validation of Self- Verification
To empirically test our theoretical findings on model collapse, we designed an experiment following the setting from Fu et al. 22025] Specifically, trained a 12 layer, 8 head GPT-2 model (with hidden size of 256) to recursively perform in context learning of linear functions from the class:
F= (f f(x) w Tuz WEEEER 15
46

---

Table 1: Distribution of Correct and Incorrect Responses by Log Probability on the MATH dataset with Phi3.5-Mini.
Log Probability Range
Number of Correct
Number of Incorrect
to -0.6
1032
83
-0.6 -1. -1.2
413
462
-1,2 to -1.8
73
386
-1.8 -2.4
24
95
2.4 to -3.0
6
18
For each prompt, we sampled 21,..., Ik; Equery and oi independently from ( Ia). The model's task was predict lquery = w/ Iquery- We compared three distinct training strategies:
Full Synthetic: The model was trained solely synthetic data generated by its predecessor.
Mixed: A combination of fresh real data and synthetic data was used, mixed in 0.5 ratio.
Verification-Bassie Filtering: For each training instance, we sampled 20 candidate re- sponses, selected the with the highest confidence (corresponding the model exclusively on this verified data.
0), and trained
The results of these experiments, measured by prediction error over recursive training rounds, are summarized in T
Table 2: Prediction Error Across Recursive Training Rounds for Different Strategies.
Strategy
Round 0
Round
Round 2
Round 3
Full Synthetic
0.2418
1.1625
1.4245
1.9523
Mixed
0.2418
0.2824
0.3178
0.3235
Verification
0.2418
0.2672
0.2949
0.3105
As observed, the error accumulates progressively with more generations of recursive training. This degradation is particularly severe in the full synthetic case, where the error grows rapidly. In contrast, both incorporating real data and applying the self verification mechanism effectively mitigate increase in loss. This outcome is consistent with our theoretical findings.
K.3 Directions for Future Experimental Work
To further validate and extend the theoretical contributions of this paper, several experimental directions could be pursued for future work:
Calibration Manipulation Experiment: initial experiment could involve selecting model for recursive training and systematically adjusting its calibration via temperature scaling or other established methods. This process would produce model versions with "low," "moderate," and "high" calibration, where the quality of calibration is quantified by the Expected Calibration Error (ECB).
2. Performance Correlation Verification:
Following the calibration manipulation, a sec- ond experiment could investigate the relationship between calibration and performance degradation. The differently calibrated models would undergo recursive training with self-verification, allowing for comparison of their error accumulation over successive generations. Such an analysis would provide empirical insights into how a verifier's cal- ibration quality (measured by ECE) correlates with the final model's test error under self-verification framework.
47

---

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54


Reviews:
[{"review_id": "WN4fghWgOx", "reviewer": "Reviewer_TBjc", "summary": "This paper tackles the problem of model collapse in recursive training scenarios where generative models are trained on synthetic data from previous model generations. The authors propose a self-verification framework that uses internally estimated confidence scores to filter synthetic outputs without requiring external validation. Their approach anchors verification to an initial model trained on real data, which serves as a fixed reference point across generations. The theoretical contribution includes finite-sample error bounds across three recursive training regimes and extends to transformer-based LLMs with convergence guarantees. The core idea is that by filtering generated outputs to stay within a margin of the highest-scoring candidates (as evaluated by the initial model), they can prevent distributional drift and maintain training stability even in fully synthetic settings.", "questions": "Again, these questions come from a place of curiosity as I was trying to understand the paper, not as formal complaints. Answering them would really help clarify the work for me.\n\nConfidence Scores: I'm still a bit lost on how the s_verify score is calculated. You define it as log G₀(y | x), which makes me wonder, is this basically like perplexity? And when you generate candidates to score, are you assuming they are all from greedy sampling, or does this account for other methods?\nFiltering Mechanism: If I'm understanding correctly, the step where you keep candidates within a margin γ of the best score seems a lot like a form of top-p sampling, but over longer generations rather than just the next token. Are you essentially using this to ensure the \"new\" model's distribution stays aligned with the baseline G₀ model? Is that the right way to think about it?\nTeacher-Student Analogy: The whole setup reminds me of a teacher-student framework, where G₀ is the fixed \"teacher\" and every subsequent model G_t is a \"student\" trying to learn its distribution. Would you consider this a fair or useful analogy for your framework?\nReusing Real Data: In Remark 3, you mention the approach relies on an \"ongoing stream of high-quality real data.\" What would happen if you just reused the same real data from the first model in every generation? Does it absolutely have to be new data to prevent collapse?\nTraining Regimes: What are these three regimes you mention? It would be helpful to briefly name or describe these in the introduction.\nData Source: Just out of curiosity, the framework assumes training on data from a model's direct predecessor. Have you considered what it would be like if the model was trained on data from a stronger or better model instead?", "limitations": "The authors do touch upon limitations, like the dependency on real data, which is good. However, the discussion feels a bit scattered throughout the paper. It might be more impactful to have a consolidated section that discusses the practical limitations more directly (e.g., the computational cost of sampling N candidates for every single training point, or scenarios where the initial G₀ model might be a poor anchor).", "rating": 5, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 4, "originality": 3, "strengths_and_weaknesses": "I should start by saying that I’m not a theory expert, so my ability to properly judge the theoretical contributions here is limited. That said, I do like the paper and the core idea.