Maximizing the Potential of Synthetic Data: Insights from Random Matrix Theory
Authors: Aymane El Firdoussi
Mohamed El Amine Seddik
Soufiane Hayou
Reda ALAMI
Ahmed Alzubaidi
Hakim Hacid
Venue: ICLR 2025 Poster (2025)
Keywords: Synthetic Data
RLHF
Generative Models
Statistical Models
Random Matrices
Primary area: learning theory

Abstract: Synthetic data has gained attention for training large language models, but poor-quality data can harm performance (see, e.g., Shumailov et al. (2023); Seddik et al. (2024)). A potential solution is data pruning, which retains only high-quality data based on a score function (human or machine feedback). Previous work Feng et al. (2024) analyzed models trained on synthetic data as sample size increases. We extend this by using random matrix theory to derive the performance of a binary classifier trained on a mix of real and pruned synthetic data in a high dimensional setting. Our findings identify conditions where synthetic data could improve performance, focusing on the quality of the generative model and verification strategy. We also show a smooth phase transition in synthetic label noise, contrasting with prior sharp behavior in infinite sample limits. Experiments with toy models and large language models validate our theoretical results.

Published as a conference paper at ICLR 2025
MAXIMIZING THE POTENTIAL OF SYNTHETIC DATA: INSIGHTS FROM RANDOM MATRIX THEORY
Aymane EI Firdoussi* Mohamed El Amine Seddik*! Soufiane Hayou?, Reda Alami', Ahmed Alzubaidi & Hakim Hacid"
'Technology Innovation Institute, Abu Dhabi, UAE Simons Institute, Berkeley, USA "Equal contribution
fi irsstaame last nestntameiii
ABSTRACT
Synthetic data has gained attention for training large language models, but poor- quality data can harm performance (see, e.g.. -gg.SShumailov et al. (2023) y [Seddik et al.]
(2024)). A potential solution is data pruning, which retains only high-quality data based on a score function (human or machine feedback). Previous work [reng et (2024) analyzed models trained on synthetic data sample size increases. Using random matrix theory, we generalize this analysis and derive the perfor- mance of binary classifier trained on mix of real and pruned synthetic data in a high dimensional setting. Our findings identify conditions where synthetic data could improve performance, focusing verification strategy. We also show a smooth phase transition synthetic label noise, contrasting with prior works sharp transition in infinite sample limits. Our extensive experimental setup validates our theoretical results.
the quality of the generative model and
INTRODUCTION
The landscape of large language models (LLMs) is evolving rapidly, with growing trend towards training models on combination of real and synthetic data. This synthetic data is often generated by previously trained models (Allal al. 2024: Ben Allal et al. 20,44 [Abdin al., 2024). However, the quality of these generators significantly impact the performance of newly trained models, potentially leading to model collapse (Shumailov et al. 2023). a phenomenon in which the model
drastically degrades in performance.
Model collapse has been extensively studied, both empirically (Guo et al. 2023) and theoretically (Seddik al. 2024)). highlighting the potential risks associated with training synthetic data. To mitigate these risks, researchers have proposed various strategies, including the verification of AI- synthesized data (Feng al. 2024). This approach aligns with the widely adopted Reinforcement Learning from Human Feedback (RLHF) technique (Kaufmann et al.) 2023). Feng et al. (2024) provided theoretical support for this strategy by analyzing synthetic data as Gaussian mixtures with noisy labels, using linear binary classifiers and scalar parameters to control verifier quality. Their findings reveal sharp performance transition: under infinite synthetic sample size conditions, model accuracy shifts from zero accuracy (due to errors in synthetic data and verification) to optimal performance as these errors decrease.
While current theoretical studies primarily focus on label noise in synthetic data (Dohmatob et al.) 2024a Gerstgrasser et al., 2024 Feng et al. (024), they often overlook potential distribution shifts in the feature space between real and synthetic data. This gap is particularly relevant in practical scenarios where generative models are trained on finite real data sets, potentially leading to imperfect learning of the underlying distribution.
Our paper addresses this gap by proposing a statistical model that accounts for both distribution shifts in the feature space and label noise. In model, we induce distribution shifts in the feature space by supposing that the statistics of synthetic data are empirical estimates of the underlying real data statisties. In finite sample size regime, these estimates may be biased, resulting in distribution

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shifts between real and synthetic data. Leveraging random matrix theory, derive the theoretical performance of a binary classifier trained on a mixture of real and pruned (i.e., verified) synthetic data in high-dimenssonal setting. Our analysis provides conditions under which synthetic data improves performance, emphasizing the critical roles of the generative model's quality and the ef- fieacy of the synthetic data verification strategy. Lastly, we show that the sharp phase transition phenomenon identified in (Feng et al., (024) in the infinite sample size limit is particular case of a general result, where smooth phase transitions can take place.
Summary of contributions. Our contributions are four fold:
We introduce statistical model for studying synthetic data that accounts for label and feature noise, extending beyond previous models that only consider label noise.
By leveraging random matrix theory, we characterize the performance of binary classifier trained on mixture of real and synthetic data in a high-dimensional setting.
When training only on synthetic data, we find a smooth phase transition in classifier perfor- mance, generalizing the work of Feng et al. (2024) on sharp transitions in infinite sample size limit.
We validate our results with extensive experiments (toy example and realistic LLM setups).
RELATED WORK
The use of synthesized data for model training has gained significant traction in recent years, partic- ularly with the widespread adoption of large language models (LLMs) that rely on large amounts of data in their training stages. Several studies have explored the impact of synthesized data on model performance, revealing both its advantages and limitations. A primary concern is the phenomenon of model collapse (Shumailov al. (2023), where the iterative use of generated data for model training results in a degradation of model quality. This issue has been explored theoretically and empirically across multiple studies (c.g. LeBrun al. (2021): 1): Alemohammad et al. (2023); BBohacek & Farid (2023) ; Bertrand et al. (2023). : Jain et al. (2024); Seddik et al. (2024); Dohmatob et al. (2024a b
Seddik et al. (024) investigated model collapse in recursive training settings, where new models are trained on data generated by previous models. They demonstrate that recursive training on purely synthetic data inevitably leads performance degradation. However, they show that mixing real and synthetic data can attenuate model collapse, though the proportion of real data must remain high to maintain model performance. Their findings support the idea that synthesized data alone cannot sustain model quality across iterations without a significant quantity of real data.
Gerstgrasser et al. (2024) argue that model collapse can be avoided entirely if data is accumulated rather than replaced across iterations. Their empirical studies on language models, diffusion models, and variational autoencoders indicate that accumulating both real and synthetic data helps maintain model performance over time, breaking the recursive degradation loop that leads to collapse. Ad- ditionally, Jain et al. (2024) introduced a weighted empirical risk minimization (ERM) approach to better integrate synthetic data the training pipeline, leading significant reduce in the test risk.
The most relevant work to our study is Feng et al. (2024) where the authors examined the effects of synthesized data on model performance in non-recursive setting, using the concept of rein- forcement through feedback to select high quality synthetic data. Their theoretical results, based on Gaussian mixture models, showed that adding feedback significantly improves the robustness of models trained on synthesized data. However, their setup assumes that only labels, not features, are noisy. Additionally. their focus is primarily on scenarios where only the number of data points, n, grows to infinity. Other (practical) scenarios where for instance the feature dimension, p, grows at fixed ratio with n are not covered.
