Strong Model Collapse
Authors: Elvis Dohmatob
Yunzhen Feng
Arjun Subramonian
Julia Kempe
Venue: ICLR 2025 Spotlight (2025)
Keywords: Model Collapse
Regression
High dimensional asymptotics
Synthetic Data
Scaling Laws
Primary area: learning theory
TL;DR: We provide an exact characterization of model collapse for mixing original and AI-generated data in the regression setting; even a small (but constant) fraction of synthetic data is detrimental asymptotically. We also study the impact of model size.

Abstract: Within the scaling laws paradigm, which underpins the training of large neural networks like ChatGPT and Llama, we consider a supervised regression setting and establish a strong form of the model collapse phenomenon, a critical performance degradation due to synthetic data in the training corpus. Our results show that even the smallest fraction of synthetic data (e.g., as little as 1 per 1000) can still lead to model collapse: larger and larger training sets do not enhance performance.  We further investigate whether increasing model size, an approach aligned with current trends in training large language models, exacerbates or mitigates model collapse. In a simplified regime where neural networks are approximated via random projections of tunable size, we both theoretically and empirically show that larger models can amplify model collapse. Interestingly, our theory also indicates that, beyond the interpolation threshold (which can be extremely high for very large datasets), larger models may mitigate the collapse, although they do not entirely prevent it. Our theoretical findings are empirically verified through experiments on language models and neural networks for images.

Published as a conference paper at ICLR 2025
STRONG MODEL COLLAPSE
Elvis Dohmatob Yunzhen Feng Arjun Subramonian Julia Kempel
Meta FAIR =Concordia University
Mila
*NYU
SUCLA
Work done while interning Meta. Correspondence to elvis comomattaoooeccccoiii..c
ABSTRACT
Within the scaling laws paradigm, which underpins the training of large neural networks like ChatGPT and Llama, we consider a supervised regression setting and establish strong form of the model collapse phenomenon, critical perfor- mance degradation due to synthetic data in the training corpus. Our results show that even the smallest fraction of synthetic data (e.g., little as I per 1000) can still lead to model collapse: larger and larger training sets do not enhance perfor- mance. We further investigate whether increasing model size, an approach aligned with current trends in training large language models, exacerbates or mitigates model collapse. In simplified regime where neural networks are approximated via random projections of tunable size, we both theoretically and empirically show that larger models can amplify model collapse. Interestingly, our theory also in- dicates that, beyond the interpolation threshold (which can be extremely high for very large datasets), larger models may mitigate the collapse, although they do not entirely prevent it. Our theoretical findings are empirically verified through experiments on language models and neural networks for images.
INTRODUCTION
The term Model Collapse refers to a critical degradation in the performance of Al models, particu- larly when a significant portion of their training data consists of synthetic data generated by other models. As detailed in Shumailov et al. (2023), this phenomenon arises the model gradually over- fits to patterns found in synthetic data, which may not fully represent the richness or variability of real-world data. Over successive training cycles, this feedback loop results in the model reinforcing errors, biases, or oversimplifications from the synthetic data. Consequently, the model's ability to generalize to real- world data is compromised, it increasingly relies the distorted distribution provided by prior Al generations rather than learning accurate representations of the real world.
This phenomenon was observed empirically (Hataya et al., 2023; Martínez al., 2023a;b; Bohacek & Farid, 2023; Briesch et al., 2023; Guo et al., 2023) and described theoretically (A (Alemohammad al., 2023: Bertrand et al., 2023; Dohmatob al., 2024a;b). The connection to the breakdown of neural scaling laws (Kaplan et al., 2020) has been pointed out and analyzed in Dohmatob et al. (2024b): as data becomes more synthetic, larger training sets do not enhance performance.
The issue is especially concerning in large-scale AI systems like ChatGPT and Llama (Touvron al., 2023; Dubey & al., 2024), which rely heavily on vast amounts of training data to maintain their performance. If synthetic data is used in training these models, even in small quantities, the model can start producing "gibberish" or nonsensical outputs, contains misinformation, or reflect stereotypes. This is because the model effectively starts to amplify its own mistakes (Shumailov et al., 2024). This feedback loop results in a gradual loss of model fidelity, reducing its ability to generalize or adapt to new, unseen test environments.
1.1 MAIN CONTRIBUTIONS
In this work, we establish a series of results which shed more light on model collapse, bringing the phenomenon closer solid theoretical foundation. We consider the following important questions:
(Q1) Is model collapse inevitable or can it be fixed by strategically mixing synthetic and real data? (Q2) Are larger models more prone model collapse than smaller ones?

---

Published as a conference paper at ICLR 2025
Our theoretical analysis focuses the solvable setting of linear regression with and without random projections, with the latter serving as an approximation of neural networks by means of random feature maps (Maloney al., 2022; Bach, 2023). Also, in accordance with the current "neural scaling laws" paradigm (Kaplan et al., 2020; Hoffmann et al., 2022) whichs underlies the training of L.LMs, where models and dataset sizes become larger over time, we focus on the setup where the total amount of data (synthetic + real data) used for training grows arbitrarily.
Let us summarize our main findings.
Result #1: Strong Model Collapse. First, we establish a robust negative result which shows that model collapse generally persists even on a mixture of synthetic and real training data, long as the fraction of training data which is synthetic does not vanish (cf. Sections 3.1 and 3.2). By synthetic data, we mean any training data from a distribution which deviates from the distribution of real data, i.e. data on which the test performance is evaluated. Thus, model collapse cannot generally be mitigated by simple adjustments such as data weighting (Jain et al., 2024). On the other hand, we show (Section 5) that sophisticated iterative mixing strategies like Ferbach et al. (2024) can mitigate model collapse. However, apart from additional computational overhead, such a strategy requires access to supply of clean/real data whose size grows at least at same rate as the synthetic data.
Test (train real synth)
synthetic dotaset size 02- 50
synthetic dataset sizo 02 200
synthetic dataset sbze n2 400 (gg
0.50
37_
JAAAAA
0.25
0.00
-0.25
0.50
0.2
0.2
0.4
0.2
0.4
Test error (train on real data only) Test error (train on real data only) Test error (train on real data only)
Figure 1: Pareto diagram: Understanding the role of model size in model collapse. We compare the test error (on the real true data distribution). for random projections model (Equation (5) of Section 2.2) when training is done on a mix of synthetic and real data (y-axis), versus real data only (x-axis); in both cases, the total amount of training data is fixed to n = 500. On the scatter plots, square points correspond to very high- quality synthetic data (i.e from distribution which is close to the true data distribution). diamonds correspond high-quality synthetic data, triangles correspond to low-quality, while stars correspond to very low synthetic data. The black lines correspond to the Pareto frontiers for each level quality the synthetic data; the higher the frontier above the diagonal in the given setting, the more serious is the model collapse. The colorbar is the log of the parametrization rate xb m/n, where m captures the size of the model.
Result #2: Model Size and Model Collapse. In Section 3.2, we disentangle the effect of a model's size on its ability to cope with model collapse. We show that in general, bigger models will suffer more from model collapse as soon as the deviation between the distribution of the synthetic data and real data is significant. Crucially, our theory also predicts that past the interpolation threshold point, this tendency can be reversed: large models become more robust to model collapse. Put together, these results predict the existence of a double-descent curve regarding the model collapse phenomenon. This is illustrated in Figures and 2. Thus, the model collapse profile depends critically on design choices like model size.
Experimental Validation. Our theoretical results are empirically confirmed with experiments in:
Toy settings, including random projections model on Gaussian data, and shallow networks fully trained on the MNIST dataset (Deng, 2012). Refer to the end of Section 3.2 and Appendix A.2. Realistic setting of GPT-2 models trained on BabiStories (Zhang et al., 2024a), a reproduction of TinyStories (Eldan & Li, 2023) using the Mixtral-8x77 open language model (Jiang et al., 2024)). Refer to Section 4.
Approach. From technical standpoint, our theoretical analysis focuses regression problems in the classical linear setting introduced in Dohmatob et al. (2024a) for studying model collapse, and also the setting of neural networks in simplified regime which can be approximated by ran- dom projections (Maloney et al., 2022; Bach, 2023). We employ the tools of aperator-valued free probability theory (OVFPT) (Mingo & Speicher, 2017) to obtain new birs-variance decomposition Etest ~ B + V + C, of the test error evaluated on the real true data distribution, of a model trained on mixture of real and synthetic data. The extra term ( then induces model collapse.
2

---

Published as a conference paper at ICLR 2025
Bias
Variance V
Extra
error Exst
1.0
1.0
6.0
6.0
0.8
0.0
0.8
4.0
4.0
0.5
0.5
0.5
0.2
0.2
0.0
0.0
0.0
0.0
0
250
500
750
1000
250
500
750 1000
250
500
750
1000
0
250 500 750 1000
Network
Network width
Network width
Network width
Figure/ 2: Illustration of our new bias-variance decomposition Exest = B + V + C for neural networks in the simplitied random projections regime (ef. Section 3.2), trained mixture of real and synthetic data. The sum B + V corresponds to the classical bias variance decomposition in this setup when data is real. The extra term 5 is responsible for model collapse when training is done on a mixture of real and synthetic data. The scalar characterizes the quality of the synthetic data (cf. Definition 1), via its mismatch where is
the training
with the real data distribution. The vertical line corresponds to the interpolation threshold the model size and is the total sample size. Notice the well-known double-descent curve in the bias.
1.2 RELATED WORK
The theoretical study of model collapse in the setting of high figh-dimensional supervised-leammnt with linear regression and kernel ridge regression was initiated in Dohmatob al. (2024a). This work derives analytical formulas that quantitatively describe iterative retraining on synthetic data in both under-parameterized and over-parameterized regimes, considering both low- and nigh-dimensional asymptotics. It places itself within an important body of works studying kernel ridge regression (on "clean" data), which serves as an effective proxy for neural networks in various regimes, for instance in the infinite-widhh limit (Neal, 1996; Williams, 1996; Jacot et al., 2018; Lee et al., 2018) or in the lazy regime of training (Chizat al., 2019) and are testbed to study interesting phenomena observed in deep learning. For instance, (Rahimi & Recht, 2008; Rudi & Rosasco, 2017; Maloney et al., 2022) study scaling laws for regression the random feature model and (Bach, 2023) analyses double descent in this setting. Scaling laws have been shown for kernel models under the Gaussian design, e.g. in Caponnetto & de Vito (2007); Spigler et al. (2020); Cui et al. (2022) for regression and (Cui et al., 2023) for classification.
Very few theoretical works tackle the analysis of models trained on mixtures of original (real / clean) and synthetic data. Bertrand et al. (2023) analyze the training process at the distribution level and provide stability results under a locality assumption in parameter space. Seddik et al. (2024) analyze the mixing of discrete original and synthetic data, and provide upper bounds the amount of synthetic data that can be included to avoid model collapse. Let us also mention the recent works (Jain et al., 2024; Ferbach al., 2024) which are potential methods for mitigating model collapse. Jain et al. (2024) analyze linear regression on isotropic Gaussian data for mixtures of clean and synthetic data by minizing a strategically weighted sum of losses (one term for each data source, real and synthetic), while Ferbach et al. (2024) can be seen as multi-step version thereof where at each stage, the synthetic data generator is distilled by interpolating with real data. These methods are analyzed in Section 5, where we outline their shortcomings regarding model collapse.
Finally, a few works go beyond the mixing scenario and analyze how to curate or filter synthetic data to avoid model collapse (Feng et al., 2024; Zhang et al., 2024b; Alemohammad et al., 2024; Gillman et al., 2024; Yang et al., 2025; eskari-Hemmatt et al., 2025), but a rigorous study of their effectiveness is still lacking.
2
THEORETICAL SETUP
2.1 DATA DISTRIBUTIONS
Consider an iid sample from D1 = (I,; M/) Sis n1 of size from the true data distribution Pi and an independent iid sample D2 { y:) ni +i+sii n) of size ny from another data distribution P2 (which we shall hereafter call the synthetic data distribution), where n := ni + the total amount of training data. Here, Ps Ps the distribution on Rd given by
(Features) (0, Ex),
(Labels) y = T "t + e, with e NN, ( independent of x.
Each Σg is a x d dositive-definite covariance matrix which captures the intrinsic variations of the input feature vector The 0g's 's control the level of label noise in each distribution.
3