\n\nThe main strength is that this paper tackles a very significant and practical problem. The idea of using the initial model (G₀) as a stable anchor is intuitive and clever. It presents a potential path forward for training that doesn't rely on a constant, expensive stream of new human-annotated data.\n\nMy main difficulty with the paper was its clarity. I had to reread sections multiple times because key concepts felt undefined. For instance, I still don't have a clear picture of what \"internal confidence scores\" actually are or how they are computed in practice. The motivation for removing external verification isn't clear either - it cost? GPU hours? There also seemed to be a lot of overlap between the Introduction and the Related Work section, which made the opening a bit repetitive. While the central idea is strong, the explanation could be clearer for a reader like me.", "was_revised": true, "final_justification": "The authors acknowledged my concerns and have mentioned that they'll update the paper to include some details that I found were missing. I already rated this paper pretty high and so I don't think there's any need to revise my reading and so I'm keeping it as is.", "rebuttal": "Thank you for your constructive comments and kind support! We have carefully addressed all concerns below and thoroughly revised the manuscript. We hope our responses resolve your questions.\n\n***\n\n**Q1: The motivation for removing external verification isn't clear - it cost? GPU hours?**\n\n**A:** We appreciate this question and have clarified the motivation in the revised introduction.\n\nAs highlighted in our revisions, external verifiers inherently depend on two scarce resources: large-scale human annotations (becoming increasingly scarce as high-quality real data is depleted) and additional fine-tuned auxiliary models. This incurs significant labor costs and computational overhead (e.g., GPU hours for continuous training/retuning), which is exacerbated in recursive scenarios where distribution shifts demand constant adaptation.\n\nOur self-verification framework avoids these issues by using the initial model as a stable anchor, eliminating the need for ongoing annotations or auxiliary models to bypass external verification’s resource constraints in recursive training.\n\n***\n\n**Q2: Confidence Scores: How is $ s_{verify} = \\log \\mathcal{G}_0(y|x) $ calculated—similar to perplexity? When generating candidates for scoring, are they from greedy sampling or other methods?**\n\n**A:** We appreciate this question, as it helps clarify a key technical detail.  \n\nThe verification score  $ s_{verify}(y|x) = \\log \\mathcal{G}_0(y|x) $ uses $\\mathcal{G}_0$ to evaluate outputs $y$ from later models: $x$ and $y$ are concatenated into a sequence fed to $\\mathcal{G}_0$, whose token-level log-probabilities (from position logits) are summed for total log-likelihood.\n\nRegarding its relation to perplexity: While both use log probabilities, perplexity measures average uncertainty over a dataset or sequence, whereas  $s_{verify}$  ranks or filters individual outputs based on alignment with $\\mathcal{G}_0$.\n\nRegarding the generation process: Candidate outputs are sampled from  $\\mathcal{G}_{t-1}(\\cdot | x)$  via stochastic decoding (e.g., temperature, top-p sampling), not greedy decoding. This randomness promotes diversity, enabling effective verification-based filtering.\n***\n\n**Q3: Filtering Mechanism: Keeping candidates within margin $γ$ of the best score resembles top-p sampling for longer generations. Is this to align the new model’s distribution with $\\mathcal{G}_0$?**\n\n**A:** We agree our filtering shares top-p sampling’s core intuition (retaining high-probability outputs while promoting diversity) but with key differences, outlined below.\n\nStandard top-p sampling selects the next token from high-probability options (token-level). Ours filters full candidate sequences (sequence-level). Also, top-p uses a fixed number of options, while our $γ$-margin filtering is score-adaptive—retained candidates depend on how many fall within the best sample’s high-quality neighborhood, adapting to input contexts and score variances.\n\nAs noted, its core purpose is aligning the new model’s distribution with $\\mathcal{G}_0$: filtering out candidates that deviate too far from $\\mathcal{G}_0$’s judgments prevents cumulative drift from real data, critical for avoiding collapse in recursive training.\n***\n\n**Q4: Teacher-Student Analogy: Is it fair or useful to view $\\mathcal{G}_0$ as a fixed \"teacher\" and subsequent $\\mathcal{G}_t$ as \"students\" learning its distribution?**\n\n**A:** We appreciate this insightful analogy, which captures a key dynamic of our framework.  \n\nYour view of $\\mathcal{G}_0$ as a fixed teacher and $\\mathcal{G}_t$ as students captures our setup: $\\mathcal{G}_0$, trained on real data, guides via verification score, indicating how well student outputs align with its knowledge.\n\nThere are two critical distinctions from traditional teacher-student frameworks:  \n\n1. Architectural parity over superiority: Unlike traditional setups requiring a stronger teacher, $ \\mathcal{G}_0 $ and $ \\mathcal{G}_t $ share the same architecture. $ \\mathcal{G}_0 $’s authority stems from its real-data training, not model size or performance.\n\n2. Indirect judgment vs. direct instruction: Traditional teachers guide students via direct supervision (e.g., distillation), while $ \\mathcal{G}_0 $ acts as a reference judge—evaluating outputs via verification scores without participating in training.\n\nThis analogy highlights how $ \\mathcal{G}_0 $ anchors the system against drift. We’ve added it to the revised discussion—thank you for the insightful framing.\n***\n\n**Q5: Reusing Real Data: In Remark 3, you note reliance on an \"ongoing stream of high-quality real data.