Our work extends the Gaussian mixture model setup to include both noisy features and labels, which is a more realistic scenario when training on synthesized data. Additionally, we consider a high- dimensional regime where both p and grow to infinity with fixed ratio, setup often used in Random Matrix Theory (RMT). This allows us to study the interaction between feature dimension, pruner error, and data size in presence of mixed data- original and synthetic prooiding a more realistic framework for studying the effect of synthetic data in practical applications.
comprehensive manner. Our approach also accounts for the
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3 THEORETICAL SETUP
Real data. We suppose that real data consists of n p-dimensional iid. vectors 2i,..., In E R' sampled from Gaussian mixture of two distinct isotropic clusters C1 and C2 of means = with I E Rr. Essentially, for a E {1,2), each data vector a; E Ca has a corresponding label yi = (-)) and is sampled as:
x; - yiju Zi, Zi I))
(I)
Generative model. To generate synthetic data, we consider the generative model corresponding to maximum likelihood which consists of estimating the underlying first and second-orde statistics of the real data with their empirical estimates. In particular, we suppose that we are given a subset á S n of the real dataset (æ;, yi)l" on which we can estimate the statistics. This setup allows us to model situation where new real data samples might be available train next-generation models and the parameter ń offers control over the generative model quality. The statistics for generating synthetic data are therefore computed using the following estimates
1 á Σ (y:z) (e) (v, ia)
U
C=
(2)
Synthetic data. We consider that synthetic data is generated as m i.i.d. vectors क ......, 2. E R.P with corresponding (noisy) labels in..... Im 11 such that ã; E Ca with true label ūi = (- for E 2} is sampled as (C, and C2 denote the synthetic clusters)
c; U.P C e
2; ~..
and the labels di are generated such that P(u: ύ.} = 1 - & where 2 O controls label noise. Essentially, the quality of synthetic data depends on the sample size and the label noise rate e.
In the asymptotic regime where á + co with 2 + O, we can generate synthetic samples that follow asymptotically the exact same distribu- tion as of the real ones, and therefore only la- bel noise is relevant to the quality of the syn- thetic data. However, in the regime when both
f = 0.1
3-
n 0.01
Density
û. - with -> n > O, while the es- timation of y with f remains consistent, the
2
estimation of the covariance is not. In fact, in this regime ||C - Ipll A O and the eigenval- ues of C spread in the vicinity of I which is described in the limit by the Marchenko-Paase law (Marchenko & Pastur 1967) as depicted in Fig. [ Eventually, such inconsistency in esti- mating the second moment in high dimensions yields a distribution shift between synthetic and real data, which might cause a drop in perfor- mance when training a new model on synthetic data generated with j and Ć. In the remainder, we describe precisely how the performance of a simple classifier is affected in this regime.
1
0
0.5
1.0 Eigenvalues
1.5
Figure 1: Illustration of the Marchenko-Pase Pastur law: The histogram of eigenvalues of the empir- ical covariance matrix Ć (as per equation 22 us- ing different values of n. The histograms corre- spond to p 500 and á = 5 x 10 (in blue) and = 10a (in red). The line plots depict the limiting Marchenko-Pan Pasttr law. As ń grows, the distribution of eigenvalues shrinks towards
Objective. Our goal throughout the paper is to study the effect of synthetic data when train- ing on a mixture of the n real and m synthetic data described above, ie., with the following objective function:
n
m
L(w) = n + m.
las ye w) +
qe Dsi w)
(4)
n + m
real data
synthetic data
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where E is some convex loss function and the q, are data pruning parameters (g: E {0, 1)), indicat- ing whether to select or drop the hth synthetic sample (æ,, Bt). In particular, the q, s are Bernoulli random variables conditionally Yi * y 9J (we recall that D: 's denote the true labels of the synthetic samples) with conditional probabilities
p = P(q: = i + ).), :: Pg:
0: = -.:)
(5)
which control the pruner accuracy (as discussed in (Feng et al. 2024).. ). As mentioned previously, we suppose training on n 2 î real data, modeling a situation where new real samples are available with á controlling the generative model quality in generating faithful synthetic features
Loss.. In the remainder of the paper we take to be the regularized squared loss as it allows us to obtain closed-form solution for the optimization problem equation 4 hence, more tractable analysis. Specifically, we take {(z, w) (w æ - y)² + 7|/2|| where y 2 0 is regularisation parameter, which yields the following closed- form solution
w + QXy. Q -- ( E
(6)
where . n + m, the matrix X (Z1...... Dn. 91 ................................
qm RPxN is the concatenation
of both real and (pruned) synthetic features, and the vector y = (1/1...,, ya. 01...., Fm) E IR is the concatenation of real and (noisy) synthetic labels.
MAIN RESULTS
In this section, we present and discuss the main results obtained through the analysis of the classifier model defined in equation 6
We start by specifying the supposed growth rate assumptions.
Assumption 4.1 (Growth Rate). We consider high-dimensional regime where p, p,n, f, m + 0o
and we recall N = n + m such that:
1) 2/ + r E 00, 20).
2) _ -
3) -ーー - [0. 1).
4) =
Role of the assumptions. The above assumptions are central to understanding the nuances be- tween real and synthetic data (as constructed above) a high-dimensional regime. Essentially,
Assumptions 1), 2), and 3) define the scaling of data dimension p and the different sam- ple sizes (n real data, m synthetic data, and ń real samples used to train the generative model). In particular, we suppose that all these dimensions scale linearly relative to each other, which corresponds to the classical RMT regime. This setting is more general than the infinite sample size regime in the sense that the former be recovered by taking 7,, f + O. Specifically, the parameter r controls the generative model quality, where lower values indicate better generative model quality. Plus, the parameter corresponds to the proportion of the real samples in the data mixture. For instance, = O models a setting where the training is done only on synthetic samples, and < highlights the fact that the number of real and m of synthetic samples are of the same order, therefore, making our results scalable to any possible proportion π.
The fourth condition about the magnitude of the mean vector μ reflects the fact that the classification problem should neither be trivial (laa4l > 1) nor impossible [[]pa]l - 0) as the dimension of data grows large. For instance, assuming || of order ((vP) would not be relevant as + oo since the classification problem becomes trivial in this regime. We refer the reader (Couillet & Benaych-Georges (016)) for a more general formulation and justifications of this assumption under an extended k-class Gaussian mixture model.
Having stated the main assumptions, we are now in place to present our main technical findings on the performance of the classifier model trained on a mixture of real and synthetic data. As corollary, we also cover the case where the model is trained solely on synthetic data and showcase a generalization of the result obtained by Feng et al. (2024).
'Technically, our results hold irrespective of the statistical dependencies hetween the data used to train the generative model in equation 2 or the classifier equation 6
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0.6
2-
m
än
0.4
0.2
0.0
01
10
10-3
10
10
10-2
10-1
109
10
10
10-1
10P
10
2
Figure 2: Behavior of (6,, 6g,6) in terms of the ratio H. For small ratio H. the values of 87, 8% 0g are close to O. (6,, 62, 07) are computed by iterating the system 7 starting from random values.