---

Published as a conference paper at ICLR 2025
Structure of the Label Shift. For conciseness, we will assume the following priors the w
True labelling function: w ~ (O,T),
Mismatch between real and synthetic: == w2 w [0, A), independent of wi.
for some d positive-seneee semidefinite matrices I' and A.
Remark 1. To ease the presentation of our results, we shall assume that the matrices Et, Eg. r, and A are diagonal matrices, and therefore commute. Furthermore, except otherwise explicitly stated, we shall assume equal covariance matrices, and take Σ₁ Σ2 Σ. as in Dohmatob al. (2024a).
The matrix I' captures the structure of the ground-truth labelling function in the real test distribution P1. Together with the label-noise levels of and oy. the matrix A = cov(wţ w() captures the covariance structure of the disparity between the true data distribution Pi and the synthetic data distribution Pg regarding the conditional distribution p()/ the marginal distribution of stays the same under Pi and P3 due the assumption Σ Σ₂ Σ. For example, the self-consuming-lee setup of Dohmatob et al. (2024a) corresponds to taking proportional to the precision matrix of the input features --1 Thus, the size of the fluctuations of each component ò, of the difference wa wI is inversely proportional to the standard deviation of the corresponding feature. Another important setup the case where the fluctuations are isotropic, i.e taking x Ia-
Quality of Synthetic Data. Due to the priori general structure of A, the label corresponding to an input x will be different for both distributions, even in the absence of label-noise. On average, the L2- L2-norm of this difference is E .u Nn,,N( [[ wi x " = tr ΣA. We therefore define
Definition 1. The quality of synthetic data is defined (A) (1/d) tr E.A, which captures the disparity between the synthetic data distribution P2 and the real data distribution Pi (small values of e (A) are better). For example, A c- as in Dohmatob et al. (2024a), then e² (A) c.
2.2 MODELS AND PERFORMANCE MEASURE
Given this training data, the goal of learner is to construct estimator w. This can be seen linear model from I + x w. Evaluated on the real true data distribution P1 (which coincides with the distribution from which the real component Dj of the training dataset D is drawn), the test error of model ʃ Rd IR defined by
Ezest ( EEE - x x=w 11
This will be main object of study, for different models f. The outermost expectation Ep is to quench the randomness the training dataset used to train the model.
We consider two families of amalytically tractable models: (1) classical linear models obtained via penalized regression the input space, and (2) models obtained via penalized regression in a feature space given by random projections. The latter allows us study the role of model size in model collapse, by varying the output dimension of the random projection mapping. This output dimension m controls the size of a neural network in simplified regime (Maloney et al., 2022: Bach, 2023).
(1) Classical Linear Model. We start with a setup motivated by Dohmatob et al. (2024a). We are interested in the penalized linear model (ridge) feL x + x: @ with parameter vector @ given by
arg min wER' 71
TTTT w n) + 1|
trained the total dataset D D, UD2, Of course, the unregularized limit À 0+ corresponds to ordinary least-squares (OLS). We shall work in the following so-called proportionate scaling limit
(Proportionate Scaling Limit for Classical Linear Model) For fixed & E (0, cc). E (0,
d, n, n1, n2 + 00, n2/n + P2, n1
= P2. d/n + +
(4)
The extreme cases P1 0 and p2 0* correspond to training on only synthetic (resp. real) data. In particular, P1 > 0 corresponds to the setting considered in Dohmatob et al. (2024a). Note that in the isotropic setting where Σ Jas controls the speed of learning on clean data. Indeed, for
4

---

Published as a conference paper at ICLR 2025
small o, the scaling law in this case is known (Hastie et al., 2022) to be Etest of O As we shall see (Corollary 1), this scaling law gets deformed in the presence of synthetic data in the training dataset, leading to model collapse.
(2) Random Projections Model. We consider neural networks in simplified regime which can be approximated via random projections (Maloney et al., 2022; Bach, 2023), i.e f(z) x= Sv. Here, a d x m random matrix with iid entries from [(0, 1/d); maps an input-vector x E IR" to a random feature vector (ax) = S x E R Only the "read-out" weights R are learned, by fitting on the dataset D. Consider the model fR : a + 2(x) E, where given by
T arg min "E n (u ((,,) v.)² + A||a||a.
(5)
Note that such simplified neural network model has been proposed in the literature theoretical testbed for studying intriguing properties of neural networks, like scaling laws (Maloney et al., 2022) and double-descent (Bach, 2023). Also see Section 1.2. It can be shown that the extreme case m/n - oo reduces to the classical linear model.
We shall work in the following asymptotic regime:
(Proportionate Scaling Limil for Random Projections Model)
d, m, n., R2, + 20, n1/n + P1, n2 + P2, d/n + o, m/d + 7 m/n + v (6) for some constants o,y, ab E (0, oo) and p1. P2 E (0, 1), with p1 + P2 = and x/ = dry.
Note that the ratio 0 = md captures the size of the network, though the number of trainable parameters (the read-out layer) is m = yd.
3 A NEW BIAS- VARIANCE DECOMPOSITION AND THE EMERGENCE OF STRONG MODEL COLLAPSE
3.1 CLASSICAL LINEAR MODELS
We begin with an analysis of the test error (frr) for the classical linear model defined in (3) trained mixture of synthetic and true real data, but evaluated test data from the true data distribution only. We will establish a new bias-variance decomposition with an additional term which quantitatively reveals the emergence of model collapse (Shumailov al., 2023; 2024).
Let us first recall some standard notations. For any 2 0 and integer k: let df 2)) be the order-k >3* (2) tla) -k. Define
degrees of freedom of Σ at defined by dff (t; 2
df (r; /n/n
u ua(n A;
1- 2x; 2)/n'
where x ({ the unique positive solution to the fixed-point equation
A df (x; /n/
(8)
The following result (proved in the appendix, alongside all other theoretical results this work) will be exploited in the sequel to show that the use of synthetic data in model training can lead to catastrophic effects regarding test error.
Theorem 1. Define o² = pirt + p20ž and let k. u 2 O be as previously constructed. In the proportionate scaling limit (4), the test error w.r. the true data distribution Pt, of the classical
linear model fel defined in (3) is given by Erest (fo)) E S, with
E B+V, V= 0² df, (k; /n/
x 1trr +k(a))
df E)/n'
df2(k; E)/n
( = p p1u tr kld) A2 kla) kla) -2
5

---

Published as a conference paper at ICLR 2025
Note that for A 0 (i.e w2 =wi,, which corresponds to assuming that the real data and the surrogate data have the same distribution, the above theorem gives Etest E ~ B+V V which is the classical bias-variance decomposition (Hastie et al., 2022; Richards et al., 2021) for ridge regression on n samples from the distribution Ps.org.2. The extra term Ç appearing in Theorem is responsible for model collapse! In Appendix D.2, we show how Theorem I recovers the main results of Dohmatob et al. (2024a) for special choices of the displacement matrix A.
Strong Model Collapse. In particular, in the "scaling laws" regime where - 0+, it holds that S <= P2 tr A. In this case, if tr A remains bounded away from zero, then so is Ç unless p2 - 0*, Lc we discard all synthetic data from the training dataset. This is strong model collapse. It hints that model collapse as exposed by Shumailov et al. (2023; 2024); Hataya et al. (2023); Martínez et al. (2023a;b); Bohacek & Farid (2023); Briesch et al. (2023); Guo et al. (2023) cannot be fixed by naively mixing synthetic and real data during training. We show in Section 3.2 that this observation continues to hold in the setting of random projections model fnr defined in (5). Finally, in Section 5 we study what happens when the synthetic data and real data are strategically mixed during training-
Proving Theorem 1. It turns out that the analysis of the classical linear model's test error Etes(ffL) in Theorem 1 amounts to the analysis of the trace of rational functions of sums of random matrices. A Although the limiting spectral density of sums of random matrices is a classi- cal computation using subordination techniques (Marčenko & Pastur, 1967; Kargin, 2015), this is not enough for the full analysis; a more involved analysis is required. For example, some of the quantities we must analyze are of the following form (where Mj- == X3 X,/n, M == M1 + M2; A and B deterministic matrices): r (A, =- E AM, + Ala--IB(M + Ala)-! My. The difficulty will be even greater in the setting of random projections fRP because it leads to more complicated terms. To the rescue, in Appendix E we shall employ operator-valued valued free probabil- ity theory (OVFPT) to compute the exact nigh-dimensional limits of quantities like the definition of r (A, B) above The tools of OVFPT have been used in the recent machine learning theory literature to obtain precise asymptotics for the test error of neural networks (trained real data only) in various linearized settings (Adlam & Pennington, 2020; Tripuraneni et al., 2021; Lee et al., 2023). The idea is to construct block matrix Q each of whose blocks is constant is propor- tional X,, X, X2. or X,, and one of the blocks Q [ of Q-! is equal to the original matrix W- AM, + Aaa) BBM Because of the structure of Q, OVPT allows us to compute the limiting value of the expectation of the traces of the square blocks of Q-!, and we ultimately extract (33) (A, lim E tr Qi, j].
AIj Such Q is referred to linear pencil for W.
Example: The Isotropic Case. To help unpack Theorem 1, consider the following concrete setup
Is. r= (I)//d)La. A= (cf//dd)aa
for some constants r,c > 0. The constant c² captures how close the distribution of the synthetic data P3 is to the distribution of the real data Pi; thus it captures the quality of the synthetic data. This gives u ~ (/((1 + x)² -),, where k > 0 uniquely satisfies the fixed-point equation K - À = k0/(1- + k)); this is quadratic equation that characterizes the well-known Marchenko-Pastte law (Marčenko & Pastur, 1967). The quantities appearing in the formulae presented in Theorem I then take the following simple forms: V 0² 0/((1 + nz) - Φ) and = 22 ((1 + R²) Φ), and
<= (p(p(1 + u1u) + (pr + x)²u) p2c /1 x)2
In particular, in the unregularized limit X - 0+ corresponding to OLS, we get K - (Φ 1)+- To further make things concrete, consider the under-parametrizer case where & E (0, in the proportionate scaling regime (4). The over-parametrized case E (1) is treated in Appendix
D.I. We deduce the following corollary to Theorem 1.
Corollary 1. Suppose & (0, 1). Then, in the limit (4) and X + O+, the test error with respect to the true data distribution Pt, of the classical linear model fcL defined (3) is given by
Ereat ~ o² 00((1 +) + (p² + P2P10/ -))) c²
Moreover, for fixed > 0 and small & E (0, 1), it holds that Erese ~ d2d/n p3e² + O In particular, if c² Q(1), i. bounded away from zero (corresponding to low-quality synthetic data). then fett) = 2(p3c2): the scaling law plateaus unless P2 - 0t. i. unless all but a vanishing proportion of synthetic data is discarded from the training dataset.

---

Published as a conference paper at ICLR 2025
20=100
2=100-
e==10>
10
10-1
0.001
0.5
10-2
0.01
1.0
0.1
10a
10r
10
10a
10a
10a
10
Total dataset sic
Total dotnent si2c n
Total dataset size
Figure 3: Strong model collapse in the classical linear model (empirical confirmation of Corollary 1). The training dataset comprises of nı + 712 samples from mixture of real samples. The real samples dataset, while the synthetic samples are from distribution with same covariance structure and label noise level = 1, but an incorrect labelling function (epistemic error). The quality of the synthetic data is controlled by the scalar with + 0 corresponding to synthetic data of perfect quality (higher values correspond to lower quality synthetic data). Solid curves correspond to experiments, and broken curves correspond thcoretical predictions of Corollary I; notice the perfect match. We that even small amount of low-quality synthetic data enough to cause model collapse, whereby the test error of the model deviates from a perfect diagonal (ideal scaling law, corresponding to p2 = O, i.e training on real data only).
p2 synthetic samples and from the distribution the real / true samples of the training
The result is empirically illustrated in Figure 3, where we see that even a small amount of low- low-quality synthetic data is enough cause model collapse, whereby the test error of the model deviates from a perfect diagonal (ideal scaling law, corresponding to p2 = 0, ie training on real data only).
Remark 2. Corollary can be extended to the the non-isotropic case, but the statement is much longer and is thus omitted here.
3.2 RANDOM PROJECTIONS MODEL
We now turn to the more challenging setting of the random projections model JRP given in (5). As mentioned before (ef. Section 2.2), such models are considered in our work as simplification of the high-dimensional dynamics of actual neural networks, which still allows us to capture the effect of model size in the model collapse phenomenon.
We will need the following scalars which capture the high-dimensional statistics of the model SRP- Definition 2. Let (e, T, w) be the unique positive solution to the following fixed-point equations
1/e = 1+ UTTEK-! 1/T = 1 K with Ko = EEE,, K == Ko + Ald,
(9)
u = 2 E(yr"L' + w- " (ywK + X2L' with L' = (1 (1+u)E. (10)
Here, fr A A/d is the normalized trace. Also define 0 = X/(TTe) 0, w/ = w/ (1 0.
As usual, denote 0² pio7 + p203. Also. for any p [0, 1] define d x d positive definite matrix T(0:p) (): := pE+AIa and T(0) : T=:: = E+ The following result is a nontrivial extension of Theorem to the case of random projections.
Theorem 2. In the proportionate scaling limit (6), the lest error the true data distribution P,, of the random projections model far defined in (5) given by Erest (frr ~ E + S, with
E = B + V. where B = (1 + w)0² tr PET + w/ wrtrrrrrr TE
V (tr (==2 + (w' - Ou) tr 2T) o2/e,
(11)
< - (p pIu) AE'T(0))) p2 2p/u0) tr AΣ + pruld tr AET(0)-
We now explore a few important consequences of Theorem 2.
A Double-Descent Curve. The bias-variance decomposition presented in Theorem 2 is empirically illustrated in Figures/ and 4 for the Gaussian setting (1) (see Appendix B. and A. for details on the experimental setup and additional results in this setting). Notice the perfect match with experiments. The shape of the bias curve Figure 2 (leftmost plot) is reminiscent of the well- known double-descent (Bach, 2023) in the unregularized setting À 0+. The divergence at the interpolation threshold m = n (i.e. 1o = 1) is because the bias term B, the variance term V, and the extra term ( (responsible for model collapse) all diverge to infinity at this point.
7