\" What if the same initial real data is reused for every generation? Is new data strictly necessary to prevent collapse?**\n\n**A:** We appreciate this question, as it highlights a critical aspect of our framework. To directly address your inquiry: *reusing the same initial real data in every generation is insufficient to prevent collapse*, and an ongoing stream of new real data is essential for long-term stability. This requirement arises from both theoretical insights and empirical evidence:\n\n**Theoretical Perspective:**  If each generation’s training data uses only original real data $ D_0 $ plus synthetic data $ D_{s,t-1} $ (generated by prior models), this leads to:\n\n1. Reduced Information Diversity:  As generations progress, the synthetic data becomes a degraded copy of the original real data, with diminishing novel information. This effectively reduces the usable diversity of the training set, gradually forcing the model to train on increasingly lower-quality data.\n\n2. Error Accumulation: Any biases or imperfections in early synthetic data are amplified in subsequent generations, causing the model's distribution to drift away from the true data manifold. Over time, this drift accumulates, leading to catastrophic forgetting of rare but critical patterns in the original data.\n\n**Empirical Evidence:**  Prior work on recursive training (e.g., [1] with diffusion models and GANs) shows reusing initial real data only delays degradation—models eventually collapse. In contrast, ongoing new real data mitigates this by injecting novel patterns and correcting drift.\n\nWe’ve clarified these points in the manuscript, with relevant citations. Thank you for prompting this deeper explanation.\n\n**Reference:**\n\n[1] Alemohammad et al. *Self-consuming generative models go mad.* ICLR 2024.\n***\n\n**Q6: Training Regimes: What are the three regimes you mention? Briefly naming/describing them in the introduction would help.**\n\n**A:** We appreciate this question. Defining these regimes in the introduction improves clarity, and we’ve revised the section as follows:\n\n1. **Unverified Full Synthetic Regime:** Trains exclusively on predecessor-generated synthetic data (no real data/verification). This naive setup is prone to rapid drift and collapse from unconstrained error accumulation.\n\n2. **Fresh Real Data Augmented Regime:** Uses mixed synthetic (from prior models) and new real data, without verification. Fresh real data delays degradation but is unsustainable long-term due to costs and data depletion.\n\n3. **Verified Full Synthetic Regime:** Trains solely on synthetic data filtered via self-verification (retaining $ \\mathcal{G}_0 $-aligned candidates). This avoids ongoing real data reliance while preventing collapse by anchoring to initial real data.\n\nThank you for enhancing clarity.\n***\n\n**Q7: Data Source: Just out of curiosity, the framework uses data from a model’s direct predecessor. Have you considered training on data from a stronger model instead?**\n\nA: We appreciate the reviewer’s question. Our choice to use the model’s direct predecessor is driven by a practical motivation: for frontier models, there may be no stronger model available, and training one would require significant compute and fresh real data, which are often already exhausted. This makes self-learning from one's own generations a necessary and realistic paradigm.\n\nThat said, if a stronger external expert model were available, our framework could naturally incorporate it—resembling a teacher-student setup and potentially improving performance. However, our focus is on the more challenging and practical case where no stronger teacher exists.\n***\n\n**Q8: The limitations (e.g., real data dependency) are scattered; a dedicated section on practical issues—such as the computational cost of sampling and a poor initial $ \\mathcal{G}_0 $—would be more impactful.**\n\n**A:** Thank you for this constructive suggestion—we agree consolidating limitations into a dedicated section enhances clarity, and have revised the manuscript accordingly. \n\nThe new \"Limitations\" section addresses key practical constraints:\n\n1. **Computational Overhead of Candidate Sampling:** Generating $ N $ candidates per training point (for verification) introduces additional computational costs, especially at scale. We discuss potential mitigations, such as adaptive candidate selection (e.g., reducing $ N $ for low-uncertainty inputs) or leveraging approximate filtering to balance efficiency and performance.  \n\n2. **Vulnerability to a Poor Initial Anchor:** The framework’s stability hinges on $ \\mathcal{G}_0 $ being a reliable encoder of real-data patterns. If $ \\mathcal{G}_0 $ is poorly trained (e.g., due to biased or limited initial data), its verification scores may misguide filtering, leading to compounding errors. We note scenarios where this risk is heightened (e.g., small initial datasets) and propose future work on robustifying $ \\mathcal{G}_0 $ via techniques like data augmentation or uncertainty quantification.  \n\nBy consolidating these points, we aim to provide a clearer roadmap for both practical deployment and future research. Thank you for strengthening the manuscript."}, {"review_id": "W06gOKsyNr", "reviewer": "Reviewer_otAB", "summary": "The authors consider a recursive training scheme in which a model is trained repeatedly on its outputs (plus maybe some real data or subject to some quality-amplifying sampling scheme). This approach is known to lead to poor model performance (e.g., model collapse) if the outputs are used naively, but there are two popular mitigations in the literature: 1) the injection of some real data at each training run and/or 2) the filtering of the synthetic data to remove low-quality contributions. The authors propose a particular approach for the latter that involves using the initial trained model to reweight the sampling distribution when the new round of training data is generated. They then present a theoretical framework for analyzing the test error of the final model after T rounds of fitting. They show that this framework allows for the study of previous approaches to mitigating model collapse and accommodates the analysis of more sophisticated black box models (e.g., transformers) so long as it is possible to bound the TV distance between the data-generating distribution and the fitted statistical model.", "questions": "* Could you please more carefully define $\\mathcal{G}_0(\\cdot \\mid x)$ - in line 117, it's defined as a function mapping $\\mathcal{X} \\to \\mathcal{Y}$, but later on, it's defined as a probability measure. I found this quite confusing.