4.1 PARTIALLY SYNTHETIC: MIXTURE OF REAL AND SYNTHETIC DATA
We start by analyzing the general case of training on a mix of real and synthetic data. As we described in the previous section, the statistics of synthetic data are empirical estimates the ones of real data. Under Assumption 4.1 the estimation of with ja remains consistent, while the estimation of the underlying real data covariance (i.e., Ip in our setting) with C is inconsistent previously discussed. As a result, studying the theoretical performance of the classifier in equation 6 demands deploying tools from random matrix theory that refines the estimation of scalar quantities depending on large random matrices. In our case, the scalar quantity of interest corresponds to the model's s accuracy which depends on the random matrices Ć and XX as per equation
In our analysis of the classifier's theoretical performance, we found that the effect of high-dimension (and that of distribution shift between real and synthetic samples) is described by three scalar quan- tities (6;, , 8g) which are defined the unique solution of the following fixed point system which is derived from Lemma D.1 in the Appendix.
a(1 T)
7
1
8g π 8,
ad, 1 + 8g
5y
5,
6g
(7)
1 8g
o(1-x) T+6, (1+6,)(1+55)
7) a(1
where a = 6(1 - + ps. These quantities will be used subsequently in our results. Intuitively, o, captures the contribution of real data, 83 corresponds to the contribution of synthetic data, and og corresponds â m,
the influence of the generative model. In infinite sample size regime where oo while the dimension p kept fixed, (6,, 8 og) (0, O, 0) as per Fig. 2 while under Assumption 4. these quantities are non zero yielding a counterintuitive behavior in hign-dimension. For convenience, we further define a set of scalar quantities that will prove useful in the next result.
A = =E[ga = Φ(1 - E) Pe, a = E(gi] = o1 - <) + pe,
π
+ ae11 T)
π
a(1 - σ) + (1 61) + +5))
c= π
X(1 x)
a = 1 + 0z
b b= +
+
1 + 8z
1 1 8z
1 8
1 + 8
a(1 T)
a(1 T)
bj
b (1 (1+ + 83) h
(1 5;) 222b2
(1 + 5z) 2 + + Ph2b2
a(1 - T)
n
h =
b2,
h =
(1 (1 57))
6²
The first set of parameters (a, X, a, b, c) pop out from the expectation of the classifier's decision function while the remaining quantities are related to second-order statistics. Essentially, the main relevant quantities to our analysis are f and E which characterize the quality of synthetic data, with and p characterizing the verification process. In an idealized scenario, we would have = =
which reflects the fact that there is no distribution shift nor label noise, while 1- corresponds to perfect (oracle) verification process. Our main goal is to study how these param- eters influence the classifier's 's performance hence providing the conditions that make synthetic data relevant for performance boost. The main result brought by this paper is therefore stated as follows.
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Oracle Supervision
Weak Supervision
0.750
Tost. Aocuracy
0.725
0.200
0.675
50
200
0.630
500
0.2
0.4 Proportion of Synthetic data.
Proportion of Synthetic data
Figure 3: Scatter plots correspond to empirical test accuracy while lines correspond the theoretical counterpart per Theorem 4.2 The parameters used in this experiments are: n = á 1000,
||인||| 0.7 and 7 = 1, (p. Φ) (o, 1) for Oracle supervision and (p, Φ) = (1, 0.5) for the Weak supervision. The parameter is variable depending the proportion of synthetic data by taking it equal to the misclassification error corresponding to training classifier on synthetic data only. As theoretically anticipated, a boost of performance is observed with synthetic data supervision while distribution shift affects negatively the performance.
Theorem 4.2 (Theoretical performance). Let w be the Ridge classifier as defined in equation 6 and suppose that Assumption 4.1 holds. The decision function w some (real) test sample Ca. with corresponding label = (-1) and independent of X, satisfies
w x 2 N (2 ,. m²)
ellee where μ bea utll
and
2( Alo)) (b allla113)
+ ai + bi
c(1 ((1+by
b2)lleal?
hb + al1
h2
bn
Moreover, the asymptotic test accuracy of the classifier is given by &b m²²) m km) where
S dt. (xx) 22
Theorem 4.2 states that the decision function of the classifier in equation i asymptotically equiv- alent to the thresholding of monovariate Gaussian random variables with respective means f and - and standard deviation v, where the statistics μ and и are expressed in terms of the scalar quantities defined above. Here, je represents the signal strength while v highlights the classifier's uncertainty or dispersion. To provide some insights into the implications of this theorem, we start by examining in a oiw-dimensional regime where p kept fixed while n, m, > co. this case, we have 11, f + 0 and o;, 5%, 5 O which yields
a(1 x) b= = n aal T). c= X(1 -)-
and a1 b1 b2 0 with h1 = h2 1. As such, the accuracy of the classifier increases with A, i... when the synthetic labels are verified (large 5 less noisy (small E). This in line with the findings Feng al. (2024) while extended by our result training on a mix of real and synthetic data. However, when the dimension scales linearly with the different sample sizes, the values of 5,,8% 88 8% f 0 yielding a lower signal strength y and higher variance v². This highlights the fact that in high-dimension, even if the synthetic labels not noisy equivalently well verified, there a performance drop due to the feature distribution shift between real and synthetic data.
Fig, 3 depicts the empirical test accuracy and the theoretical prediction per Theorem 4.2 when varying the proportion of synthetic data. As theoretically anticipated, adding synthetic data does not boost the classifier's performance unless it is verified accurately (oracle supervision versus weak supervision). Morcover, our results show the effect the distribution shift which heavily affects performance in the case of weak supervision (Fig. right).

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4.2 FULLY SYNTHETIC: TRAINING ON SYNTHETIC DATA
In this section, we study the fully synthetic setting which corresponds to training solely on synthetic data (i.e. 0 in equation 6). For simplicity, consider only label noise and ignore feature noise in the synthetic data. Essentially, this allows us to exhibit the smooth phase transition of the classifier's accuracy in terms of label noise. which extends the result of Feng et al. (2024). Specifically, we obtain the following corollary of theorem 4.2
Corollary 4.3 (Performance when training only on synthetic data). wg be the Ridge classifier described in equation 6 trained only on synthetic data with only label noise (i.e., Ö Ip). Under Assumption 4.1 the decision function w x on a test sample a E Ca with corresponding label y (-1)" and independent of X. satisfies
2, N N Max Us - 122)
w. x
where
6(1 - 8) - ps
|11112
al1 2 + 7(1 5g)
14
aa/l4/22 |||l +1
2(1 - h))
1 1-
+
h(a 1112 7(1 85])
7(1 (6)
h
with
5, = 7,0 - -
V(a + 7- n,0 + 4n,07
= lim p-000 m'
h 1-
(a ((1 8))2
2
Corollary provides explicit formulation
of Theorem 4.2 with synthetic data only and ignoring distribution shift (yielding an explicit expression of 5,). This setting provides clearer interpretation of the effect of label noise since the classifier's performance is directly re- lated to the quantity = @(1 e) pee The breaking point of the classifier's performance occurs at À = O, which corresponds to the ac- curacy of random guessing. yielding to the crit- ical value of label noise ex (1 + -) This critical value is equivalent to the one obtained by Feng et al. (2024), however, extend their result to the high-dimensional setting which ex- hibits smoother phase transition depicted in Fig. 4 Essentially, the sharp phase transition of Feng et al. (2024) covered by our result by taking 7). O. In this sense, the predicted smooth transition better mirrors real-world sce- narios where finite sample sizes introduce grad- ual changes in performance rather than abrupt shifts. This makes theoretical findings more applicable and reliable for practical scenarios.