---

Published as a conference paper at ICLR 2025
Strong Model Collapse. Observe that the first term in the expression for & given in Theorem is lower-bounder by p pr AE pla)=, which scales linearly with the square of the proportion P2 nz/n of synthetie data in the training dataset D. However, unless P2 + 0+. ie unless the proportion p2 of synthetic data in the training dataset vanishes, the performance of the model eventually plateaus above the baseline F. (corresponding the setting where all training data is real, ie no synthetic data in training dataset). This is strong model collapse.
Since the factor A Eg ( + 0ld)) only depends on the design choices of the model (via the scalar 0 defined pre- viously), we expect that different design choices (e.g.. model size) will lead to different model collapse profiles.
5
Test error Elest
0.0
0.1
3- 0.5
Are Larger Models More Prone or Less Prone Model Collapse? Figure shows the results of small experiment to investigate this. The input dimension is d 600, and the covariance matrix is identity la (isotropic features). The total number of training exam- ples is fixed to 500. The A matrix is taken to be of the form A = (c² (d)dd==1 (similar results are observed for different covariance matrices) for different values of c² as follows: c² = O (synthetic data of very high quality), represented with square markers; 2² 0.1 (high qual- ity synthetic data), represented with diamonds; e² = 0.5 (low quality), represented by triangles; and c² (very low-quality synthetic data), represented by stars. As indi- cated on the figure, the leftmost plot corresponds the regime where there is much fewer synthetic than real samples (n2 = 50 synthetic samples versus n = 450 real samples). Here, for both very high-quality and high-quality (squares and diamonds), the optimal tradeoff is struck by larger models (i.e, larger values of v/-). For lower-quality data (triangles and stars), the frontier shifts upwards and from left to right; intermediately sized models become optimal for coping with model collapse.
1.0
2
250 500
750
1000
Network width m
Figure 4: Impact of model size (network width m) on model collapse. As usual, solid curves correspond to experimental re- sults (5 runs), while broken curves corre- spond to predictions of our theory (here, Corollary 4). Error hars correspond to in- dependent runs. Also Figures 2 and 7.
In the middle plot, size of the synthetic dataset comparable to the size of the real dataset (n2 200 versus ι4 = 300). For high-quality synthetic data, larger models are still better than smaller models. However, for this setting, the frontier shifts upwards and from left to right, and the optimal model size is intermediate. For the rightmost plot, the size of the synthetic dataset is considerably larger than the real dataset (n2 = 400 versus ni 100). The results are similar to the case n22 200 except that the Pareto frontiers are higher over the diagonal (indicating more serious model collapse). In all cases, very small models never optimal: they are not good even in the classical sense when training is done only on real data, and the presence of synthetic data only makes this worse
Special Cases of Theorem 2 In the limit p2 0+ (i.e., no synthetic data; all the training data is real), - 0 in Theorem 2, and we recover the main result of Bach (2023) as special case, namely Ftest (fRP) ) = B V. with B and V as given in the theorem. Note that even in this special case, our result is more general since it covers the entire regularization path while the formulae in Bach (2023) are only for the unregularized case x 0+. On the other hand, Theorem 2 is a generalization of Theorem 1, as can be seen by taking tb + co. Refer to Appendix G.2 for details.
4 EXPERIMENTAL RESULTS
Our theoretical framework is developed within the context of high- nigh-dimensional linear regression and random projections models using Gaussian data. Our first departure from the confines of our theory are experiments with two-layer neural networks trained on the MNIST dataset (Deng, 2012) both in the random feature model (with ReL.U activations) and with fully trained networks. These are presented in Appendix A.2. We find that the general trends observed in asymptotic theory still hold: (I) there is significant model collapse, which only diminishes as the fraction of synthetic data approaches 0: (2) larger models exhibit more severe model collapse (Figures 8 and 9).
We now provide evidence that our theory is applicable to large-scale problems, particularly in the context of language modeling with GPT-2 models. The BabiStories dataset (Zhang et al., 2024a), reproduction of TtnyStories (Eldan & Li, 2023) using the Mixtral-8x7B open language model (Jiang et al., 2024) enables us to study language modeling with relatively small models in compute-
8

---

Published as a conference paper at ICLR 2025
Pr
0.0
Number of Layers
14.17
0.001
12
0.005
18
Loss
1Px1
24
0.02
13
0.05
0.1
16
0.2
17039
0.5
13 8.10
10
2x
1-38 3+39
1.0
180.7
102
10:0
10ah
Number of Tokens
Number Tokens
Figure 5: Results on BabiStories with GPT-2 models. Synthetic BabiStorics generated with trained GPT- 2-small with the same set of prompts. (Left) Impact of the proportion synthetic data pa on model collapse in a language model with 12 layers. (Right) Impact of model size (number of layers) on model collapse. Here the model is trained synthetic data only (i.e pa 1). The loss is evaluated on the TinyStories test set.
efficient and environmentally friendly way. It comprises stories generated by prompting large mod- els to create narratives in simple language that three-year-old child could understand, effectively simplifying the underlying language model.
Setup. We train GPT-2-small model with 124 million parameters on the BabiStories dataset as the generator. Using the same story prompts, which include the story beginning and character names, the generator creates our synthetic dataset. We then mix this synthetic dataset with the original BabiS- tories, train, and evaluate model perplexity on validation set derived from the original BabiStories. Detailed information on optimization and model configurations is provided in Appendix B.3.
Impact of Synthetic Data Proportion. We investigate the effect of varying the synthetic data proportion (f2) on the model's scaling in Figure (left). Here, the x-axis represents the number of tokens used to train the model. In this experiment, the synthetic data is of high quality, as evidenced by the low training loss and coherent text generations, corresponding to the small c² (cf. Definition 1) case in illustrative Figure 1. Consequently, even moderate amounts of synthetic data delay the progression of the scaling laws, and we expect this to eventually lead to plateaus or at least very bad bad (i.e small) exponents in the final scaling laws as predicted in Dohmatob al. (2024b) in the special case of training on synthetic data only.
Impact of Model Size. We next examine the impact of model size on training with synthetic data. In addition to the GPT-2-small model (12 layers), we introduce two larger models: one with 18 layers (166 million parameters) and another with 24 layers (204 million parameters). The embedding dimension and number of attention heads remain constant across all models. We generate synthetic dataset 10 times larger than the original one to support the scaling of tokens. As shown in Figure 5 (right), larger (deeper) models maintain a lower test loss until the dataset size increases-ikee likely exceeding the interpolation threshold which point smaller models begin to exhibit lower loss and reduced overfitting, This aligns with the predictions of Theorem 2 (also refer to Figure/ 1, 2, and the discussion just after the theorem), which suggest that larger models tend to amplify model collapse beyond the interpolation threshold. In Figure 5, observe this amplification when the number of tokens exceeds x 1000_ Conversely, the theory predicts that over-parameterized models help mitigate collapse, a trend we observe when the number of tokens is below x 10'0, leading to improved performance of larger models.
CAN STRATEGIC DATA MIXING SCHEMES PREVENT MODEL COLLAPSE?
Having established the occurrence of strong model collapse both theoretically and empirically, we now explore strategies to mitigate it and leverage synthetic data under stronger assumptions. We begin by assuming clear information about the data source and consider the following strategic iterative mixing, inspired by Ferbach et al. (2024). In this approach, model is fitted mixture of synthetic and real data. In the next iteration, the labels for the synthetic data are replaced with the labels predicted by the previous iteration of the process, and so
For concreteness, take E1 = Σg = Σ = La for the covariance matrices, and A = (c2/d) (c² La. In this setup, the proposal of Ferbach et al. (2024) then becomes: At iteration + mix n2 = P2n samples of synthetic data from a source having quality parameter c² = 2, with ni = n n2 samples of real data to construct a penalized linear model ((++1) according to (3). This trained model generates the synthetic data with c² 6241-

---

Published as a conference paper at ICLR 2025
10
<0=10-1
</= 10
10
error
20
Mixing
Reeraiive
Clean
Single
10
10
10? Total detasel size
10
10a Total dalasel size
Total dalasel size
Figure 6: Iterative vs Single-Step Mixing, Solid lines represent the experimental results (5 runs). while dashed lines correspond to the theoretical predictions of Corollary 2. The iterative mixing is repeated 5 times, with pr = p2 = 0.5. "Clean" refers to the scaling when using solely the n1 = pin real data in the dataset.
Thus, the idea iteratively enhance the quality of the synthetic data through bootstrapping.
Corollary 2. For large it holds in the limit (4) and then o, À + 0+ that Erest ((0) ~ E/(11
pś) Op?"), where E o² d/n, o² = p + 2202
Let us explore some important consequences of Corollary 2.
Itteativve Mixing Recovers Scaling Laws but Might Not be Feasible in Practice. If the practi- tioner can curate a sufficient amount of data from the original distribution, the training dataset will include non- non-vanishing proportion of real data, ensuring that p1 remains bounded away from zero. By comparing E with p3" we observe that iterative mixing over iterations, where f is of the order of (n)(n/d), results in a scaling law proportional to E,, as empirically confirmed in Figure 6. How- ever, this comes the cost of significant bootstrapping, large volume of real data, and the need to clearly distinguish between real and synthetic data across Merations-conditoonn that are all too computationally expensive and challenging to implement in practice.
Ittraative Mixing Relies Mostly on Real Data. In Figure 6, we compare the scaling of iterative mixing with the scaling obtained using only the Pi real data portion from the same training set ("Clean"). While the scaling rate remains consistent, iterative mixing consistently underperforms compared using real data alone. This suggests that iterative mixing may primarily neutralize the synthetic data and heavily rely on the real data to recover the scaling. Even when the original synthetic data is of high quality (i.e., when Co is small, rightmost plot of Flgure 6), the iterative method fails to effectively leverage the synthetic data, resulting in worse performance than single mixing. Thus, although iterative mixing recovers the same scaling rate, the model still collapses to some degree, and no significant performance improvement is observed.
Iterative Mixing with Little Real Data is Bad. If we consider the setting where we only have limited real data where there is faster accumulation of synthetic data, which corresponds to p2 1 (the real data in the training set is diminishing), then it holds that for any t 2 1, Etest (1(()) = of + tEE This is an increasing function of f, meaning that there is still catastrophic model collapse.
DISCUSSION
Our work systematically characterizes the effects of training models mixtures of real and syn- thetic data, showing that model collapse is a robust phenomenon that persists even with small frac- tions of synthetic data, in the asymptotic regime. By introducing new mathematical tools, we extend prior work to analyze more complex mixing settings and models (random projections). broadening the scope of theoretically tractable problems. Experiments confirm our theoretical predictions across large language models (LLMs) and also fully-trained feed-forward neural networks.
Going beyond the prevalent "neural scaling laws" paradigm (Kaplan et al., 2020; Hoffmann et al., 2022) which is at the basis of the current trend in training LLMs, this study emphasizes the impor- tance of preserving and labeling real data, either by curating it or avoiding unintended synthetic data in training, reflecting shift as Al-generated data becomes prevalent. Our work also delineates the impact of model size on the model collapse profile. Future work will explore the effect of other model design choices like activation functions, depth, and optimization hyper-parameters like learn- ing rate and momentum. To this end, we can leverage "Gaussian equivalents" (Goldt et al., 2022) to extend our theory to wide, fully-trainee networks in the neural tangent kernel (Jacot et al., 2018) and lazy (Chizat et al., 2019) regimes, using operator-valued free probability theory (Mingo & Speicher, 2017), like we have done in our analysis.
10

---

Published as a conference paper at ICLR 2025
7 ACKNOWLEDGEMENTS
YF and JK acknowledge support through the NSF NRT training grant award 1922658. Part this work was done while YF and JK where visiting the Centre Sciences de Donnees (CSD) the Ecole Normale Superieure in Paris, France, and YF and JK wish to thank the CSD for their hospitality. YF would like to thank Jianyu Zhang for his help with the experiments involving GPT-2 on BabiStories.
REFERENCES
Ben Adlam and Jeffrey Pennington. The neural tangent kernel in high dimensions: Triple descent and multi-scale theory of generalization. In International Conference on Machine Learning, PP- 74-84. PMLR, 2020.
Sina Alemohammad, Josue Casco-Rodriguez, Lorenzo Luzi, Ahmed Imtiaz Humayun, Hossein Babaei, Daniel LeJeune, Ali Siahkoohi, and Richard G. Baraniuk. Self-consuming generative models go mad. arXiv preprint arxiv:230.008850 2023.
Sina Alemohammad, Ahmed Imtiaz Humayun, Shruti Agarwal, John Collomosse, and Richard Baraniuk. Self-improvin diffusion models with synthetic data, 2024. URL https://aaxttii org/abs/ 2408. 10.08.16..0 16333.
Reyhane Askari-Hemmal Mohammad Pezeshki, Elvis Dohmatob, Florian Bordes, Pietro Astolfi, Melissa Hall, Jakob Verbeek, Michal Drozdzal, and Adriana Romero-Soriano Improving the scaling laws of synthetic data with deliberate practice. arXiv preprint arXiv:2502.15588 2025.
Francis Bach. High-dimensional analysis of double descent for linear regression with random pro- jections. 2023.
Quentin Bertrand, Avishek Joey Bose, Alexandre Duplessis, Marco Jiralerspong, and Gauthier Gidel. On the stability of iterative retraining of generative models their data. arXiv preprint arxiv:2310.00429, 2023.
Matyas Bohacek and Hany Farid. Nepotistically trained generative-ai models collapse, 2023.
Martin Briesch, Dominik Sobania, and Franz Rothlauf. Large language models suffer from their own output: An analysis the self-consuming training loop, 2023.
Andrea Caponnetto and Ernesto de Vito. Optimal rates for the regularized least-squares algorithm. Foundations of Computational Mathematics, 7:331-368, 2007.
Lenaic Chizat, Edouard Oyallon, and Francis Bach. On lazy training in differentiable programming. Advances in neural information processing systems, 32, 2019.
Hugo Cui, Bruno Loureiro, Florent Krzakala, and Lenka Zdeborová. Generalization error rates in kernel regression: the crossover from the noiseless to noisy regime. Journal of Statistical Mechanics: Theory and Experiment, 2022 (11): 14004, nov 2022.
Hugo Cui, Bruno Loureiro, Florent Krzakala, and Lenka Zdeborová. Error scaling laws for kernel classification under source and capacity conditions. Machine Learning: Science and Technology, 4(3):035033, August 2023. ISSN 2632-2153. doi: 10.1088/2632-2153/acdos
Li Deng. The mnist database of handwritten digit images for machine learning research [best of the web]. IEEE signal processing magazine, 1916)1141-142, 2012.
Elvis Dohmatob, Yunzhen Feng, and Julia Kempe. Model collapse demystified: The case of regres- sion. In Advances Neural Information Processing Systems, volume 37. Curran Associates, Inc., 2024a.
Elvis Dohmatob, Yunzhen Feng, Pu Yang, François Charton, and Julia Kempe. A tale of tails: Model collapse as a change of scaling laws. Forty-first International Conference on Machine Learning, 2024b. URL https: opwnreviieeeeeeeeee/ net fow.mmdd- wiw.mikkWKWKuukk....
Abhimanyu Dubey and al. The llama 3 herd of models. arXiv preprint arXiv:2407.2178. 2024.