\n\n* I find Assumption 1 hard to parse - in particular, what is the order of quantifiers (is this tau the same for all gamma or is it gamma-dependent?) and how credible is this assumption? I am not particularly compelled by the citation to Huang et al. 2025, but the practical validity of this assumption seems crucial to this framework's value. I would add more exposition here to both parse the assumption and explain how the reader should think about it. It also seems like this assumption rules out randomness in y | x given that y*(x) is seemingly defined as a deterministic function?\n\n* Please define the coverage coefficient outside of the statement of Theorem 1 - it seems like an important object in your paper and it's too hard to find. It is also sometimes indexed by t and sometimes implicitly indexed (I assume by T). I would be more consistent with this.\n\n* I am completely open to adjusting my score upwards if the authors adjust their comparison to previous work (or convince me that their description is fair) and/or introduce some additional discussion of what this framework enables. I think this work would be a clear accept if it had empirical confirmation of a new claim resulting from this theory.", "limitations": "Yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 4, "strengths_and_weaknesses": "**Quality**\nI think the submission is certainly technically sound and is a substantial and interesting manuscript. It is always difficult to separate criticisms neatly into these four buckets, but I am somewhat concerned about the set of testable predictions that this theory enables. I will elaborate more on these concerns within the significance subsection and in the questions/comments.\n\n**Clarity**\nI think the paper is generally clearly written, but I would encourage the authors to use fewer paragraph headings/remarks. If every paragraph begins with a bold title/remark, it no longer really draws the reader's attention. There are also a few notational concerns that I will raise in the questions/comments section.\n\n**Significance**\nFirst, I would like to acknowledge that the paper includes non-trivial technical contributions and nicely generalizes some observations made in the previous literature. I would also disclaim that I am not a researcher in the area of synthetic data / model collapse (though I do have a theoretical background and am familiar with the technical tools used by the authors). To understand the significance of this work, I  familiarized myself with two papers cited by the authors (Feng et al. and El Firdoussi et al.). To that end, I take some issue with how the authors compare and contrast this manuscript with previous work. First, while the authors instantiate a somewhat more complex verification mechanism for improving the training data at each iteration, it is no less external (the first model is used as the verifier at all iterations - for all practical purposes, we can think of it as an external verifier in the recursive context). In addition, for T = 1, Feng et al. also consider self-verification and argue that it can be surprisingly effective (both theoretically and empirically). The criticism of the use of Bernoulli random variables also strikes me as misplaced. It's simply a tool used in the other work to determine when the synthetic data agrees with the real data (this agreement is also crucial to the authors' results: see the prominent role of eps(gamma)). For the second and fourth contributions, I also agree that the theoretical framework introduced here accommodates more models and is ostensibly finite-sample, but in doing so, the authors have introduced terms that are harder to get a practical handle on and make credible statements about. In a nutshell, the authors have avoided the restriction of specifying a data-generating distribution and model-fitting procedure by defining a bound in terms of a TV distance between the DGP and the learned model. To me, this object is harder to reason about and their analysis is subsequently predicated on Assumption 1 (which is also very hard for me to think about or test - if the authors could establish that such a tau appears to exist empirically for popular models that would certainly help). I think the contributions labeled as (2)/(3) are the core contributions of this paper and make it worthy of publication. However, to justify the framework's generality (and claim that as a positive), the authors should make testable predictions about complicated models that previous theory failed to do. The authors do not provide such predictions or empirical evidence here, so the value of this generality is not as clear as it could be. Nevertheless, while my criticisms appear lengthy, I would like to emphasize that I still believe this work to be substantial and I expect it would be of significant interest. \n\n**Originality**\nThough I am not an expert in synthetic data/model collapse and I would defer on this to someone with more expertise in this exact area...judging from the two papers I carefully read, this paper is certainly sufficienty original for publication at NeurIPS.", "was_revised": true, "final_justification": "I don't feel totally comfortable raising my score further (the validation presented is not thoroughly convincing to me), but I am supportive of the paper's publication. I think relative to the other papers I read in this area (and also published in venues like NeurIPS), the contributions here are of a similar magnitude. It also seems that many of my clarity concerns were addressed by the authors.", "rebuttal": "Thank you for your constructive comments and kind support! We've carefully addressed all concerns and revised the paper accordingly.  \n\n---\n**Q1: Although the verification mechanism is more complex, it still relies on the first model as a fixed verifier—functionally external throughout the recursive process.**\n\n**A:** We appreciate this insight. Though $\\mathcal{G}_0$ originates within the system, its frozen status makes it effectively an external verifier in the recursive process.\n\nThat said, our design offers a practical and principled alternative to traditional external supervision. Standard verification methods typically depend on large-scale human annotations or auxiliary fine-tuned models—both costly in terms of labor and computation, and increasingly impractical as high-quality real data becomes scarce.\n\nIn contrast, our framework avoids such overhead by reusing $\\mathcal{G}_0$ as a stable verifier anchored in real data. This preserves the benefits of external verification (e.g., consistency and filtering) while maintaining the self-contained, scalable nature of self-learning.\n\n---\n**Q2: The criticism of Bernoulli variables seems misplaced—they're just a tool to measure synthetic-real data agreement, also central to the authors' results (see $ε(γ)$).**\n\n**A:** Thanks. We have revised the manuscript to clarify the role of Bernoulli variables in Feng et al., and to position our $ε(γ)$ as an analogous construct within our framework.\n\nTo clarify, both approaches share analogous roles: Feng et al. use Bernoulli random variables to model synthetic-real data agreement, abstractly capturing alignment likelihood with ground-truth labels. In our setting, $ε(γ)$ is analogous—it characterizes the probability that when the model assigns most probability mass to high-quality verification-filtered data, its prediction matches the true label.\n\n---\n**Q3: The authors avoid specifying a data-generating process, using TV distance instead—though this quantity is harder to reason about.**\n\n**A:** Thanks. We agree TV distance is hard to operationalize, but we use it deliberately as it has become a standard analytical tool in recent model collapse theory (e.g., Dohmatob et al. 2024c; Fu et al. 2024, 2025; Seddik et al. 2024), enabling study of recursive training dynamics without restrictive assumptions on the data-generating process.\n\nThat said, we fully recognize the importance of grounding such abstract quantities in practical settings. Specifically, in Theorem 4, our manuscript leverages PAC-Bayesian analysis tailored to transformer-based LLMs to derive a finite-sample bound for the TV distance term, connecting our theoretical bounds to state-of-the-art architectures with concrete guarantees.\n\n---\n**Q4: Could you clearly define $\\mathcal{G}_0(\\cdot | x)$ in line 117? It is described as a function mapping but later as a probability measure, which is confusing.**\n\n**A:** Thanks. We have revised the definition of $\\mathcal{G}_0(\\cdot | x)$ to clarify that $\\mathcal{G}_0: \\mathcal{X} \\to \\Delta(\\mathcal{Y})$ denotes a conditional distribution mapping an input $x \\in \\mathcal{X}$ to a distribution over responses—i.e., $\\mathcal{G}_0(y | x)$ is the probability assigned to response $y$ given $x$. This correction aligns the formal definition with its use as a probability measure. Line 117 has been updated accordingly.\n\n---\n**Q5: Assumption 1 is hard to parse: Is $τ$ the same for all $γ$ or $γ$-dependent? How credible is it? Also, since $y^*(x)$ seems deterministic, the assumption appears to rule out randomness in $y | x$.**\n\n**A:** Thanks. We address each point and summarize the revisions below.\n\n### 1. Intuition and Credibility of Assumption 1\n\nThis assumption encodes a confidence-calibrated agreement: predictions with sufficiently high confidence are likely to be correct. Specifically, if the model assigns probability mass $\\ge 1 - τ$ to the verified high-quality set, then the probability that its prediction matches the ground truth $y^*(x)$ is at least $1 - ε(γ)$.\n\nThis aligns with standard self-training assumptions (e.g., Assumption 2 on page 14 of [1]; the abstract of [2]) and active research on LLM confidence calibration (surveyed in [3]). It is realistic: modern LLMs often show strong calibration, with [4] reporting small expected calibration errors for Llama-3-8B (3.52%), Qwen-2.5-7B (5.41%), and DeepSeek-V2-Lite (3.39%) on MMLU, indicating close confidence-accuracy alignment.\n\n### 2. Parameterization and Determinism in Assumption 1\n\nAssumption 1 treats $τ$ as a fixed constant for notational simplicity, but our analysis holds for any $τ \\in (0, 1)$. Specifically, the bound for\n$$\nP [ \\mathcal{G}_T ( y_γ^+(x) | x ) \\geq 1 - τ]\n$$\n(see Equation 31, page 22) remains valid regardless of the choice of $τ$. Moreover, $τ$ does not need to depend on $γ$, since variation in verification strictness across $γ$ is already captured by $ε(γ)$.\n\nAssumption 1 also implicitly assumes a deterministic ground-truth label function $y^*(x)$. This reflects a realizable setting commonly used in self-training theory (e.g., [2]), enabling a clean and tractable convergence analysis. This clarification has been added to the manuscript.\n\n### 3. Empirical Support for Assumption 1\n\nTo strengthen the credibility of Assumption 1, we conducted additional experiments analyzing the relationship between model confidence and prediction correctness. Using the Phi3.5-Mini model on the MATH dataset (following [2]), we sampled responses and measured log-probabilities:\n\n| Log Prob Range   | Num. Correct | Num. Incorrect |\n|------------------|--------------|----------------|\n| 0 to -0.6        | 1032         | 83            |\n| -0.6 to -1.2     | 413          | 462          |\n| -1.2 to -1.8     | 73           | 386           |\n| -1.8 to -2.4     | 24           | 95             |\n| -2.4 to -3.0     | 6            | 18            |\n\nThese results confirm that correct responses consistently receive higher log-probabilities, validating the positive correlation between confidence and correctness. Formally, the log-probability distribution for correct responses stochastically dominates that for incorrect responses.\n\n**References**\n\n[1] *Self-evolved reward learning for LLMs.*\n\n[2] *Self-improvement in language models: The sharpening mechanism.