0.8
Test Accuracy
0.6
=1.0
0.4
=0.1
-0.01
0.2
0.001
0.9
0.2
0.4
0.6
0.8
1.0
Figure 4: Phase transition in terms of label noise
as predicted by Corollary 4.3 The critical value for predicted
-)- We fix p 100 and vary m. The remaining parameters are || = 1, 0.3 and ø = 0.8, i.c. 5" = 0.73.
EXPERIMENTS
In this section, we present our experiments conducted on different real-world tasks and datasets in order to illustrate theoretical findings presented in the previous section.
5.1 EXPERIMENTAL SETTINGS
Amazon Reviews. We use the Amazon Reviews datasets (Bitzer et al. (007)) which include several binary classification tasks corresponding to positive versus negative reviews of books,

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2
3
2
Class 3
Class
Class 3
Class 2
Cluss
Class 0
Class
Class
Class 4
Class 6
Class
Class 3
Class 3
Cluss
Class 9
Class 7
Class 5
Class 3
Figure 5: Illustaaiion two different generation schemes for the MNIST data. Top figure: Generat- ing MNIST-like data samples by only estimating the mean of each class faa for E [10] and without estimating the covariance matrix, i.e samples here are generated through the distribution N W( Ip)- Botton figure: Generating samples by estimating both the mean and covariance of each class, as of our considered generative model defined in equation
elect ronics and kitchen. We apply the standard scaler from sklearn (Pedregosa et al., 2011] and estimate ||||| with the normalized data. The synthetic data is generated following the described generative scheme (see equation 2). We use the Ridge classifier in equation 6 for this data.
We also conducted experiments on the MNIST (LeCun Cortes (0110) dataset to illus- trate our theoretical insights, by training simple neural network with one-hidden layer and ReLU activation function. Concerning the synthetic data, we used different values of to generate new samples in order highlight the importance of the generation quality, and introduced a label noise to highlight the importance the pruning. Figure shows some examples of MNIST-like synthetic data that has been generated and used in experiments.
MNIST.
LLM Safety Alignment. We also investigated the impact of synthetic text data for the task of alignment of LLMs with direct preference optimization on safety datasets, using the approach as in (Alami et al., 2024). We finetune the Falc Falcon 2-11B Instruct model (Malartic et al.) 2024) 5000 human data from Anthropic's HH-RLHH datase which correspond to real data, while synthetic data are extracted from the PKU safe RLHF datase which are generated using Alpaca3-70gf We increase the amount of synthetic data by injecting gradually five batches of 7000 samples per batch, to study the performance of the fine-tuned model as we add more synthetic data. In this experiment, we focus only on label noise by randomly perturbing the synthetic dataset.
Each entry from the synthetic dataset includes a prompt 0, safe response (3) (safety- response), and less safe response yاب (safety-rejectee dataset by swapping safe and less safe responses with a probability (label noise), and selecting the prompts according
response). We, therefore, perturbed this
verifier of parameters p and &i described earlier in this paper.
For the evaluation, we use the ALERT (amaaseF](TTdesssee er al. (2024)) to test the safety of responses of the finetuned model after being judged by J.Lama-Guaren 3-BB (Dubey et al. 2024). As in (Alami et al.) 2024). we compute the safety score as the percentage of safe answers labeled by 8B. We report the results in figure 8 for strong supervision (p,6) = (0.2, 0.9) and weak supervision (p, Ø) = (0.5, 0.5) for both E = 0.1 and E = 0.5.
LLM Q&A Safety Generation. This experiment aims evaluate the impact of synthetically generated prompts (i.e. feature noise). To construct the generative model for this experiment, we fine-tune an LLM (M) with supervised fine-tuning (SFT) on pairs of question-answer (Q&A) sen- tences extracted from a safety dataset. Initially, we fine-tune M on 12k human annotated Q&A as safe or unsafe (Ji et al., 2024), yielding a fine-tuned model on human data denoted as M₁-- Then, MA is considered as the generative model to generate large dataset of synthetic Q&A prompts
ama
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BOOKS
ELEC
KITCHEN
Oracle Supetvision
0.85
Test Accuracy
Sapervision
0.85
0.84
0.80
0.86
0.82
0.0
0.5 Pruportion of Syuthetic data
0.0 Proportion of Syuthetic data
0,5
0.0
0.5 Proportion of Synthetic data
Figure 6: Results of the Amazon Reviews setting: Test Accuracy with the proportion of synthetic data evaluated on Amazon Review Blitzer et al. (007)) dataset. The number of real data sample used and = 0.2 (fixed). The pruning parameters are (p, () = (0. 1) for Oracle supervision and (p. の) = (1, 1) for No supervision.
800, the dimension
400, 7
Label noise 0.3, with minor Feature Noise
No Label Noise
90
Oraele supervision
90
Best state estimation
Test Accuracy (96)
Strong supervision
Test Acouracy (%)
Wiols stats estimation
85
Very
No supervision
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No covaricace estimattion
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0.1
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Proportion of Syuthetic data
Proportion of Syuthetic data
Figure 7: Results of the MNIST setting: Training an NN with one hidden layer and ReLU activa- tion function on mixture of real 500) and varying the proportion of synthetic Gaussian data.
(around 120k samples) that were further annotated as safe/unsafe using Mistral-Nent and Qwen2 7BB-Instrcct (Yang et al. (2024), which incorporate further label noise. To verify the data, we use Llama Guard-3.3 3.1 (Dubey et al. (024).. We conducted this experiment using two LLMs (M) of different sizes (to vary the generative model quality) which are the 1aama-3.1---. and Cemma-2-2B-- (Team et al. 2024) instruct models.
5.2 EFFECT OF LABEL NOISE
Figures 6 (left plot) S reflect the effect label noise. Essentially, as theoretically anticipated, the trained models do not benefit from synthetic data unless it is accurately verified. Specifically, in the case of weak supervision, model performance drops significantly, and the improvement from using synthetic data is only visible with very high synthetic sample sizes. On the contrary, with strong supervision, we observe a monotonous performance boost as the proportion synthetic data increases.
5.3 EFFECT OF FEATURE NOISE
In this section, we discuss the experiments related to feature noise. In Fig. (right), we depict the performance of one-hidden layer MI.P trained mix of real and synthetic MNIST data following our theoretical framework. As we can observe from the figure. the performance boost from synthetic data heavily depends the generative model quality predicted by our theoretical results. We further observe the same trend using LLMs as depicted in Fig. 9 where we observe that the synthetic data generated by Li ama 3 1-88- Inst ruct yields a better performance boost compared to Gemma 2 2-2B-it as we increase the amount of synthetic samples, which means
https::/mistraa hww.///w///iimmmmmmm stral nemoo

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0.1
0.5
Strong Supervison
Strong Supervison
90
Safery Soore (%)
Wenl Supervision
Safety Svore (9)
Wisak Supervision
60
80
70
0.0
0.2
0.4
0.6
0.8
0.0
0.2
0.4
0.6
0.8
Proportion of Synthetic data
Proportion of Synthetic data
Figure 8: Results of LLM Safety Alignment: Strong supervision corresponds to (p. σ) (0.2, 0.9) and weak supervision (o, Φ) (0.5, 0.5).