---

Published as a conference paper at ICLR 2025
Ronen Eldan and Yuanzhi Li. Tinystories: How small can language models be and still speak coherent english? arXiv preprint arXiv:2305.07759 2023.
Yunzhen Feng, Elvis Dohmatob, Pu Yang, Francois Charton, and Julia Kempe. Beyond model collapse: Scaling up with synthesized data requires verification, 2024. URL org/abs/ 2406. 07515.
Damien Ferbach, Quentin Bertrand, Avishek Joey Bose, and Gauthier Gidel. Self-Consuming Gen- crative Models with Curated Data Provably Optimize Human Preferences. ArXiv, 2024.
Damien Ferbach, Quentin Bertrand, Avishek Joey Bose, and Gauthier Gidel. Self-consuming gen- erative models with eurated data provably optimize human preferences, 2024. URL https:
// //arxiv..rg//bailib http://aas/22000.9// 09499.
Nate Gillman, Michael Freeman, Daksh Aggarwal, HSU Chia-Hong, Calvin Luo, Yonglong Tian, and Chen Sun. Self-corecting self-consuming loops for generative model training. In Forty-first International Conference on Machine Learning, 2024.
Sebastian Goldt, Bruno Loureiro, Galen Reeves, Florent Krzakala, Marc Mézard, and Lenka Zde- borová. The gaussian equivalence of generative models for learning with shallow neural networks. In Mathematical and Scientific Machine Learning, pp. 426-471. PMLR, 2022.
Yanzhu Guo, Guokan Shang, Michalis Vazirgiannis, and Chloé Clavel. The curious decline of linguistic diversity: Training language models on synthetic text, 2023.
Trevor Hastie, Andrea Montanari, Saharon Rosset, and Ryan J. Tibshirani, Surprises in high- dimensional ridgeless least squares interpolation. The Annals of Statistics, 50(2), 2022.
Ryuichiro Hataya, Han Bao, and Hiromi Arai. Will large-scale generative models corrupt future datasets? In Proceedings of the IEEE/CVE International Conference on Computer Vision (ICCV), pp. 20555-20565, October 2023.
Jordan Hoffmann, Sebastian Borgeaud, Arthur Mensch, Elena Buchatskaya, Trevor Cai, Eliza Rutherford, Diego de Las Casas, Lisa Anne Hendricks, Johannes Welbl, Aidan Clark, Tom Hen- nigan, Eric Noland, Katie Millican, George van den Driessche, Bogdan Damoc, Aurelia Guy, Simon Osindero, Karen Simonyan, Erich Elsen, Jack W. Rae, Oriol Vinyals, and Laurent Sifre. Training compute-optimal large language models, 2022.
Arthur Jacot, Franck Gabriel, and Clement Hongler. Neural tangent kernel: Convergence and gen- eralization neural networks. In S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa- Bianchi, and R. Garnett (eds.), Advances in Neural Information Processing Systems, volume 31. Curran Associates, Inc., 2018.
Ayush Jain, Andrea Montanari, and Eren Sasoglu. Scaling laws for learning with real and surrogate data, 2024. URL https://arxiv..rr wwwp:xiiwrrggbbb..ooo
04376 6.
Albert Q Jiang, Alexandre Sablayrolles, Antoine Roux, Arthur Mensch, Blanche Savary, Chris Bam- ford, Devendra Singh Chaplot, Diego de las Casas, Emma Bou Hanna, Florian Bressand, et al. Mixtral experts. arXiv preprint arXiv:240J.04088. 2024.
Jared Kaplan, Sam McCandlish, Tom Henighan, Tom B. Brown, Benjamin Chess, Rewon Child, Scott Gray, Alec Radford, Jeffrey Wu, and Dario Amodei. Scaling laws for neural language models. arXiv preprint arXiv:2001.08361. 2020.
V. Kargin. Subordination for the sum of two random matrices. The Annals of Probability, 43(4), 2015.
Donghwan Lee, Behrad Moniri, Xinmeng Huang, Edgar Dobriban, and Hamed Hassani. Demys- tifying disagreement-on-tth-iiiie in high dimensions. In International Conference Machine Learning, pp. 19053-19093. PMLR, 2023.
Jaehoon Lee, Yasaman Bahri, Roman Novak, Samuel S. Schoenholz, Jeffrey Pennington, and Jascha Sohl-Dickstein. Deep neural networks gaussian processes. In 6th International Conference Learning Representations, ICLR 2018, Vancouver, BC, Canada, April 30 May 3, 2018, Confer- ence Track Proceedings. OpenReview.net, 2018.
12

---

Published as a conference paper at ICLR 2025
Alexander Maloney, Daniel A. Roberts, and James Sully. A solvable model of neural scaling laws, 2022.
Gonzalo Martínez, Lauren Watson, Pedro Reviriego, José Alberto Hernández, Mare Juarez, and Rik Sarkar. Combining generative artificial intelligence (ai) and the internet: Heading towards evolution or degradation? arXiv preprini arxiv: 2303.01255, 2023a.
Gonzalo Martínez, Lauren Watson, Pedro Reviriego, José Alberto Hernández, Mare Juarez, and Rik Sarkar. Towards understanding the interplay of generative artificial intelligence and the internet. arXiv preprint arxiv: 2306.06130 30, 2023b.
V.A. Marčenko and Leonid Pastur. Distribution of eigenvalues for some sets of random matrices. Math USSR Sb, 1:457-483, 01 1967.
James A. Mingo and Roland Speicher. Free Probability and Random Matrices, volume 35 of Fields Institute Monographs. Springer, 2017.
Radford M. Neal. Priors for infinite networks. In Bayesian Learning for Neural Networks, pp. 29-53. Springer, New York, 1996.
Ali Rahimi and Benjamin Recht. Weighted sums of random kitchen sinks: Replacing minimization with randomization in learning. Advances in Neural Information Processing Systems. Curran Associates, Inc., 2008.
Dominic Richards, Jaouad Mourtada, and Lorenzo Rosasco. Asymptotics of ridge(less) regression under general source condition. In Proceedings of The 24th International Conference on Artificial Intelligence and Statistics, volume 130 of Proceedings of Machine Learning Research. PMLR, 2021.
Alessandro Rudi and Lorenzo Rosasco. Generalization properties of learning with random features. In Advances in Neural Information Processing Systems. Curran Associates Inc., 2017. ISBN 97815 110669944
Mohamed EI Amine Seddik, Suei Wen Chen, Soufiane Hayou, Pierre Youssef, and Merouane Deb- bah. How bad is training on synthetic data? a statistical analysis of language model collapse. arXiv preprint arX4v:2404.05000 2024.
H. Shimodaira. Improving predictive inference under covariate shift by weighting the loglikelihood function. Journal of Statistical Planning and Inference, 90(2):227-244, 2000.
Ilia Shumailov, Zakhar Shumaylov, Yiren Zhao, Yarin Gal, Nicolas Papernot, and Ross Ander- son. The curse of recursion: Training on generated data makes models forget. arXiv preprint arxiv:2305.17499 2023.
Ilia Shumailov, Zakhar Shumaylov, Yiren Zhao, Yarin Gal, Nicolas Papernot, and Ross Anderson. Ai models collapse when trained on recursively generated data. Nature, 631, 2024.
Stefano Spigler, Mario Geiger, and Matthieu Wyart. Asymptotic learning curves of kernel methods: empirical data versus teacher-student paradigm. Journal of Statistical Mechanics: Theory and Experiment, 2020(12):124011 December 2020. ISSN 1742-5468. doi: 10.1058/1742-54688 abc6ld.
Hugo Touvron, Louis Martin, Kevin Stone, Peter Albert, Amjad Almahairi, Yasmine Babaei, Niko- lay Bashlykov, Soumya Batra, Prajiwal Bhargava, Shruti Bhosale, et al. Llama 2: Open founda- tion and fine-tuned chat models. arXiv preprint arXiv:23307.09288 2023.
Nilesh Tripuraneni, Ben Adlam, and Jeffrey Pennington. Covariate shift high-dimensional random feature regression. arXiv preprint arXiv:21 11.02244, 2021.
Robin Vogel, Mastane Achab, Stéphan Clémençon, and Charles Tillier. Weighted empirical risk minimization: Sample selection bias correction based on importance sampling. ArXiv, 2021.
Christopher Williams. Computing with infinite networks. In M.C. Mozer, M. Jordan, and Petsche (eds.), Advances in Neural Information Processing Systems, volume 9. MIT Press, 1996.
13

---

Published as a conference paper at ICLR 2025
Pu Yang, Yunzhen Feng, Ziyuan Chen, Yuhang Wu, and Zhuoyuan Li. Spend wisely: Maximizing post-training gains in iterative synthetic data boostrapping. arXiv preprint arXiv:2501.18662 2025.
Jianyu Zhang, Niklas Nolte, Ranajoy Sadhukhan, Beidi Chen, and Léon Bottou. Memory mosaics. arXiv preprint arXiv:2405.063% 2024a.
Jinghui Zhang, Dandan Qiao, Mochen Yang, and Qiang Wei. Regurgitative training: The value of real data in training large language models. arXiv preprint arXiv:2407. 12835, 2024b.
14

---

Published as a conference paper at ICLR 2025
Appendix
Table of Contents
A Further Experimental Results
16
A.1 Additional Results for the toy setting of multivariate gaussians
16
A.2 Experimental results for Neural Networks on MNIST
16
A.3 General Picture for Neural Network on MNIST
17
B Experimental Details
B.1 Toy Setting: Random Projections Model
................................
18
B.2 Two-layer neural networks
18
B.3 Language Modeling
18
C Static Single-Step Data Mixing
19
Some Omitted Theoretical Results and Comments
20
Classical Linear Model in Over- Parametrized Isotropic Setting
20
D.2
Connections Classical Model Collapse in Regression
20
E Deterministic Equivalents
21
E.1 Classical Linear Model
21
E.2 Random Projections
22
E.3 Proof Proposition
22
Computing
22
Computing
23
Computing (1)
25
Computing (2)
25
Proof of Proposition
25
F Proof of Theorem 1 and Corollaries
26
F.1
Proof of Theorem
26
F2
Proof of Corollary 1
27
F3 Proof of Corollary 3
27
Proof Corollary 2
28
F5 Proof Corollary
28
G Proof of Proposition 2 and Theorem
29
G.I Proof of Proposition 2
29
G.2 Recovering Theorem from Thcorem 2
31
G.3 Proof of Theorem
31
Phase-Diagram for Random Projections Model
31
The General Regularized Case
................................
31
H.2 Unregularized Limit
32
I Raw Man--Varannce Decomposition
33
L.1
Classical Linear Model
33
1.2 Random Projections Model
34
15