*\n\n[3] *Uncertainty quantification and confidence calibration in large language models: A survey.*\n\n[4] *Your Pre-trained LLM is Secretly an Unsupervised Confidence Calibrator.*\n\n---\n**Q6: Please define the key coverage coefficient outside Theorem 1 (hard to find) and standardize its indexing.**\n\n**A:** Thanks. We now define the recursive coverage coefficient early in the technical section as\n\n$$\nC_{γ,t} := E_{x} [ 1 / \\mathcal{G}_t(y_γ^+(x) | x) ],\n$$\n\nwith $t$ indexing the training iteration. To clarify the notation in Theorem 1: The coefficient $C_{γ}$ referenced there corresponds to $C_{γ,0}$. These revisions improve clarity and consistency.\n\n---\n**Q7: I am completely open to adjusting my score upwards if the authors adjust their comparison to previous work. I think this work would be a clear accept if it had empirical confirmation of a new claim resulting from this theory.**\n\n**A:** Thanks. We clarified our comparison and conducted additional experiments to support our theoretical claims.\n\n### 1. Clarifying Our Comparison with Prior Work\n\nWe have revised the manuscript to emphasize two core contributions and clarify three key comparisons:\n\n- General and Practical Framework: Unlike prior work (Feng et al., 2025; Firdoussi et al., 2025) focused on Gaussian mixtures or linear classifiers, our framework extends to Transformer-based LLMs, addressing realistic settings.\n\n- Recursive vs. Non-Recursive Training: While prior work studied non-recursive setups ($T = 1$), we focus on recursive training—where model collapse primarily occurs—explicitly modeling its iterative dynamics.\n\n- On the verifier $\\mathcal{G}_0$: As noted in Q1, it acts as an external verifier but avoids the need for additional annotations or stronger auxiliary models, enhancing the practicality of the framework.\n\n- On Bernoulli variables: As discussed in Q2, we now clarify their alignment with our $ε(γ)$—both quantify agreement—while acknowledging their utility in prior work.\n\n- On TV distance: As explained in Q3, we recognize its practical challenges but emphasize our finite-sample bounds (Section 5) tailored for Transformer-based LLMs, bridging theory and practice.\n\n### 2. Experimental Validation of Theoretical Results\n\nTo validate our theory, we follow Fu et al. (2025)’s model collapse setting, training a 12-layer, 8-head GPT-2 (hidden size 256) to recursively in-context learn linear functions:\n$$\n\\mathcal{F} = \\\\{ f | f(x) = w^\\top x, w \\in \\mathbb{R}^d \\\\},  \n$$\nFor each prompt, we sampled $x_1, \\ldots, x_k, x_{query}$ and $w$ independently from $\\mathcal{N}(0, I_d)$. The model predicts $y_{query} = w^\\top x_{query}$. We compared three training strategies:\n\n- Full Synthetic: Trained solely on synthetic data.  \n- Mixed: Fresh and synthetic data were mixed in a 0.5 ratio.  \n- Verification-Based Filtering: Sampled 20 candidates, selected the highest-confidence prediction ($γ = 0$), and trained on verified data exclusively.\n\nThe results of these experiments are summarized below:\n\n| Round              | 0      | 1      | 2      | 3      |\n|--------------------|--------|--------|--------|--------|\n| Full Synthetic     | 0.2418 | 1.1625 | 1.4245 | 1.9523 |\n| Mixed              | 0.2418 | 0.2824 | 0.3178 | 0.3235 |\n| Verification       | 0.2418 | 0.2672 | 0.2949 | 0.3105 |\n\nAs observed, the error accumulates progressively with more generations, particularly in the full synthetic case, where the error grows rapidly. In contrast, incorporating real data and applying the verification mechanism effectively reduce the loss, consistent with our theoretical findings.\n\nFurthermore, we promise to include additional experiments to further support our results in the final version."}, {"review_id": "S9fYVL67CH", "reviewer": "Reviewer_pDf5", "summary": "This paper provides a theoretical investigation into the problem of model collapse in recursively trained generative models, particularly focusing on large language models (LLMs) trained predominantly on synthetic data created by preceding model generations. The authors formalize the recursive training process and demonstrate, via rigorous finite-sample analysis, that naive recursive training can lead to exponential error escalation and performance degradation.", "questions": "1. Empirical Support and Simulation: While the theoretical results are promising, the main paper does not present any simulated or empirical results to demonstrate error control in finite-sample settings or to visualize error growth/collapse. Are there toy simulations or practical experiments (even in the appendix) that support the tightness and realism of your bounds? What are the expected practical limits when deploying self-verification in real large-scale LLMs?\n2. Robustness to Calibration Errors: The key error bounds hinge on the base verifier model $\\mathcal{G}_0$ being well-calibrated. In practice, language models often exhibit calibration issues, and distribution shift may degrade this further. How sensitive are the theoretical guarantees to $\\mathcal{G}_0$'s miscalibration or to adversarially bad initial models? What guidance can you provide for identifying when self-verification will fail due to poor calibration?", "limitations": "Yes", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 4, "strengths_and_weaknesses": "**Strengths**\n- Strong Theoretical Foundation: The paper offers a well-motivated and thorough theoretical framework analyzing recursive training and self-verification's effect on error propagation. The development of error bounds and the explicit, stepwise derivations in Section 4 show technical rigor and completeness.\n- Explicit Theoretical Guarantees for LLMs: Section 5 translates the general theory to the specifics of standard transformer architectures, complete with explicit error bounds involving practical quantities such as sample size, model capacity, and verification parameter settings.\n- Actionable Theoretical Results: Theorems 1–4 provide concrete quantitative conditions (e.g., on the minimal verification sample size $N$ and confidence threshold $C_0$) that practitioners could, in principle, guide real-world recursive training protocols by tuning.