Ilama3. I-8t-Instruct
Gemma-2-2BA4
75.0
Test Accuracy (96
Accuracy
72.5
70.0
70
67.5
68
Oracle Supervison
Oracle Supervisea
Supervision
Supervision
65.0
66
0.0
0.2
0.6
0.8
0.0
0.2
0.6
0.8
Proportion Synthetic data
Proportion Synthetic data
Figure 9: Results of LLM Q&A Safety Generation: Evaluation of two LLMs trained as pre- sented in section 5.1 is depicted for both (left) M= Llama3. Inst-Innttuucc and (right) M= Gemma-2-2B- it. The test accuracy is computed over the testing dataset extracted from Ji et al. (2024), with 2.8k Q&A samples, The results shown are the average over runs.
that Llama3. 1--8---I-nee Inst generates better synthetic samples (less distribution shift) than Gemma-2-2B-1
CONCLUSION AND LIMITATIONS
In this work, conducted comprehensive theoretical and empirical analysis of models trained on mixture of real and synthetic data with verification. By leveraging random matrix theory, we identified critical factors such as distribution shifts and label noise that significantly impact model performance. Our findings demonstrate that synthetic data can enhance model accuracy under spe- eific conditions, particularly when the generative model is of high quality and the verification process is accurate. Additionally, we extended previous research by showing that performance transitions are smooth rather than sharp when synthetic data incorporated
high-dimensional settings.
Despite these advancements, our current setting is limited to label verification of synthetic data. Incorporating feature verification represents promising extension for future research, which could provide further insights into the reliability and effectiveness of synthetic data in model training. Another possible extension our work is to study distributions beyond the Gaussian model and analyze how higher-order statistics can be incorporated into current framework.
In conclusion, this work provides a foundational understanding of the conditions under which syn- thetic data can be beneficial for model training in sigh-dimensional settings. By integrating both theoretical insights and empirical validations, this study provides new insights into the effective utilization of synthetic data, paving the way for more resilient and performant Al models.
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APPENDIX
Contents
A
Useful Lemmas
13
A.1 General lemmas
13
A.2
Deterministic equivalents
14
A.3 Resolvent identities)
16
Random Matrix Analysis of the general model
19
B.1 Test expectation:
19
B.2 Test variance:
20
Particular Isotropic covariance matrix
25
C.1 Resolvent identities in the case of C =
25
Test Expectation
26
Test Variance
26
Random matrix analysis of distribution shift
29
D.I
Deterministic equivalents)
29
D.2
Deterministic equivalent of QAQ
30
D.3
Useful results:
................................
D.4
Test Expectation:
31
D.5
Test variance:
32
E
Details about experiments with Safety LLM Alignment with IPO Hyperparameters
34
34
F
Details about experiments with LLM QA classification
35
F.1
Prompting LLMs
35
F2
Hyper-parameters
A USEFUL LEMMAS
Notation:
For {1,2}, denote by I, {i | belonging to class Ca. Furthermore, we denote D uu + Ip E x "] for C.. and Σg dade
C.). i.e, the set of indices of vectors
Here we will list the most useful lemmas and results used in our analysis.
A.1 GENERAL LEMMAS
Lemma A.1 (Inverse identity). For invertible matrices and B, we have that:
A A--B--1 A A A)B-1
Lemma A.2 (Woodbury). For A RPXP, U, Roxk such that both A and A UVT invertible, we have:
+ (A+UVT)
A U [ VTA- - -VTA-
particular case this lemma A.2 in the case of L, is called Sherman CarmaannMorrioon'' identity.
Lemma A.3 Ahherman-Morisson For A E RPxP invertible and u. E RP, A + wo is invertible and only if : 0 Au 0, and:
A uw A -1
(A + um 7) 1
1 A u
Besides,
A u
(A +w0 T)
1 A-1 -1u
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A.2 DETERMINISTIC EQUIVALENTS
Let us state here the deterministic equivalent of the resolvent matrix Q defined in the general model's equation (6) for any general covariance matrix C and mean pa Bp f that define the statistic of the synthetic data, as in equation 13
Lemma A.4 (Deterministic equivalent of Q). Under the 4.7 assumptions listed above in the main paper, a deterministic equivalent for Q = Q(7), denoted Q is given by:
T Tup
+ +1p) + a(1 x (1sp + +C) + 1
Q
1 1
+ 85
where:
n
a = Φ0 - -3) + pe, 8 Tr(Q), ds a Tr(CQ)
1 N
π
Tt + m
N
Proof.
We want to find Q such that for all bounded a, b R":
a (E Q)b 0
Let Q = (S + + We want to determine an S that satisfies the above property. We have that:
[[ Q (S N
E
) (lemma A.1
N
E EQQ(S - 0 Q
m
E[Q(S - xæ Q + N
EIQ(S q: QQ]
1 m
E[QS-
Q
T.C Q
E [QQ
qi
Q
x, u. JQ
N
1 + δn
N
1 5g
+EEQ
QS
Q 2. C
1Q + (1 (1- E
4i
Q a. 3 .. 1Q
- åR
1 åg
E Q
SS-
2
Q + -1- 11- , (S
q,
0 + O(N-1)
1 + 8n
1 8g
Thus, it suffices to have:
S T(U + Ip)
a(1 - π) ( + +C)
1 8R
1
to get the desired property.
Lemma A.5 (Deterministic equivalent of QAQ). Let A Rpxy be any deterministic symmetric semi-definite matrix. We have that:
a(1 m)
QAQ + QAQ + N(1 + +82 T(EQAQEE(QQQ)
TIE,QAQQE
N(1 + +Z)
Thus, we get that for A E, and for A Σg:
Tr T(( + a(1- x) π)
QEQ QEQ
N(1 + 8)2
N(1 + 6z)²
QE Q
π
T=XQEE + a(1 T)
N(1 + 8)2
N(1 d5)
And by denoting:
b a(1 T) Tr(:QQEQ).
TT(EQ))2)
a1 N(1 + 8)2
N(1 + 55)
by a(1 R)
0g N(1 ThE:; QVQ).
Tr((2g
N(1 + +6g)
h (1 (2) an) azb1
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We get that:
QIQ + 1- b
QIQ + Q
h
h 1 di QI
" QEQ
Qu + h
Proof. Recall that:
πΣ a(1 T) + 1
Q(7) =
1 1+ 8
1 + 6s
Let us denote by:
1+8 + a(1-r)Ea so that: Q - (S + ,Ip))
1+6s
We have that:
E E[QAQI
E[QAQ + Q Q)AQ
QE|AQ] E Q)QQQQQQQQQ
=Q [| Q))
=[(QQQQAQ
=QAQ
1 VV
QAQ + EQ
N AQ
N
QAQ + QSQAQQ N
E QAQ]
QAQ + EAQ+EQQQQQO TE Q QAQI - (1 2,
And we have that:
QAQ
u. QAQ]
I+
1
Q-2, Q-
B
Q
n. x.