---

Published as a conference paper at ICLR 2025
A FURTHER EXPERIMENTAL RESULTS
A.1 ADDITIONAL RESULTS FOR THE TOY SETTING MULTIVARIATE GAUSSIANS
Figure provides additional plots for various data quality parameters c² showing model collapse a function of model size in the toy setting of multivariate Gaussians with random projections (experimental details Section B.1).
e -0000
-21011
c² 0.5
1.0
25-
2.0
2.0
15
1.5
0.6
0.02
0.4
10-
1.0
0.1
02
0.5
0.5
05
3.5
0.0
400
502
0.0
200
600
800
1020
0.0
420
3.0
800
1000
Network wichh
Network withh
Natwork wish
Networre witth
Figure 7: Impact of model size (network width m) on model collapse. Same setting as for Figure 4, but with quality parameter e² (smaller better) as shown on top of each plot and proportion of synthetic data P2 in the legend (Figure 4 showed the reverse).
A.2 EXPERIMENTAL RESULTS FOR NEURAL NETWORKS ON MNIST
Setup. For two-layer neural networks, we consider two scenarios: (1) learning with random projection model as in Section 3.2, where the first layer of the network is fixed randomly, and only the second layer is trained, and (2) learning with fully trainable neu- ral network. The first setting directly corresponds to our theoretical results from Section 3.2, but with ReLU activation functions. In the case of fully trained neu- ral networks in the second setting, our theory does not apply directly. However, we hypothesize that the gen- eral trends observed in asymptotic theory will still hold: (1) there will be a significant model collapse, which only diminishes the fraction of synthetic data approaches 0; (2) larger models will exhibit a more se- vere model collapse.
Kioden Dimendion
10
Number Samples
Figure 8: Fully trained two-layer network on MNIST data. Impact of model size (hid- den dimension, aka network width) on model collapse. Here, the model is trained solely on synthetic data (i.e p2 +
To align with the theoretical setting, we employ a (multivariate) regression approach where labels converted to one-hot vectors and the model is trained using mean squared error. The synthetic labels were generated by another two-layer network, with Gaussian label noise (standard deviation of 0) added. A validation set is used select the best checkpoint, and evaluation is conducted on the test set using the clean labels. Further details of the training are provided in Appendix B.2.
Results. Figure presents the results for both random feature models (left) and fully trained neural networks (right). In these experiments, we mixed synthetic and original data in the training set with varying coefficients, P1. As the proportion of synthetic data, P2, increases, the scaling laws slow down and eventually plateau. We observe strong model collapse: only when f2 approaches 0 does the collapse subside. The results are consistent across both cases, validating our theoretical predictions and demonstrating the applicability of our insights more complex scenarios.
We also investigated how model size, specifically the hidden dimension fully trained neural net- works, affects model collapse. As shown in Figure 8, models with varying hidden dimensions were trained exclusively on the synthetic dataset with P2 = 1. For training sets ranging from 10,000 to 50,000 samples, our results indicate that larger models more susceptible to model collapse under the same validation and evaluation protocols. Notably, all these models remain in the interpolation regime, aligning with our theoretical predictions.
16

---

Published as a conference paper at ICLR 2025
The proportion synthetic data,
-F-
0.001
00011
0.005
+
0.01
-
0.02
+
0.05
+
0.1
+
02
+
0.5
0.9
Loss
10"
10
10"
Number Samples
Number Samples
Figure 9: Model collapse function of the proportion of synthetic data. We use the MNIST dataset with regression loss. Error bars correspond 5 runs. Left, Random feature model with hidden dimension 100,000. Right, Two-layer neural network width (i.e hidden dim.) m = 2000.
Quality of Synthetic Data (Generator Performance)
0.021
0.057
0.095
0.148
Synthetic size n2 2000
Synthetic size
10000
Synthetic size n₂ 40000
0.15
MSE Loss
0.10
0.05
10a
102
10a
10
10a
10
Hidden Dimension
Hidden Dimension
Hidden Dimension
Figure 10: Understanding the role of model size in model collapse under varying qualities of synthetic data and dataset sizes. The quality of the synthetic data is evaluated using the MSE loss on the test set. The model trained solely on synthetic data (p2 -> 1).
A.3 GENERAL PICTURE FOR NEURAL NETWORK ON MNIST
To provide a comprehensive understanding of how the quality of synthetic data, the quantity of synthetic data, and the network size impact performance, we conducted a large-scale experiment varying these factors, as shown in Figures 10 and 11. Figure 10 uses the MSE loss an indicator of synthetic data quality, while Figure 11 uses accuracy as the indicator. To simplify the analysis, we focus on pure synthetic data (p2 = 1).
The synthetic data with the highest quality already achieves accuracy close to the optimal. the quality decreases, we observe that the shape of the curve begins to resemble a double descent curve, similar to the changes in the Pareto frontiers shown Figure the number of synthetic data 722 and hidden dimension d, the figure captures various segments of the double descent curve depicted in Figure 4. When n2 is small (as seen in the left subplots), corresponds to large parameterization rate vh. placing in the second descent region of the double descent curve. Conversely, when n2 is large (as shown the right subplots), captures the up-and- down behavior characteristic of the double descent phenomenon.
With different combinations of
17

---

Published as a conference paper at ICLR 2025
Quality of Synthetic Data (Generator Performance)
97.9
92.9
85.8
74.6
Synthetic size
2000
Synthetic size
10000
Synthetic size 40000
90
Accuracy
80
70
10a
10a
10a
10a
10a
10a
10
Hidden Dimension
Hidden Dimension
Hidden Dimension
Figure 11: Understanding the role of model size in model collapse under varying qualities of synthetic data and dataset sizes. The quality of the synthetic data is evaluated using the accuracy on the test set. The model trained solely on synthetic data (p2 -> 1).
B EXPERIMENTAL DETAILS
B. TOY SETTING: RANDOM PROJECTIONS MODEL
Setup. As sanity check empirical confirm our analytical predictions from Theorem 2, we con- sider setting with multivariate Gaussian data (1). The feature covariance matrix Σ is constructed have power-law eigenvalues AÚ C/j, where C such that = Å + ... + Aa ground-truth labelling weights wi of the real data distribution Pi sampled from NO0, (1/d)Ta), while the ground-truth weights w for the synthtic data distribution are sampled from N Nww,, A) with A = (c" /d)E-1 for different values of e² ranging from {0, 0.1, 0.5, 1} which controls for the quality of the synthetic data. We run a small experiment with label noise levels σ₃ 02 = 0.1, input-dimension d 600, number of real samples n 300, and synthetic samples n2 = 200, for a total of п1 + n22 500 samples. We fit a random projection model fRP according to (5) and for different values of the width parameter m (to control the size the of the model), and report the results in Figures and The regularization parameter is set to a very small value (10-8). We also consider variation of this experiment with different values of the synthetic dataset size ne and report the results in Figure!
The
B.2 TWO-LAYER NEURAL NETWORKS
The two-layer neural networks trained using stochastic gradient descent (SGD) with batch size of 128 and a learning rate 0.1. The models are trained for 400 epochs to fully converge. We employ held-out validation set from the training set select the best checkpoint to evaluate.
B.3 LANGUAGE MODELING
The generation process for the BabiStories dataset is detailed in the GitHub repository of Zhang (2024a). The dataset comprises a training set of 2,200,000 stories and a validation set of 22,000 stories, created by prompting the Mistral-8x7B model. Fach prompt includes description of the generation task, character names, specific words be used, and the beginning of a story. The dataset stores the beginnings of these stories along with their generated continuations.
In our experiments, we trained GPT-2-small model on this dataset to generate synthetic data. The model was trained using next-token prediction, utilizing the beginnings and continuations of stories to have good story generation quality. To maintain consistency with the original prompt distribution, used all the prompts that were initially employed generate BabiStories. During story genera- tion, we applied a temperature setting of with top-p decoding where = 1. After generation, we filtered stories of poor quality, such those containing unwanted symbols, following the same
18

---

Published as a conference paper at ICLR 2025
procedure in Zhang et al. (2024a). The filtered beginnings and synthetic continuations were then collected to form the synthetic dataset.
We used GPT-2 model with embedding dimension of d 768, 12 attention heads, and context length of 512 tokens, which typically encompasses one to three stories. During training, we applied learning rate of 5 10-3, dropout rate of 0.05, L.2 weight decay of 0.1, and warm-up phase of 2,000 iterations.
C STATIC/ SINGLE-STEP DATA MIXING
For the purposes studying scaling laws for learning on mixtures real and surrogate (e.g synthetic data), the setting considered in Jain et al. (2024) consists in the following optimization problem:
1 Σ
arg min wER"
(x) w yx)²
Σ
(x. w y1)² + A||w||2. (12)
n1
122
(EL.))),,
(EL.
This is instance of weighted weighted empirical risk minimization (Shimodaira, 2000; Vogel et al., 2021) where the weight the sample weight ny is constant across each group: T, = (1
a)n/n, ~ (1 (l-a)pp aa/pi pı for real samples real samples vs π. an/na a/p2 for synthetic samples. Thus a E (0, 1) is a mixing coefficient for the two the two source of data; in particular a - 0 corresponds
group 1) for training, while 1 corresponds to only using surrogate data (group 2). Formula (12) replaces the formula for the weights vector w of the classical linear model Ser. (3).
only using real data (which corresponds
For conciseness, as Section 5 we focus on the isotropic case considered in Section 3.1 where the feature covariance matrices are Σn Σ² Ia and the shift matrix A = cov(w" - wx²) has the form (c² /d) la for some scalar с > 0. Further, let us consider the regime where d/n - 0. In the language of our paper, one should think of this corresponding to the proportionate scaling regime given in (4), and then letting ø + 0 (extremely under- under-parametrized regime). We have the following result.
Test esror on real data
D2 1000
= =000
f): 4000
0.6
0.4
0.3
0.4
0.2
0.1
0.00
0.25
0.50
0.75
1.00
0.00
1.00
0.50
0.75
1.00
Dataset mixing confficient a
Dataset mixing confficient a
Dataset mixing coefficient
(a)
1000, d = 500.
0? =0:
n2 1000
f2 10000
0.3
0.2
0.1
That
0.0
0.00
0.25
0.50
0.75
1.00
0.00
0.25
0.50
0.75
1.00
0.00
0.25
0.50
0.75
1.00
mixing confficient
Dataset mixing coefficient
mixing coefficient
(b) n2 10000, 100, so that @ d/n < 100/10200000 110200 < 0.01 (small). Corollary correctly predicts that the optimal strategy mixing coefficient a, Rs 0, to discard surrogate data altogether.
Figure 12: Failure of naive real+surr data mixing to solve model collapse. For this experiment, we use different several different values for the size of the real data and the syathetic data nı2 Solid curves correspond experiments while broken curves correspond Error-bars correspond to independent runs.
theoretical prediction give in Corollary 3.
Corollary 3. Consider the proportionate scaling limit (4). For small 6 = d/n, it holds that
Frest (FL.) ~ -22a² A+ ((1 atppoof + ap2o3) O(
(13)
19

---

Published as a conference paper at ICLR 2025
The formula given in (13) represents a U-shaped function of a, minimized when a,, with
a, - clip|o.1] (1 (1- Piof 1200 1 2r A
(14)
It should be clear that if tr A = (11) and 01,02 = ((), then a, + 0; this corresponds to only using real data for training! In contrast any fixed value a [0, 1). leads a positive lower-bound on test error Eteat Z tr A; this is effectively model collapse. The situation empirically confirmed in Figure 12.
D SOME OMITTED THEORETICAL RESULTS AND COMMENTS
D.1 CLASSICAL LINEAR MODEL IN AWERR-ARAMETRIZER ISOTROPIC SETTING
We now complement the analysis presented at the end Section 3.1 with an analysis for the case
(1, x0). Plugging into Theorem gives - Φ 1 and + 1/(1 of o/d²) 0/(0 1) in the
limit À - 0+ We obtain the following corollary.
Corollary 4. For Φ E (1, in the limit (4) and X + 0+, holds that Etest
E + C, with
E B. = (1- y.
o²
c2 $2
Φ - p22 + ( -22)
0-1' 5= Moreover, for large ø (1, 00), holds that Fres((fcr)) E (1 2/ p2c + +(1/00))
P2
(15)
Φ
Thus, for any fixed 0, strong model collapse occurs: the RHS vanishes only if P2 - 0+ ie only if we discard all but a vanishing proportion of synthetic data from the training dataset. This is strong model collapse. Combining with Corollary 1, we conclude that (at least in the isotropic setting, strong model collapse occurs both the under-parametrrieed and over-parametrized settings.
D.2 CONNECTIONS TO CLASSICAL MODEL COLLAPSE REGRESSION
In the setting of classical model collapse (Shumailov et al., 2023; 2024; Alemohammad al., 2023; Dohmatob et al., 2024b;a), we have w2 wi + N X E,, where N is the number of iterations (i.e self-loops) in the synthetic data-generation process. Let ny be the number of samples available for training stage with training data (Xe, Xyw, + Fey) Raxd x RR, where the noise vectors Fg independent with iid components from N (7) In the proportionate scaling regime RI,,..., NN + oo with d/ne + de (0, 00) for all f, the situation is equivalent taking
=== +
of d (Kg3 Σ)
A = Σ E(x))
Σ n4 -
E(E + kela) with K2 K(ne, O)
In particular, if max; de S. 1 (so that there is enough samples perfectly fit the training data at each stage of the iterative process), and for simplicity we set a, do for all f, then the above expression simplifies A (E, /ne d)) E-1 More generally, consider the generic setting where A (62 (d))))-1 for any e 0, that the previous setting corresponds c² o Φt).
In the particular case where P1 - 0+, i.e only synthetic data is available for training. Theorem then gives
<= <= (ds tuk² -) = =22df2 (1 (1+ n? n -ddf
dl2 tr(E + kTa) mT))=
In particular, taking o² = of0/ Φt) gives
2² 1++12 n n df,
df,
tr(E + kId) +[I+kIa))+
df, Σ
di
(16)
d 12
This is recovers the main result of Dohmatob al. (2024a).
20