\n\n**Weakness**\n- Lack of Empirical Validation in the Main Paper: The paper provides extensive theoretical analysis but does not include any experimental or simulation results in the main body that demonstrate the practical efficacy or tightness of the bounds. This omission makes it difficult to judge the real-world practical impact or applicability beyond the theory.\n- Limited Discussion of Assumptions' Strength and Realism: The theoretical results depend on confidence calibration and internal agreement assumptions (e.g., Assumption 1 in Section 4.1). There is little empirical or interpretive discussion in the main text about how easily these are met in complex, real-world LLMs, or what happens if calibration fails.\n- Potential Fragility to Model Miscalibration: The self-verification guarantees rely on the reference model $\\mathcal{G}_0$ being well-calibrated and robust to distribution drift. The implications of an initial poor model, selection of the $\\gamma$ margin, or the existence of hard adversarial examples are not explored, nor is there a sensitivity analysis included.\n- Absence of Empirical Figures, Tables, or Visualizations: There are notably no figures or tables of results (such as error curves, ablation studies, or schematic diagrams) in the main body, which impairs intuitive understanding and makes it harder to connect theory to practice or to evaluate claim substantiation visually. For example, a figure demonstrating the exponential error growth in Theorem 1 or a table comparing theoretical vs practical error rates under various $N, C_0$ would add substantial value.", "was_revised": true, "final_justification": "Frankly, I am not very familiar with theory-oriented papers, so I initially provided an inappropriate score. Considering the opinions of other reviewers and the author's efforts, I am willing to offer support, but I will lower my confidence level accordingly.", "rebuttal": "Thank you for your thorough review and constructive comments. We have carefully addressed all your concerns and thoroughly revised the paper, and sincerely hope our responses fully answer your questions.\n\n---\n**Q1: Empirical Support and Simulation: Though theoretical results are promising, the main paper lacks empirical findings showing error control in finite samples or visualizing error growth. Are there toy simulations or experiments (even in the appendix) supporting the tightness of your bounds? What are the practical limits for deploying self-verification in LLMs?**\n\n**A:** Thank you for recognizing the value of our theoretical contributions, which are central to the learning theory area. Following your suggestion, we conducted experiments during the discussion period to validate our results. We also promise to include additional, more comprehensive experiments in the final version.\n\nSpecifically, we follow Fu et al. (2025)’s model collapse setting, training a 12-layer, 8-head GPT-2 (hidden size 256) to recursively in-context learn linear functions:\n$$\n\\mathcal{F} = \\\\{ f | f(x) = w^\\top x, w \\in \\mathbb{R}^d \\\\},  \n$$\nFor each prompt, we sampled $x_1, \\ldots, x_k, x_{query}$ and $w$ independently from $\\mathcal{N}(0, I_d)$. The model predicts $y_{query} = w^\\top x_{query}$. We compared three training strategies:\n\n- **Full Synthetic:** Trained solely on synthetic data.  \n- **Mixed:** Fresh and synthetic data were mixed in a 0.5 ratio.  \n- **Verification-Based Filtering:** Sampled 20 candidates, selected the highest-confidence prediction ($\\gamma = 0$), and trained on verified data exclusively.\n\nThe results of these experiments are summarized below:\n\n| Round       | 0     | 1     | 2     | 3     |\n|------------------|--------|--------|--------|--------|\n| Full Synthetic   | 0.2418 | 1.1625 | 1.4245 | 1.9523 |\n| Mixed            | 0.2418 | 0.2824 | 0.3178 | 0.3235 |\n| Verification     | 0.2418 | 0.2672 | 0.2949 | 0.3105 |\n\nAs observed, the error accumulates progressively with more generations, particularly in the full synthetic case, where the error grows rapidly. In contrast, incorporating real data and applying the verification mechanism effectively reduce the loss, consistent with our theoretical findings.\n\n**Regarding the practical limits of self-verification**, we agree that scaling self-verification in LLMs presents practical challenges. A key constraint is the computational overhead introduced by candidate sampling—for instance, generating $N = 20$ outputs per input to enable verification.\n\nTo address this, we outline potential mitigations in the revised manuscript, such as adaptive sampling (e.g., reducing $N$ for inputs with low predictive uncertainty) or approximate filtering strategies, which aim to strike a balance between efficiency and verification performance. \n\n---\n**Q2: Limited Discussion of Assumptions' Strength and Realism: The theoretical results rely on confidence calibration and internal agreement assumptions (e.g., Assumption 1), but the main text rarely discusses empirically or interpretively how easily these hold for real-world LLMs.**\n\n**A:** We appreciate the reviewer’s focus on Assumption 1. We have strengthened the manuscript to address this by clarifying both the intuition behind the assumption and its alignment with real-world LLM behavior.\n\nAt its core, Assumption 1 formalizes a confidence-calibrated agreement condition: high-confidence model predictions are likely to be correct. This is consistent with standard assumptions in self-training (e.g., Assumption 2 on page 14 of [1]; the abstract of [2]) and aligns with active research on confidence calibration in LLMs—a well-studied area (see [3] for a survey).\n\nCritically, empirical work shows modern LLMs often exhibit strong calibration. For example, [4] reports small expected calibration errors for pre-trained models like Llama-3-8B (3.52%), Qwen-2.5-7B (5.41%), and DeepSeek-V2-Lite (3.39%) on MMLU—demonstrating close alignment between confidence and accuracy.