QA QQ_,-
1+
N(1 +
1
1
E[Q x QAQ
N(1 8)2 E[Q-r, x. QAQ 2,2 Q-2.]
1+0
1
QQQQAQ
E
TH(EQAQ)E 2, Q
I+
N(1 + 8)2
1
E[QEQAQ
T(XQAQ
1+0
N(1 8)2
And:
QAQ]
Elq. Q -. QAQ
1+
Q--2- Q. 2, C. Q B1
E
Q 2, ã.
QA
1+ òs
N(1 8g)
Elg.Q .. QAQ 5.]
Elgs Q ,, म.
QAQ s, ب, Q r.]
1+ 8s
N(1 + 85)2
EIQEQQAO
TME ã. Q m-]y
1 ōg
N(1 5z)
EQQ
Tr TTEE&QAQEE
1+ 8g
N(1 63)²
Which concludes the proof by summing all these separate terms.
Corollary A.6 (Trace identities). Using the above lemma A.5 we get that:
wh N TEEE)QU = (1+6)* (a1 (1 b2) + azbi),
= (1 (1)
N E(2
a(1 - T)h bi
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And in the case of isotropic covariance matrix: C I'Ip:
N (TEQ) = o²
TrEE(QQE)) N
" (1 b2 + +b1),
N Tr(E(QQQQ)
h02
A.3 RESOLVENT IDENTITIES
Let Q be the resolvent matrix defined in equation (6). from the dataset V by removing the 2th sample v; Le::
Denote by Q-o; the resolvent matrix obtained
q
1
-1
Q
N Yiu
Then, using lemma A.3 have that:
Q 0.U; Q
Q Q
1 + N
Q v; Vi
and,
Q am D2
Q Bi
Qz: =
Qū: =
(8)
1
1 + ōs
where:
8 N
TT(QQ)= = N Tr(Q), 65 = (6( -e) pe)
1
Tr(Eş
= &
TT(CQ)
(9)
N
N
Let us recall the expression of Q defined in lemma A4
π Tuu
+ +Ip)
a(1 T) ( + +C)
-1
Q
1+0 +
+ 7
1 0s
(A+UA --U
where:
A a11 1) C+ ( 1 p. U
a(1 - π
1+ 5g
1++5
14 1 ōs
(10)
Since C is symmetric and real valued, then it diagonalizable, and can be written as:
CDPDP
where: P-1 P is the matrix containing the eigenvectors of C in its columns, and D (d, the diagonal matrix of the eigenvalues of C. Hence, can be written as:
A = PAPT, A = Diag (x
a(1 π d:
(11)
1+8
1 dg
And using Woodbury's 's identity in lemma A.2 we get that:
Q A- A 'U
A - A-
where: A -1 = PA T and A 1 Diag
113
1135
i=1
Let M- = [1 + AUTAA-TA -1 and denote by Mej its coordinate in the ith row and j column. We
have that:
Q A -1 A -UMUUA A-
= A -1 A (Gr14TT +2ppp + Cs(Hu + pd )) A
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where:
TM1,
C a(1 T)22.
az x)
11 1+8
0)(1 os) 63 V (1 (1+0)(1
M2.
1 + 0g
Thus,
Q - A A (Siee + C2MSH3 + Cs(pe + MaH }) A
(12)
can further show that:
1+ a11 T
M1,1 det(AM-1)
M1 A 10gA-1pp. 148
òg
aT TT)
M1.2 det(M-1)
k A-1 1B
(1 + 6) + 8g)
aT - T
M2 = det(M-1)
Fe ATA-'H
(1 () + +8g)
π
M2.2 = det(M
+
+ o t A
det(M-)) = 1
π
t A-
1) (1+ ax(1 - n)
p4
A p4
(1 5) + 6s) (# A 2B)
an (1 T)
+
+ óg
Lemma A.7 (Delta). The parameters 8 and ög defined in equation identities:
g are given by the following
d;
8
8s
N 1=1 1
al al-m) m)
d
N
a(1- m) d.
I48
1t65
140
140s
where: (d.)%=1 are the eigenvalues of the covariance matrix C.
Proof. Let: M = (L2 + UU and denote by Mr, its coordinate in the 7th row and 7th
column. We have that using the expression of Q in equation 12
Q-A-1-A- A- (Cih + Capaua + Calung + Map 1) A -1
Then:
1
8 = Tr(Q)
N
1
Tr(A-1)
N Tr(A- (SiupT + Cauaur + Co(uuu + Hau J)) A -1)
N
We have that, when N + oo:
Tr(A -1 Ap A A-1)= A
= N PA-1)
N
since |e||| O(N-1) 1) by assumption 4.1
The same applies for p4g. Thus:
1
1 Sap (A-1)
2
8 N Tr(A -1)
N Sim T (A 2)2
N San (A -1)2
N
1
Tr(A (
N
1
N
1
+ +NN-11
N
1=1
7
I43
+
a(1 d,
146g
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Hence we have the desired result for ó in the regime N > 1 which considered in our assumption
Similarly for ōg, have that:
- ds
Tr(CQ)
a
n1
2 Tr(CA NIN CCA- A- 1 A -CA---AAA pB N Sap A -CCAAAA1 "pp
N
1 Tr(CA ) + +
N
1 Tr(DA-1) +N(N7)
N
d
+ +N
N == 7
o(1=))
de
148
14ds
Which concludes our proof.
Now let us compute the trace identities that will be useful the next sections.
Lemma A.8 (Trace identities). We have the following trace identities:
2
d
(+(())
Tr((Z,Q)3)
N
N
i=1
7
1+8
a(1-aidd) ds)
Itos
2'
N
= N
7+
148
a
T+ōg
d
di
TEQQE,
N
7 148
a(1-n) 1465
di
Proof. We have that:
TT(())) = Tr (mM T + +1,)(uHpTT + Q1)))
N
1 TT(QP) +
N
I
Tr((A (T(((-1))) O(N-))
N
TT((PA-'PT) -
N
Tr((A 113 N
+ -(N
+ ONN-1)
7 1 T
Thus we demonstrated the first identity. For the second one, we have that:
N Tr((EsQ)3)
1
Tr (mHa)) C)Q(ung + C)Q)
N
1 Tr((CQ))==((TT
N
1
N Tr((CA Tr((CA-1))) -12 O(N-1)
I Tr((DA -113 + O(N
N
1
d;
+ ++0(N-1)
N
1=11
7
1+8
011 Rt)
1485
d
And the same spirit of the proof applies to the last identity.
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B RANDOM MATRIX ANALYSIS OF THE GENERAL MODEL
In the generalized model, we consider that the synthetic data follow the following distribution:
ã; ~ N (p C)2 HB = 3p + n
(13)
where E defines the alignment of the synthetic mean with the mean of real data, and f is vector orthogonal to μ. Now we will analyze here the performance of the classifier given by equation (6), and prove generalized theorem B.l in the paper.