---

Published as a conference paper at ICLR 2025
E DETERMINISTIC EQUIVALENTS
Let Ly (resp. Yj) be the design matrix (resp. response vector) corresponding dataset D,. Thus, the design matrix X1 E xd for the real dataset has rows given by 2; for i E [nn] and Y R with components yi for i E [n1], with X2 E R xd and 22 E R defined analogously for the synthetic dataset. Let X RRx (resp. (resp. Y E IRt) be the design matrix (resp. response vector) corresponding to the total dataset. We temporarily drop the condition Σj Σg = Σ, and instead consider generally different covariance matrices En and En for the marginal distribution of the features x under the real data distribution Pi and the synthetic data distribution P2.
E.1 CLASSICAL. LINEAR MODEL
Note that the weights w of the model For given in (3) can be written explicitly w RX Y. where R == (X X + ndda) = (X, Xi + X2 X2 + ndla) -1 a random matrix. Its test error (x,r
= Ex,y l| fu--w wi II, In Proposition 4, we shall show that the RHS in the above can be decomposed into a sum of simply random quantities (k) of the form r (k) (A) and r; (A, B) that now describe and analyze.
Etest (fel) writes Evest(fc))
Let A and B be d x d positive-definite matrices with well-behaved spectral (this will be made precise latter) and let A > 0. In analyzing the slas-variance decomposition of the test error, we are ultimately led to consider the following quantities
(4) = E tr AM, (M + Ala)
(17)
, (2) (A, B) E tr A(M Ma) B ATa)
(18)
(3) (A, == E tr (M + Ad)) + Ala MM,,
(19)
(4) (A, = E tr AM, (M + ATa) B(M Xla)
(20)
where we recall that M == M1 M2 and M, == XJ X,/n.
Let e2) be the unique negative solution to the following pair of fixed sred-point equations
1 otr KI 62 øtr - with K =- pie EΙ P262 + Ala-
(21)
Also, define (u1, U2) to be the unique positive solution to the pair of fixed-poing equations
U1 p T'K-2 u2 = E2TK with L' = Pru En + P2 + XB.
(22)
Consider the following deterministic matrices
C = -
+ Ala)
(23)
D, = e Xu, + py ey
where :' and 2' = 1.
The following will be crucial for proving Theorem and its corollaries.
Proposition 1. the proportionate scaling limit (4), it holds for j = 1, that
1) (A) x tE,
(24)
(A
(25)
(A, AE
(26)
r3 (A. A5,
(27)
21

---

Published as a conference paper at ICLR 2025
E.2 RANDOM PROJECTIONS
For d x d deterministic matrices A and B, define the following quenched quantities
3 ( (A): := tr ASRS M, (3) : B
SBSRS M,,
(28)
0 (A, tr AM,SR' SBSRST, (A, E AM SBSRS M2, "
where we recall that R. := (8 MS + +m)) )1, M == +1+ +2, My == X3 Xy/n. These quantities will be useful because we may write
2
o
V&= Σ JoJLE lr ",SRS" ETS
Σ j=1
2
(4)
(Ia: Ex),
n
j
B& tr + ESRS M - 2tr SRS M tr A ESSRS M₂
TE + (1 (r, Σx) ( + -r (TE,) (2) (TEx) +12 (
(3) (A, Ex).
Each term in the above decomposition can now be analyzed via operator-valued free-probability theory. The following proposition will be heavily exploited in the prove of Theorem 2.
Proposition 2. In the proportionate scaling limit (6), it holds that
( (A) = PyYTe AEK-1
(3) = tf
rj
]j
(29)
rj (4) (A.E) pyytr AEDK
(5(5) (A, tr EK-
where the constants 61 and e2 and the matrices C1 C2 D, and E. are in Theorem 2.
E.3 PROOF OF PROPOSITION
WLOG, we only consider the case j = 1, and suppress this subscript henceforth from all the rj (k)s.
Computing ,/ We only do j = as j = 2 is completely analogous. One can obtain a minimal 9 x 9 linear pencil Q for the random matrix R = AM1 + Ala) such that Q is 9 x 9 block matrix (not shown here") and I asymptotic limit, one has
- [1, 5 X (using zero-based indexing). It follows that in the
w Fr R G1,5,
(30)
where (id @Er [Q-1] MC) is the matrix containing the limiting expected values of the normalized traces of the blocks of each of the 9 x 9 = 81 blocks of T- (we define the trace of each rectangular as zero). Using classical operator valued free probability theory (OVFPT) (Mingo & Speicher, 2017), we have the following fxed-poin equations which define G1,s implicitly
G1,5 p1 Ga,fr (AIa + PiGa, + P2G7,
(31)
X
G3 X - oG42²'
(32)
X
G7,7 = X - $Gs,6
(33)
G4,2 - Afr En (Xl + pr G. + +2,77,7
(34)
Gs.6 - -Xtr Σg + P1 21 + 22)
(35)
We deduce that G3.3 = 11, G7,7 = E2, and
()) /d G1,5 tr AL (Ala + 21 p2022'2)
where E2), is the unique pair of nonnegative solutions to the system of equations
E1 = 1+ 2t (Nd + 1211
(36)
c 1+ Φtr Ex
(37)
12c
'lll the linear pencils in this work are very hig and omitted for hrevity.
22

---

Published as a conference paper at ICLR 2025
Putting things together gives (1) ~ d- G1,5 piej tr AM p2e2202 + (a.) p1 tr AM In particular, in the limit p2 - 0+ (i.e single data source), the first equation becomes
a/Ki 1 - 7 X = Φi E (Ia pimE
+1)
or cquivalently,
df df1 Σ1)
K1 A N1
(38)
11
Furthermore, (() is now given by
(12 = e tr AET +
+ k1
(39)
Computing r(4), Here, the minimal linear pencil for the random matrix involved R = AM +
ATa)-1B(M + A1 is a 16 x 16 block matrix Q such that R = Q (1,99/ Thus, r(4) /d = G1,16/X, where G = (id & tr "1= E Mcc(C).
First consider the special case P2 + 0+ (i.e ne is negligible compared to n1). The fixed-point equations defining G1,9 are given by
G1 Atr AΣ (G3,3 B Ta G3 'AT EI.
(40)
A
G3,3 X D-2422
(41)
A
GI111 X @G12,10
(42)
A0G4,10
OGa.GII.IGAII
G3.11 = (A - (G1,2 @G12110)
(43)
X
G12.10 = - Xti Ej (Ala + E
(44)
G4,10 -A E, (AB G3,1 + - GH,L
(45)
G4 - E (Av 13,
(46)
Observe the equations for G3,11 and G4.10 further give G3,11 = -, where o solves the equation
v = Gn Σ (wΣ, + (AIa + G3, (Ala + G1I.1E1)
(47)
Now, let e be the unique non -eggaivte solution to the equation
(48)
1 dir E, (N + eE,)
is easy to see that we must have G3.3 = G11111 = and
,(4) G1
(EB vla) E1)
- ABE, (kla + En) -2 -e 2t AΣi (kay + Ei)-
(49)
uR²
K fr ABE, (KIa + Σn) X²
fr AΣ, (kIa + En)
x
where x == A/e. Furthermore, defined earlier now satisfies
2 (vE, + ABT +
= 2i (+2) + AB)(mla+ +1)
Solving for gives
@xtr BY, (nia + E1)
Atr BE, (KIa + Σn) -2
1- øtr (i +eE,)
n - dfz(k)
23

---

Published as a conference paper at ICLR 2025
In partieular, if Σn and A = Id, then
Ad
n df2(k)
and so we must have
G1,9 A
R E
(KIa + Σ1} -2
uk
(niTa - + E1)
A
X²
x²
tr Ei (kld + )1) -2 (x2
dx2(x)
A
X d
(12
K
x
df2(k)
df2(x)
(df df2(k))
(50)
X
Ad
- df2(k)
dfz(x)
À d (n df)
n (f2
(k)
df2(k)
d n - (k)
Φ n (x)
where, in the last 2 steps we have made use of the following identities which follow from the defini- tion of
K - x k df
tt En + E,) (x) - dx2
We deduce that the variance term in the bias-variance decomposition of the test error is given by
Var = 0² ( (4) 0² n n df2(k)
df2(x) =² = df2(x)/n
(51)
1
df)((m))m
Let us now compute the limiting value of (4) for any values of the proportions pl, P2 E (0, 1) with
+ P2 = 1. The fixed-point equations defining G1,9 now become
G1 = p.tr AE + 2i + P2
(52)
with S 2G B G2,10la) + P2(e2G2,10 e1 G6,11a)
(53)
G2,2 = e1,
(54)
G C22
(55)
G2,10 G3,11 G4,10
(56)
A G6,13 = G7.11 DE2Gs,13
(57)
X
G8.13 = Ati E2(AB Pı - Saa.aaa--GCccii//MMMtoaoaaoaaaa
(58)
G4,10 = Air EAB Pı G3,11 + (3
(59)
where e 2 0 and es solve the following system of equations
1+ otr (NLa + EI + D262 el
(60)
= I+ øtr E2 + ple + 2222)))
(61)
Furthermore, we deduce that Go.13 12 and G2,10 = U1, where i and U2 solve the equations
= pef (p E1 + P2 + XB) + Pie E1 +2222
(62)
= pestr Ep(pre + 2202L + XB) + 1i + P2e
(63)
Putting things together gives the formula for (4) proposed in Proposition 1. In partieular, taking P2 + (i.e P1
recovers the formula special case.
24

---

Published as a conference paper at ICLR 2025
Computing (()) A minimal linear pencil for the corresponding random matrix R = AMMMM +
Ala) B(M Ma) M1 is a 17 17 block matrix Q such that R Q [[1, This gives
,13) /d 11,16:
where G = fr)
MIr(C). The fxed-point eqautions that determine G1,16 are G1,16 = piti AE, Pie +2c
fr
with S pief D - P2 - G2, M p2eg2a)?
G7.14 = G.,13 222
G3,11 = G2,10 -1
We deduce the formula given in Proposition 1. In particular, taking the limit P2 - (i.e Pt - 1) gives
3 ~ e + = + 23 Au
U1 = peftr Σ1 (V1 E1 (B)( + ATa) 22 ptt
E(1 + + -2, i.e
11
Φir BE [1 + kTa)
tr BE (E) + =kTa)
(64)
X
1- øtr Σ [1 k1ld)
d- (k))
Finally, recalling that #1 = a/e1 by construction, we get
(3) d G1,16 eî tr ABE (6) + XTa) 2 + Xa AE + +Ma))
2 22u
= AB + 11
tr Ar [Σ1 + x1
ei
tr BEi [Σ] + kIl)) -2
= lr ABE +m)l)) 2 tr AΣ
E1 + 1
n - n-dr (x1)
Computing (2), (2)) A pencil for the relevant matrix R = X2 -A imal linear pencil Q of size 15 15, where R = [1, We deduce that (2) /d tr R/X² G1,s/A2, where G = (id tQ E (C). The fixed-point equations defining G1,8 are given by
has min-
G1,8 Atr AS + + Aaa)
(65)
with S = XB P2 - 12,
(66)
G2.2 = 61,
(67)
G5.5 = 622
(68)
G2 G3,10 Gr9
(69)
G5,12 Ga G
(70)
X
G1,9 = Atr En (AB - pi - Po -&G + Aa)
P2G5,5
(71)
G7,12 = - Afr E - G1. - 12
- P2Gs, Eg + Ala) -2
(72)
Based on previous calculations, we deduce that G2 = - and G5 = -12, and so
G1,8 = (2) ~ d
X X tr Ar-AppuE P2uEE
+ & + + 22 - tr AŻK-2
claimed. This completes the proof of Proposition 1.
E.4 PROOF OF PROPOSITION
In Section G.1 we will establish more general result which implies Proposition 2 as special case.
25

---

Published as a conference paper at ICLR 2025
F PROOF OF THEOREM 1 AND COROLLARIES
Let note that the results in Bach (2023) were obtained in two-stage approach, where random matrix theory is applied on the raw (unquenched test error || - wills, with the projection matrix more the resulting expressions but treated like deterministic matrix, and then RMT is done now treating S as random. The case general case P2 E (0, 1) is much more difficult; the key technical difficulty can be pinned down to the problem of obtaining analytic deterministic equivalents for the trace of the and derivatives of the resolvent of sum of random matrices. To circumvent this, we employ the tools of operator-valued free probability theory.
F.1 PROOF OF THEOREM
From Proposition and applied with Σ₁ E2 E, we know that
Erest (Fer) Vi + Bi, with
2 V13 =
Pjoj Σ pjjo tr D Epjo?" 1 j=1
tr - -x - kTa)-
j=1 x n
0² 2 - I(- kul kIg)
B P2 tr A K 22
P2 tr (p2 + pIu)E² + u(p E kTaa²) (I+ + ( +
Now, for the Vi term, first observe that
df
(r Σ( + 11
Ix)(E + kla)
df
df2
dfg
Ktr
E(E + kIa)
df2
dfa
dfa
(dl) df2)
n df2
df
(n df1).
dfg
We deduce that
V1 0². (1 df, /n) K df2
0². dl2
V,
n df,
n df,
where we have used the identity A = ddf, /n, which defines w.
We now handle the Bj term. First observe that + n/(n. df2), and so one computes
n: + 1)TT))(EE(++ kIa) = x² n df, triE XL2 B.
which is the classical formula for the bias term.
To finalize, observe that
(E+ - tr AEC2, K tr AΣ (p2(1 +( (E+
P1 + piu) tr A (
" u +E =:
which concludes the proof.
26