\n\nTo further validate this, we conducted experiments on the MATH dataset using Phi3.5-Mini, analyzing log-probabilities of correct vs. incorrect responses:\n\n| Log Prob Range   | Num. Correct | Num. Incorrect |\n|------------------|--------------|----------------|\n| 0 to -0.6        | 1032         | 83           |\n| -0.6 to -1.2     | 413          | 462        |\n| -1.2 to -1.8     | 73           | 386          |\n| -1.8 to -2.4     | 24           | 95           |\n| -2.4 to -3.0     | 6            | 18            |\n\nCorrect responses consistently show higher log-probabilities, with their distribution stochastically dominating that of incorrect ones. This confirms the positive correlation between confidence and correctness, directly supporting Assumption 1.\n\n---\n**Q3: Robustness to Calibration Errors: The key error bounds depend on well-calibrated base verifier $\\mathcal{G}_0$, but LLMs often have calibration issues, worsened by distribution shifts. How sensitive are theoretical guarantees to $\\mathcal{G}_0$'s miscalibration or adversarial initial models? What guidance exists for identifying self-verification failures due to poor calibration?**\n\n**A:** We thank the reviewer for raising this important point. Our theoretical framework explicitly controls the dependence on the calibration quality of the base verifier $\\mathcal{G}_0$ via the term $ε(γ)$, which quantifies the correctness of high-confidence predictions. For instance, in Theorem 4 (*Error Bound for Transformer-based LLMs with Self-Verification*), the leading error term is:\n$$\nε(γ) +(\\frac{1}{τ} + \\frac{1}{NC_0}log\\frac{1}{τ})\\cdot \\frac{1 }{\\sqrt{n}} log\\frac{1}{\\delta} + \\frac{1}{N}log\\frac{1}{τ},\n$$\nwhich implies that poor calibration in $\\mathcal{G}_0$ (i.e., large $ε(γ)$) will directly degrade the final bound. In this case, the guarantee gracefully degrades to an $O(ε(γ))$ bound, which signals when self-verification may become unreliable.\n\nNotably, model collapse research focuses on scenarios where *the initial model performs well* (to isolate degradation from recursive training), rather than adversarially poor $\\mathcal{G}_0$. Assuming a severely miscalibrated $\\mathcal{G}_0$ would conflate inherent model failure with collapse due to recursive dynamics—contradicting the core motivation of studying collapse.\n\nIn practice, recent studies have shown that modern pre-trained LLMs are often well-calibrated. For example, [4] reports small expected calibration errors (ECE) on MMLU: LLaMA-3-8B (3.52%), Qwen-2.5-7B (5.41%), DeepSeek-V2-Lite (3.39%), indicating strong alignment between confidence and accuracy. As noted in Q2, our additional experiments with Phi-3.5-Mini on MATH further validate this assumption, showing that high-confidence predictions are typically correct.\n\nFor extreme miscalibration, while our bounds loosen, confidence calibration in LLMs is well-studied with effective solutions (see [1] for a survey). To guide practice, we recommend measuring ECE on a validation set and applying calibration (e.g., temperature scaling) if it exceeds a reasonable threshold. These steps have been incorporated into the manuscript to help identify and mitigate calibration issues prior to deploying self-verification.\n\n---\n**Q4: Absence of Empirical Visualizations: The main text lacks figures/tables, hindering intuitive understanding and theory-practice connection. For example, a figure on Theorem 1’s exponential error growth or a table comparing theoretical vs. practical error rates under varied $N$, $C_0$ would add significant value.**\n\n**A:** We apologize for the absence of figures and tables in the main text, which hinders intuitive understanding. Per NeurIPS rebuttal policies (prohibiting PDF/link uploads), we cannot include visualizations here, but we firmly commit to adding the requested elements in the final version.\n\nSpecifically:\n\n- For illustrating error growth in Theorem 1:  The experiment in our Q1 response shows rapid error escalation in the full synthetic setting (consistent with predicted collapse) and that our verification mechanism effectively prevents this. We will present these as figures (error curves across training rounds for different regimes) and add more comprehensive results.\n\n- For clarifying mechanisms: We will add a schematic diagram of the verification architecture to the main text, aiding intuition about how filtering operates.\n\n**Regarding error rates under varying $N$ (number of candidates) and $C_0$:**\n\n- $C_0$ is a data- and model-dependent quantity (not tunable) characterizing worst-case confidence, consistent with standard model collapse literature.\n\n- $N$ is adjustable, and we conducted additional experiments (same setup as Q1: 12-layer GPT-2 on regression) with $N = 1, 10, 20, 30, 40, 50$.  Note that $N=1$ means using only synthetic data per round without verification, equivalent to the Full Synthetic baseline (naive recursive training without filtering).\n\n| Round   | $N=1$  | $N=10$ | $N=20$ | $N=30$ | $N=40$ | $N=50$ |\n|---------|--------|--------|--------|--------|--------|--------|\n| 0       | 0.2418 | 0.2418 | 0.2418 | 0.2418 | 0.2418 | 0.2418 |\n| 1       | 1.1625 | 0.3438 | 0.2672 | 0.2598 | 0.2647 | 0.2520 |\n| 2       | 1.4245 | 0.3803 | 0.2949 | 0.2815 | 0.2798 | 0.2889 |\n| 3       | 1.9523 | 0.4052 | 0.3105 | 0.3193 | 0.2967 | 0.3051 |\n\nThe table confirms that error decreases with increasing $N$, validating the efficacy of self-verification. The drop is steepest from $N=1$ to 20, then plateaus, suggesting $N=20$ balances cost and effectiveness.  These results will be visualized as error curves in the final manuscript, showing the relationship between $N$ and performance.\n\n---\n**References**\n\n[1] *Self-evolved reward learning for LLMs.*\n\n[2] *Self-improvement in language models: The sharpening mechanism.*\n\n[3] *Uncertainty quantification and confidence calibration in large language models: A survey.*\n\n[4] *Your Pre-trained LLM is Secretly an Unsupervised Confidence Calibrator.*"}]

Author rebuttal:

Meta-review: This paper presents a set of theoretical results on avoiding model collapse through self-verification. Even though this is largely a theoretical paper, I would encourage the authors to incorporate some of the empirical experiments that they ran during the rebuttal phase into the body as it would make for a stronger contribution.
Decision: Accept (poster)