Q()) (Xy XD(G)O). Q(7) VV +rIp)
Wly
N
N
The performance of 6 are fully determined by the first two order moments: 1 x)
Elun, a] and
B.1 TEST EXPECTATION:
We have that:
1
1 t=1 N
Qua
+
Σ
Quaia;
N
==1
Let ~~N((=11))m Ip) be test sample independent of all the training samples (v:) Then:
Elw æc
E|ya
Qæ:
Elaaa.0 Qæ]
N
First sum: We have that, using the same lemma
A.3
1
a Qal
=
E3 [ysar T Q==,
N
1+
5
1=1
1
n
E[x;]
N
1
1
(-1)a
1+8 p JQU
N
(-1)am
1+8 # JQu
Thus,
1 n
(-1)ax
E 3 Qxi
4 Qu
(14)
N
1+6 8
Second sum:
Using the same lemma
A.3
1
E a
Qæ
E
Elg.ū, T. Q_
N
85
i=1
Elg.ū E(2. TE[Q=,,
N(1 + 8s)
(-1)a
\1 Qu
N(1 + 85)
(-1)
14 Qu
1 ōs
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where (here ys means the true label of z;))
E(aūi = yı P = 1 | y/l >iPllg = 110r + vil
= y/: (0 11 - E) -pp) = Ay
Therefore,
Blwe T 2d] = (-1)
(T+5""
π T
A(1 - π)
11
Q₁
(15)
1 ôs
B.2
TEST VARIANCE:
To determine the variance of w, T it only remains to compute its second order. We have that:
2
[ x²
1
E
Σ yoa
Qx
N²
j=1
27
27
N22
1
E
Σ
y:
Qæ
N2
1
E
Σ U
Q
N
2
E
Σ
y
Qæ
Σ U5
Qz
Let us compute each sum on its own, and then group the results at the end.
First sum:
We have that:
No E
Σ via
Qæ
Blynyaa
x T Qaa T Qæ)
N2
k=1
a Qwæy Qal +
E a Quæ Qæ]
N
N2
For i o k. we have that:
B Q Qæ)
Elyaya. Qwx Qux]
QEQaa]
E 2
Q- QQ -2 2k
(1 + 8)2
(Q Q-ari
QQ m.. T Q mit
Q
a.. TA Q mi.
(1 + 8)2 E Diyiao
Σ
Q x.
1 8
1 5
(A1 Ag
Ag + A4)
(1+ 8)2
And we have that:
A = Eluin Q- ΣQ 10,
μ EIQEQ/
bi ( 1 -T (1~ b2 Q
h h
And:
1 E Ely.yr. Q- Q- EQ _Q-2,,z1
A N(1 + 5) TM(EE(QEQ)
/1 TQU
N(1 +
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Since, by concentration laws:
N 2b Q 2
EQ
m.a 2%
Ela x Q -. _Q-=1,
wi, m3c
N
Q =,, EQ =.,,
N
Tr
D wie IQ -2
N
-. IQ- m.,,))) Tr(EE
1
TrEE(
N
And we can easily verify that:
A3 A2
A4 O(N-)
Thus,
n
/ [B[QQEQ (TEEQQQE))
Ely.ua. Quay Qœ:
((OH)
N2
N2
N + 8)
π²
[ITEQQQQ|= (TTEE(QEQ)) TQQ
(1 8)2
N(1+6)
And then, for i {I,..., n):
E E|x Quæ,
Qæl
Blæ QEQa
1
E(a Q m; =Q-n.mi)
(1 8)2 1
a Q- _Q
(1 + 8)
1
(1 + 8)22
Thus:
Eix Quæ, Qæ
N(1+8) 8)2 Tr(EE(QEQ)
n
N2
i=1
Hence, the first gives
2
n [E 2 (TEEEEQQQ)
N2
Σ
ya
Qæ
(1+ 8)2
N(1+6) 4 T 1Qu +
N(1 + 8)2
T(EE(QEQ))
(16)
Second sum:
We have that:
27
m
m
E
Σ
Q
E Elggūüū Qua Q=)
N2
N2
E El::0;ū:ū, Quæi Qæ) +
Elg:a, a.. Qa:
N2
N2
For Fori (1, m). we have that:
0.000 Q Q=) = E QEQa,
E Elg.q, ū.ū, _. Q-a 2Q
85)
Q e.,, 1 [. Q Q-
Q -.., q. Q-ár.
. T Q Q-
Σ Q a
(1 + 83)2
1 85
1 8s
(Ai A2 Ag + A4)
(1 + 82)
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And, have that:
A1 Elgg 20
B Q --,,, 2Q_2,
A² B(QEQ
And:
1
A2
Elq:q;ū:00 _. Q -., 3, Q a.,, EQ-a.,
((1 + 65)
1
N(1 + 65))
X²
N(1 0s)
And, can easily observe that:
Aa = A2,
A4 = O(N-1
Thus:
12 T)²
2 N(1 +8))
Fla.q, Queij Qæ)
[1: [
(Q=Q)) Q114
N2 itj
(1 + 6g)²
And for i {1 f1,...m..
E E T Quæ Qæ:
Elg:3 Q -, 2Q ă.
(1+ 8s)
O
EQ -. T.]
(1+ 0g)
7 [
क. EQ a:
(1 + 6g)
Tr Tr QEQQ)
(1 + 8z)²
Hence, by grouping the terms, the second sum gives us:
2
1
Σ
m
a. Q
(17)
N2
2 7)²
14 ETJEQQQQ 14
2 N(1 + 5s)
([ Qu4s)
a(1 x)
E
(1+ 85)2
N(1 + 8g)²
(18)
Third sum: Let us now compute the remaining term in the sum that given by:
E a Quq, 0,q Q
N
Let [l,..,n] and j {1, ...; m. have that:
Blyaq,0a0 Qux T Qū,)] E( Q
B 1, (1+ 8) 8s)
mc JQ 2Q-
Q 2 _ Q-ij
5(Q-.15
Q ii Q-ij
y:
Q Q-ig
(1+ + 5)(1 + 5s)
N(1 + os)
N(1 + 5)
(Ai Ag Ag As)
(1 + () (g)
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We have that:
A o Q [ XE y QQ-y----ij0jp1
= Ay E
And:
1
A2 N(1 + 8)
Ely.q Q- -- Q-1
X
Elyao EQ [
N(1 8)
À
T({E)[QEQ) TQpp
N(1+
And also:
A3 = N(1 + 6s)
Ely.q, x; T 15 IQ
X
Tr QEEQ)) Qus
N(1 + 6s)
Hence:
2
B a
Qæq, Ú0 _ Qæ)
(19)
N2
1=
21
N (1+t
Σ + 1
Eg E(QUa]] 4 "uua
QNQQ/
(20)
(1 + 6)(1 + ōs)
Grouping all the sums: Denote by: Ti N Tr(EE(QEQ)) then: T- N Tr(E)
Now let us group the terms in T in the three sums, and those that do not depend on T. We get that:
2
+ 2 2)
2AT(1 T) + (1+ 8)(1 + os)
B[(w" x0
EQQQQU
(1 (1+55)* u MiQQQIu
EEQQQ ps
(1+ 8)2
+
π
2T
Qu (1+ 8) + 85)
2AT - T
1 Quu
(1+ 8)2 a(1 π)
(1+ 8)3
21
7)
2 - σ)
+12
11 Qu4
(1 8) + 8z)2 1 Q44
(1 + 0z)²
(1 + 65)3
n22
2 7)²
14 JEQQQ 443 + 2AT (1 π
TEQQQQI +
E[QEQ|us
(1+ 5522
(1 + 5z)
5)(1 ōz)
TTi
2T
2x(1 - T)
1
Q1
TQUB
(1+
1 8
1 5g
(1 T)
21
T)
2AT
1 85 149
T
1 / "Qug
(1 + 2
This leads to the following theorem:
Theorem B.1 (Gaussianity of the General model). Let wy be the Mixed classifier defined in equation 6 and suppose that Assumption 4. holds. The decision function w some test sample C2 independent of X, satisfies:
w T
2, N ((-1))mg: V4 m²))
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where:
X(1 (1--7)_ T7) T
my
1++ 8 4
1+ ōs H/ Que.