---

Published as a conference paper at ICLR 2025
F.2 PROOF OF COROLLARY
Indeed, here we have A + 0 and u + 6/(1 Φ) in the limit À + 0t. Theorem I then gives
o²6
Frest (ett)) ~ V + B 5, where V 1-6"
VV
B 0,
22c
(p2(1 - + + p300) 22c²
Ç = 1-
(p2(1 - p20) + pî
1 σ
1 (D2 + (p1 - p2)00 = p2c² + P2p1c
22c
1- Φ
For small 6, this further gives Evest (Fo)) 0'0/(1 6) + p2c² + 0 dd + p2c² + ((²²).
F.3 PROOF OF COROLLARY
The setup can be seen as special instance of the setup considered in the proof of Proposition (cf. Appendix F.1), since it corresponds to taking Mn = (1 - a)LipI,, and Dا = aLi/p2. We must have
1 1 + ptr Σ K =1
(1 a00/p1
(73)
61
(1 a) + Q62 X
1
aΦ/ Pp2
+ ptr
ΣgK = 11+
(74)
62
(1 a)e1 ae2 + X
At least for x= 0 and 0 < Φ < these equations can be solved explicitly to get e1, 61 2 0 but the resulting formulae are rather complicated, and therefore are omitted altogether. In any case, heorem correctly predicts the test error, as can be seen in Figure 12.
A particular case where things are solvable to give simple expressions, is when Φ - 0+. In this limit, it is easy to see that 61 = C2 and /1 = 22 0. This gives
K = Σ + Al::
(75)
L' E,
(76)
C1 (1 a)E,
(77)
C2 = =ω
(78)
D₁ E, E,
(79)
(2, = (M
Xaa)
(80)
(81)
(8) ( (1-)1) AE Xld)
(82)
- +
AE
(83)
r2 (A,E) P2 -
-pakak--=pp---- (2 tr AΣ Ala)) -2
(84)
We deduce that
2
0f (4) = (01- - appoo) df2();
2
Vi
Σ
Pj
(85)
n
n
j=1
B = 12
(3) (A. Σ) (3) (2,(1) メ
1311 - a)
(86)
Putting things together then gives
Erso Fot) ~ B, + VV ~
(apio) + (1 a)p203) d
as claimed.
27

---

Published as a conference paper at ICLR 2025
F.4 PROOF OF COROLLARY 2
Applying the first part of Corollary recursively gives for any iteration t 2 1,
1- p2 E, with E
o²6
Erest P = ot E
p2
Iterating the above gives
G+ = o² 0r
+ P2
Φi = d/Ne,
N₂ - n,
+/ =
(87)
Setting E E o66(1 x)) o n, get
0²
Etest( (ctt
Etest
= ب = 22 + 1 0e
02 p-
2²0r
= 15 P2 +
+
1- 1-
1 Φ1
0 02
0² ot
2(+1) 2
= P2
+ 22
+
1 -1--1
2(++1 2
Σ
2(-5))
P2
6
+
P2
Øj
2(++1
② E
2j
P2
P2
05jSi
2(++1)
P2
2(+1)
6 +
1- P2
E.
11
In particular, we if pa is bounded away from 1 (i.e if p1 == P2 1((1),, we get
1
Erest ( (et)
E P2' 2
p2
for large f. The first part is just constant multiple of the scaling law would have with training on a dataset comprising of n units of clean data.
On the other hand, we have
lim Etest (FL ~ 2 + tE.
p2-1
This is an increasing function lower-bounded by c² + E. We recover the classical picture, in which model collapse prevails (depending on the size of c², as per Corollary 1).
F.5 PROOF OF COROLLARY
From Theorem and the observation that + 1 and + 1/(1-6/03)
=0(á0 1) in the
limit À + 0t, we have Erest (w) ~ E S, with
2 2 1)2
V σ² Φ -
E V-V+B,
B
= 1/0),
0 1- 1166
< = P2 c
P2(1+
- + (p1 + 0 -
P1
d²
and the first part of the result follows after some algebra.
The second part then follows from expanding the above around oc.
28

---

Published as a conference paper at ICLR 2025
G PROOF OF PROPOSITION 2 AND THEOREM 2
G. PROOF OF PROPOSITION 2
We state and proof more general result without the requirement E1 DS = Σ.
Let 62, be the unique nonnegative solution to the following fixed-point equations
e1 = + 1T K
(88)
62 1 ØTtr EzK-1 1-
(89)
1
7 1+ tr tr KoK
(90)
with Ko 1ie p2e2 K == + Ala-
(91)
Also, let (1i, 02 w) to be the unique nonnegative solution to the following fixed-point equations
Ui veftrr \wla)
(92)
1/2 wežft +
(93)
w = (g + >-
(94)
with L PIV1L1 + p2 + XB.
(95)
Finally, define d matrices C₁, C2, D1 D2, F. by
C "iiii + p2 Mn) + wl/ E1 + + Xaa)
(96)
C2 "p2c3 (yr² (B + p141 1 w/a Σg + u2 Ala)
(97)
D1 re + (etw Tv 4a + y p2 - e2u1)
(98)
D2 = 22e2 + (ezw - Tv2) La + 7 (e2u1 e142) E,,
(99)
E == + wla);
(100)
Proposition 3. In the proportionate scaling limit (6), it holds that
nT)) A
(101)
(3) "
(102)
(4) (A, Ir AE D
(103)
(9) (A AEK
(104)
Observe that if we force 7 1 and 0, then we recover the corresponding formulae given in Proposition 1. On the other hand, taking 2 = Dg = 2 gives Proposition 2.
Proof. WLOG, we only consider the cases where j 1.
Computing ( There is a 11 x 11 minimal linear pencil Q such that ASRST Mi = [1, 10]
(zero-based indexing). We deduce that
( = E ASRS M1 d. G1,10-
(105)
where G (id@EE))-- Clix1, Moreover, G1,10 is given by the following fixed-point equations
Gı,10 Gr K-
(106)
with K = 72 + Ma. L = pl Gr,5221 + G2,
(107)
1
1
G5,5 1 G2 Σ K-K1
(108)
+ &2,22 Σ,
1
1
Gs.8 I G2
(109)
E EgK-1
+ GG EgK
22
(110)
1 1
29

---

Published as a conference paper at ICLR 2025
Then, one deduces that
tr ASRS M1 = d G1,10 = Pie AE
(111)
Computing
'i (4) deduce that
Here, the pencil Q is 20 20 and AMSRST SRS 11,1/1133/A We
(4) B tr AMS BSRS -d G1,13A.
(112)
where G (id & Q- C20x20 Moreover, G1,13 is given by the following fixed-point equations
11,13 p1ytr AE,TK where
(113)
T : X(7*e) (e) G8,12 + (G3,55)a) perr"(e2G,,16 - - 19,18)
(114)
G12,12 Go,6 T,
(115)
ge G4.14
G3,15
(116)
G4,14
xytr En (pr + p2G9,1822) X(yr2 Go,12Ta)) K-2
(117)
Oc Gu.11
G9,18
(118)
A
G10.17 - -yytr E2 (p
>(yr> Go, K-2
(119)
G6 -
(120)
G7,11 (7K + X(AB - P2
(121)
We deduce that G3,15 -11, G9.18 = -22 and Go.12 = w, where 12 2 solve the
following fixed-point equations
cheetrr En (yr En P202 A(772 B + wTx)) K-2
phoftir E1( X
U2 oveatr Me (7T
p2U2 1(77>(721)) wla)) K-
whestt Eg(7T> x
w 76 (7K² + A(AB + pı Ei + p202 K- with - E1 + p1 En + XB. Putting everything together then gives
K-)
r/ d G1,13 = P1y tr AE, TK- where
x
7 : T/> 2 B (ea T + P2T7 ( E2 =: D1-
Computing (3) The matrix of interest AMSRSS BSRS Mi admits minimal linear pencil Q of size 21 x 21, such that the formal equals (1, 20). follows that
Ti (3) E tr SMS BSRS M1 ~ d G1,20.
(122)
where G = or QQ 22121 The fixed-point equations defining G1,20 are
11 fr AΣ (T/X)K - where
T' : p1 ( 22G AG C3.n((7p29Ggg2'' Ald)²
G3 é1
Gg,9 = e2:
Ga W,
Ga,15 = -11;
Gg,18 UU2-
Putting things together gives
(3) ~ d 2 tr AΣ, A2,
where - = - P242 wT + u1 + A1a)² =: C1
which completes the proof.
30

---

Published as a conference paper at ICLR 2025
G.2
RECOVERING THEOREM 1 FROM THEOREM 2
Indeed, we have
(1))
df2(k)/n
+ 1 2122(())
1 dfz(k)/n
for any regularization strength À > O, where K is as defined equation (8). Refer to Lemma 1. Plugging these limits into the formulae provided in Theorem 2 then recovers Theorem 1.
G.3
PROOF OF THEOREM 2
This follows directly from Proposition 2 and the computations in Section I.2,
H PHASE-DIAGRAM FOR RANDOM PROJECTIONS MODEL
H.1 THE GENERAL REGULARIZED CASE
Lemma 1. The scalars u and w' which appear in Theorem 2, and described in Definition solve the following pair of linear equations
u 212,2
(123)
qu' (2,2(0) + 0² (0) u).
Furthermore, the solutions can be explicitly represented as
Φz
0 (2)
u 7 7- -- (0)) where: = = 12.2(0)(7 - (0)) + 0 (0)
w'
(124)
6z - (0))
In parficular, in the limit 7 00, it holds that
2122(k))
df2(x)/n
0 K w - u 1 212,2(1)
(125)
1 1- (k))k/
where K >> 0 as defined in (8).
Proof. The equations defining these are
u = the E(y7 2T, + K-a) w = -' K? + X
(126)
(127)
where Ko eLi, K Ko + Ala. and L' uZ + B. Further, since B E, we have u.). Now, we can rewrite the previous equations like so
L' = (1
wh E(772 " o w = t (ywe? "² + (1+ xr X2 (1+ w)tE =K-
- ory wir ΣK-
This can be equivalently written
u = o(1 u) ,= na' =w
(128)
(129)
Now, observe that
(130)
-1,
(131)
-= == =0 (
(132)
- ==xx(++)))=/mm (1,2(0)((77)3,
(133)
- = (,2(0)((7)2
(134)
31

---

Published as a conference paper at ICLR 2025
where have used the definition 0 X/(yre). Thus, and is have limiting values and w respectively, which solve the system of linear equations
12 (0) w) vy - 2 1,22' 21222 u) u) @1),200)'' u = w 7 7 2
yw' = 1,2(0 + 0² (1) + u) = 122(0)w + + 2,2 uu)
where we have used the identity Φ7 = x/. These correspond exactly to the equations given in the lemma. This proves the first part.
For the second part, indeed, 5 = no/7 - in the limit y - co, and so = A/(re) which verifies the equation
0 ~ A + Aptr (Iye2
_ +E+ - 012) /n,
ye
YE
ic 0 ~ À f d (0)/n and 0 > 0. By comparing with the equation = df) satisfied by A > in (8), we conclude = K.
Now, the equations (123) become w' 0, and (I2,2(1) u), i.
2122((K) df2(m)/n 1 - 212,(16) 1- (f))(())/
as claimed.
H.2 UNREGULARIZED LIMIT
Define the following auxiiliary quantities
k:=- x
7TE X
(135)
where c, u, and previously defined in Section 3.2.
Lemma 2. In the limit X + 0+, we have the following analytic formulae
X Xo (1 v/) + 700.
(136)
K + Kg = (vb 1) + 0
(137)
1 1070 770/7 7,
(138)
+ 6g = 6770-
(139)
Proof. From equations (9) and the constraint Σ₁ = Eg = E, we know that é1 e, where and T are unique positive solutions pair fixed point equations. Observe that K₂ eΣ and K = yr Ko + Al yre. (E + OTa). Defining = F1,1 (0), one can then rewrite the equations defining and as follows
ア E(E + (Ia) A one'.
VT X
(140)
YTE
Ae
x+
Xti Xtr KoK = X+
tr E + 01a) -1 X (n/7)7'
(141)
TTE
We deduce that
X
T' -- -n/" d'é' = X70.
(142)
1- on
In particular, the above means that 7) < min(7, 1/6). The last part of equations (142) can be rewritten as follows
11 (1-01)(1 n/7) =70, ie on² - (0r + 1)m +
0 = 0.
(143)
32

---

Published as a conference paper at ICLR 2025
This is quadratic equation for 7) function of X and 8, with roots
+7+ ( + 172 4107 (0/0)X)
& ++++ v 1bb + 1)2
1(1b - 6/0")
(144)
26
2Φ
Now, for small À > 0 and vi * 1, we can do Taylor expansion to get
1+1
1
0(1²).
2Φ
01
More explicitly,
[1/6 + 1/(11=100
<
n 0(2)
1/66-((6111) 1)0),
17-1/((1- 1-(-A/((10))))) - (())) 11/6 2/((e) - 1)0),
i
7- = 0(2))
> 1,
Because S min(1,1/0, 7). we must have the expansion
17 = 0(23) + ? 1/((1 +))),
if <
1/((0> 1)1)m
1
(145)
(1 - 00 \ + O
1/0
provided 0g > 0, i.e no # 1. in this regime, we obtain
A
[1/(1 1- A/((1111111))))111 0)(1) = (1--))66
if < 1, if > 1,
7'
1 1/7
[1/(1 - 1/11 + o(1) +0
6' À
[1/(1 - o(1) +
if $ if > 1,
Φ1)
{1/(1 + A0/((1) 1)0o) (1b 1)00/4.
7/7 70/7 = (1 1//)++
= 1 01) droo (1 -
On the other hand, if 0o 0 (which only happens if 1/ < and > 1 OR + 2 and Ø S it easy to see from (142) that we must have + 0, e' - 0
1/7, e 2 0.
Next, let's compute the limiting values of and w/ = w/72
I RAw BIAS- VARIANCE DECOMPOSITION
I.1 CLASSICAL LINEAR MODEL
Proposition 4. Evaluated on the distribution Ps = Pr the test error of model Fer. defined in (3) is given by
Exett FoL))))L)= Bx Vv.
(146)
2
o
where V4
Σ
(4) Ex),
Ek,,
(147)
j=l
12
(3) (A, + (3, 2n,,
if 1, + (9), Ez), (A, Σ2) + x2, (T +
(148)
= 2.
33

---

Published as a conference paper at ICLR 2025
Proof. Indeed, one computes
ED ED|lū - w = No TT.TT x (x= --= [)
Xx.,V || w1
ExI,Vi.X..12 [I + Aaa) -1 x Y/n w
-22, [I + +Xaa) [XIW]= + E1, X2 + E2)/ n will
= N2, [I + Ala) (M + M2ws) -wfllie + Ví + 1/2
= Bx + V2,1 + V4,2-
where
By == E [I (1a) (Mawz + M-&w"x) w 1
(149)
V& == of E tr (M + ATa) IM + Ma)
of
(4) n ; (Id: Ex).
(150)
remains analyze the bias term BR. To this end, observe that
(M + + Mx = Ia - (M + X (M- x+ + (a) = Ia - (M + AMM -MM
Denoting M-1 = M2. M = M1. w" = w2, w-2 = wt and 6% (-1)* 8, where 8 == w - we deduce that
wi,
(M + ATa) - M + (M + ATa) - 1M - - w
(M + -a)-~MMkuu A(M + Ala) --MM -A(M+ATa) M -
= (M +
A(M + ATa) -'u
Since wf and 01 = 6 == w - are are independent with distributions (0, (O) and (0,, A) respec-
tively, we deduce that
B = II(M + ATa) -1 M28 - X(M + -'a)-'wi 122. tr (M + - (M + Xla) AM + A + -
= 10
E + (T,
=
On the other hand, we have B2 B2,1 + B2,2, where
B2 - (M + Aaa) M, X(M + Ala) -wa)
= - (M + M1 - X(M + X1a) (wo] + +)
(M + XIa)- + +1a) 100--XM----- + -wil
tr A(M + AM) A1a)
A1a) -M + ALd E Ala)
Ir AMM Ala) 2E, -MM AT) + +
42
+2ArAM -\a)) - 2 XL, +la))
(3) (A, 22, A, E,) 22r (4) (A)
This completes the proof.
I.2 RANDOM PROJECTIONS MODEL
We now expand the test error Erest (FR) of the random projections model fRP defined in (5). For convenience, we recall the definition of the model here. Let be a m random matrix with iid entries from (0, 1/d). The model FR defined by (RP(x) : ((x) T, where (xx) == ST e Rm defines random feature map, and & 1 is given by
arg min L(w) = Σ [xx) v + A110112- VER
(151)
n
34

---

Published as a conference paper at ICLR 2025
Note that the gradient VL(v) of the regularized loss L is given by
VI(e)/2 Σ S x (X&S0 - Yk)/n + du Σ S M&So E X/
Hu Σ S Ya/n,
k
where H == ST MS + A E Rmmxm with M = M1 + M2 and M' = X Xx/n. Thus, setting
R := 11 we may write
I = RS (X,T1 Y1 + 12/nn RS (Mww) + M2 + RS X En/n + RS x2 Ez/n.
Now, one deduces the bias decomposition
Etest (RF) = E.DEENNO ((fFr))) - x wt) = E,,, 1150 w = Bk + Vis
of Ex Ex, S ",SRS'EESRS' where V&= V&,1 Vk,2, with Vk.j
E&SRS
B₁ = Xx,,XXISSSSS (Mww) + M2w2) w
The variance terms VR.5 can be directly handled via FPT computations. We now look at the bias term Bg. We first treat the case k = 1. One has
SRS (Mww) M2w2) will. 113
= ESRS" (M1 + M2) - La)w] + SRS M201
(SRS" M- - 1a)w1 12 EIISSST M20l3
=E Ir M 1.)E(MSSS - la) + +E ur SAMSSRSSSSSS M2
=tr ISRS MEMSRS 2E tr IESRS M tr AM ESRST M2
trIE + E tr EMSRS' TSMSRSTSRS" ISRS 2I TESRS M + tr AM2RRS ESRS M2.
classical term (B)
extra term (C)
where we recall that := (ST MS + Mmm) and M = Mi + M2 with Mj = Xk.
For the purposes of FPT computations, it might help to observe that Mj = 12%1/ 2 Zx>ix 1/2 where
7x Xx
1//2 /./.1//(((()) ((n) is ni x d random matrix with iid entries from N(001/(nA)) Thus,
Mr = AM
(152)
M = Σ
I1/2 2 Zx=12
(153)
M = XM,
(154)
M = M1 + +M₂ = 12 2122 +
+ 11 2//22, 22211/1)
(155)
R = R/X,
(156)
R (S MS + I,m) (sT 2, 2,
ST 2g 1//
1/2 Im
(157)
We need minimal linear pencils for the random matrices
SM, BSRS
(158)
AMSRS BSRS M
(159)
ASRS M, AM2SRST BSRS M2,
(160)
(161)
_//² in terms of the set of free variables (A, B, 3 ,22 S,,,,, ST, J. Observe that
1/2
tr AMSRS BSRST M
tr AMS BSRS M1 + tr AM2RRS BSRS M2 2tr AMSRS BSRS M,
tr A M = ASRS M1 tr ASRS Mz.
35

---

Published as a conference paper at ICLR 2025
For our business, it is therefore sufficient to only compute (minimal) linear pencils for
ASRS Mt,
(162)
AM, BSRST,
(163)
SM,S BSRS M..
(164)
AM,SRS BSRS M
(165)
where Ms = 11/2 2 Z : (S MS
MS + Im) M == M + M2.
Observe that without the matrix (i.e taking m = and Id), the four matrix expressions above reduce to what we had in the classical case.
36


Reviews:
[{"review_id": "CnIJKSy1P1", "reviewer": "Reviewer_Wc8A", "summary": "This paper studies model collapse, which relates the performance degradation of ML models due to synthetic training data. For linear regression and random projections model, the authors establish scaling laws that characterize this degradation, as a function of the proportion of the synthetic data as well as the quality. Their findings suggest that larger models become more susceptible to this degradation. Experiments pertaining to training GPT-2 with synthetic data are used to demonstrate validity of their results beyond the limited theoretical models.", "questions": "- Model Assumptions: Regarding the label shift assumption, what are the authors thoughts on how the results would change if the domain shift were to be accounted for as well? \n- Experimental Setup: \n    - I understand that it might not be computationally feasible for GPT-2 in the given timeframe, but could the authors provide an ablation on the MNIST experiment with regards to the quality of synthetic data? That is, models trained on labels taken from increasingly worse synthetic classifier and the corresponding trends. \n    - Minor: Regarding the GPT-2 experiment, if BabiStories is itself synthetic, why not use TinyStories as real data and BabiStories as the synthetic dataset for training the models, as opposed to creating a new GPT generator?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The paper is very well written; the problem statement and the contributions are clearly established. \n- The theoretical results seem sound and intriguing, and I appreciate the authors efforts to present simplified examples to help the reader understand the implications better.\n- The findings seem to transfer to practical learning settings fairly well.", "weaknesses": "- Model Assumptions: I am unsure of the modelling of synthetic data and how well it translates into practice. Specifically, the authors model synthetic data using a label shift, assuming that the data (X) marginal remains the same. However, it seems unrealistic for autoregressive training (a key experiment in the paper), where the input tokens for next token generation come from the synthetic distribution. \n- Experimental Details: The theoretical results establish a strong dependence on the quality of synthetic data. However, the experiments with real data (MNIST/GPT-2) do not provide quantitative metrics to measure the degradation in the synthetic data source (either accuracy of MNIST classifier or perplexity/goodness scores of the trained GPT-2 generator), which makes it hard to ascertain which paradigm (in fig.1) the experiments align best with or what level of degradation in practice results in the observed trend. \n- Minor Issues\n    - Typos: Line 225 (v \\in R^{m}), line 299 (synthetic data P2), line 398 (represented by stars)\n    - Suggestions for Clarity:\n        - Akin to theorem 1, is it possible to present a simplified version of theorem 2 for the general audience? As it is, definition 2 and theorem 2 are hard to digest just on their own.\n        - Line 481, before stating the result, can the authors explain in words the process of iterative mixing proposed in Ferbach et al., 2024? It would make the manuscript more self-contained.  \n    - Missing Citation: Line 424, for the MNIST dataset, please include the citation for \"Gradient Based Learning Applied to Document\n Recognition\", LeCun et al, 1998 \n    - Visualization: For fig.1, please consider changing the y-axes test error to the same scale. Right now, it is hard to compare the error values or the slope in subplot 1 to those in 2 and 3.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "JcjN6Q4Uci", "reviewer": "Reviewer_5qx9", "summary": "This paper considers the questions: is model collapse inevitable when training with synthetic data? Are larger models more prone to collapse than smaller ones? The paper concludes that model collapse is inevitable if training with some non-zero about of synthetic data. Larger models may suffer more from model collapse in the low quality data setting. The paper contains theoretical analysis for linear models and random projection models, and empirical results for GPT-2 models trained on the BabiStories dataset.", "questions": "- For the empirical results, it is said that the curves are expected to plateau in Fig 5. Can this be shown empirically by further scaling?\n- Can the double descent phenomenon be showed more clearly in the experimental results?\n- Separately, I would be interested in more analysis specifically in the high quality data setting, ie what can be said about the amount of model collapse as the synthetic distribution approaches the real distribution?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The paper shows model collapse in theoretical settings of the linear and random projection models\n- The paper shows nice alignment between the experiments and theoretical analysis (eg Fig 3)", "weaknesses": "I'm a bit unclear on the takeaway from the experimental results with GPT2. Specifically, the paper seems to make mixed claims for the case of large models beyond the interpolation threshold; it is said that large models may mitigate the collapse beyond the interpolation threshold and that large models tend to amplify collapse beyond the interpolation threshold in the experimental results.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Qu45j77KAp", "reviewer": "Reviewer_g7eD", "summary": "This paper studies the model collapse phenomenon in a supervised regression setting. The authors study whether model collapse is inevitable or whether proper mixing strategies can mitigate it. They also study the connection between model size and robustness to model collapse.\n\n*Setup and tools*: The theoretical analysis focuses on linear models and random projection models (simplification of neural networks). The authors derive their proofs using linear pencils and operator-valued free probability theory (OVFPT).\n\n*Contributions*: The authors provide two main theoretical results: 1) they show that even the smallest fraction of synthetic data can lead to model collapses where increasing the model's size does not improve the performance. This is dubbed \"strong model collapse\". 2) They study the impact of the model's size on the severity of the collapse and show that bigger models are more prone to model collapse when the distribution shift between synthetic and real data is important. They also show that an interpolation threshold exists (model size = sample size) beyond which larger models are more robust to collapse. This is akin to a double descent phenomenon. The theoretical results are experimentally verified on synthetic data and the authors also conduct experiments with neural networks on MNIST and GPT2-small on BabyStories data that corroborate their results.", "questions": "See weaknesses.\n\nI list the potential typos found in the text below. *\"a-->b\"* means I think a should be replaced by b.\n\n*Section 2.2*\n- Paragraph **Random Projection Models** (l224 & l225 ): it seems that dimensions of $S, x, v$ do not match. According to previous notations, $x \\in \\mathbb{R}^d$ implies $S \\in \\mathbb{R}^{m \\times d}$ (instead of $d \\times m$ as it is currently (l224)) and $v \\in \\mathbb{R}^m$ (instead of $v \\in \\mathbb{R}^k$ as it is currently (l225)).\n\n*Section 3.1*\n- Paragraph **Example: The Isotropic Case** (l299): *\"the synthetic data $P_1$\"* --> it should be *\"the synthetic data $P_2$\"*\n\n*Section 3.2*\n- Paragraph **Are Larger Models More Prone...** (l398 & l401): *\"represented by crosses\"* --> it should be *\"represented by stars\"* (l398). I am not sure but according to Figure 1 *\"for both high-quality and low-quality (squares)\"* --> it should be *\"for both very high-quality and high-quality (squares and diamonds)\"* (l401)", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "- The problem tackled by the authors is of great interest for the training of large-scale generative models.\n- I find the derivation of the proof elegant and the usage of the OVFPT novel in this setting.\n- The derived results are insightful and well-analyzed by the authors, along with the empirical validation on synthetic data that helps to convey their implications to real-world scenarios.\n- The authors conduct experiments on real data with (small) neural networks and LLMs that corroborates their theoretical analysis.\n- The paper is well-written and logically structured, making it easy to follow the authors' arguments and methodology.", "weaknesses": "1) Could the authors detail more the key steps/ideas of the proofs in the main paper (paragraph **Proving Theorem 1**)? I believe such tools could be beneficial to the reader, even in other applications.\n2) Could the authors discuss the estimation of the quality of synthetic data (parameter $c^2$) on the real applications with MNIST and BabiStories? How would one assess it when training a large-scale model like an LLM  or a VLM?\n3) Could the authors discuss in more length the \"cost\" of approximating neural networks as random projections? In particular, what is lost in terms of generality, does it approximate any architecture (e.g., can LLMs be studied in the NTK or infinite-width limit?)", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Spotlight)