(1 + 8)2 k E + 12 x²²
TT²
2A -x) 1 TE "EQQQQ pg
(1 + 65)2 Mg E Q
(1+ 8)(1 + 6g)
TTi
2
2X(1 π)
+ (1 + 5)2
1 + 5
1 T '21
1 Qui
+ 6s
+ (1 T2
21 T)
2
a
pg Q11
+ 8 1u '113)
(1 +1+00 0g)
1 òs
where:
1
T N
Ta Q(OQ), A= -= PE
24

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C PARTICULAR CASE: ISOTROPIC COVARIANCE MATRIX
Here, we consider a simple covariance matrix of the form C = I²Ip p for some o > 0. So
ōs = 20
(21)
C.I RESOLVENT IDENTITIES IN THE CASE OF C = I0pp
We have that by lemma A.4
Q
I+8 (mp
π
+ +1p)+ + a(1 - x)
1- a028 (Hann) + +0²1p) + t7lp)
-1
a(1 π
1 1
1 a020 H + 01p
where:
π
ao T)
1 0=
+
(22)
1+8
1 2028
Define by:
(a(1 ( + 2028 H8p + 8Ip )
-1
R1
R2 =
1 8 84 T + HII
(23)
such that:
Q
1 0 up + RR
π
1 + a020 H8p +R²"
a(1 - π)
Thus, using lemma A.3
Rip Q1
R2p4
Que
(24)
1+ 148 R.4
1 1+ Itao28 11 R2
a(1-m)
And:
RRuu R
a(1 - π R2 M R
R 1 + Tp Rip
RR
(25)
1 ag28 a(1 Ji T) ReM3
Lemma C.1 (Delta). The parameter as defined in equation
9 is given by the following identity:
11
1)
8
8 7 I+8 + 00
1-a025
Which gives us third order equation:
a0²763 + a
Lemma C.2 (Resolvent identities). Using the first identity in Sherman-Morisson lemma A.3 we have that the expressions of R2 and R₂ are given by:
a(1 π 1334
R1 I, (2 + 20 + 0a(1
R2
22
8
82
8) + θm p
And we also have the following identities:
143
R1p3
REH
a(1-5) It0s
+
N 14.
|1|2
|||l22 (1 (1- 0 8s) + ax(1
a(1 T) 8)2 ||
|u|| 0(1 + 85) + a(1 - -1 3 1
R4
T) |
0
0 88) + +x π) 14,
3(1 + 8011ma112
μ Ræjs==
8(1 + 8) + क
25

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xT32 |e
113 R2f43 = ||
0(1 + 6) + π x1|pe|?
|1ali
2a(1 T)332 |1|1
(2(1-x)2p) |0
1
A2
0(1 08) + a: T 148
(0(1 + 65) + a(1
14g 112
=|11112
ax11 - n7 n/ ||||
a(1 - x) p4
82
(2 + 6s) + a(1 -) p4 (3)
8(1 + 65) a
p&
a(1 x |1
n T||
143 R2
B² (1 ( + 68) + a TT) 13
8(1 + 6) + 1
on xz x) |=|
+ (0(1 + +6 + x ++ + 68) + a(1 7)) -1
|
TB TB3 |4|
=||1112
62
(2 + 6) + π || 11 12)
0(1 6) + ग |1
|
n,3²
|||| (Tr ||| + 20 + +))
82
(2(0(1 + 8) || 12)2
Lemma C.3 (Trace identities). Let (4,..,,). and j {....., m., such that: Σ E/x E/x
pp + Ip and Σg 1=0==((0jj E[ pale + I
We can prove that:
y0t 1 " TT(EE,)))) g2 N
n10
N (T((E)))
TT(E,
Ø2 N
f2
The performance of wy in is fully determined by the first two order moments: B[w, æ] and [ x)
C.2 TEST EXPECTATION
We have that using the calculus in the past section:
1 + M1--) π)
E [ x = (-1)
1 a028 14 Qu
(26)
And finally we use lemma C.2 and the following identities to obtain the result:
Rij
1g R2f
μ Qu 1 1 14 RRRR
pg pTQuu=
1
a(1-)) I+ds 1 R2443
C.3 TEST VARIANCE
To determine the variance of w. it only remains compute its second order. We have that:
E[w x) = .22
E
Σ y:
Qx
Qx
27
27
No E
Σ y
Qx
+
E
Σ
ش Qu
2
E
Σ y
Qx
Σ T Qæ
No
N2
And using the same computations in the past section, we get:
First sum:
The first sum gives us:
1
4 E[QQQQ 2 QEEQ)
N2
E
Σ a
Qx
= (1 8)2
N(1+
"u
N(1 + 6)2 THEE((QQ))
26

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Second sum:
the second sum gives us:
)
21
1
E
Σ a Qu)
N2
A² 7)²
2 N(1 0s)
[T=22( Qua)) + a(1 x) TT
(1 61+
N ds}²
Third sum:
The third sum is given by:
E y Q Qæ]
N
E j=1
2Am
rt)
I B'EQQEQu 1 TT
N (1+10+
Σ
Σg
)QEEa) u (TQus))))))
+
(1+ (1 + o5)
1 òz
Grouping all the sums: Denote by: T= N Tr(EEQQQ)), then: # E[ = o²T. Now let us group the terms in I in the three sums, and those that do not depend T. We get that:
n²
(² +)²
2AT(1 T) + (1+ 5)(1 + 6z)
B((w) w 2;
E EQQQQ
+
(1+ 5z)² 1 LEQQQIs
1 TEEQQQQ
(1+ 8)22
2T
2AT TT)
+7
u T Quu
(1 + 8)2 + 85) ui Q11
(1 (1+0)2
(1 +1+0 0)3
a111 - T)
21 - T)
2)T (1 - π
+ 2T
140 Q14
(1 + 6) + 6s) 1 Q/13
(1 + ōs)²
(1 55)3
(1+ 8)2 1 "EQUQ) + 2² x)
n
11 [E[QQQQ M3 + 2A (1 π
EEQUQ
(1 + 6z)²
(1 + 8)(1 8g)
TT
(1- 2TT
1 1+8 H Qu 21(1 - T)
TQuB
(1+ 6)2
1+0s
(1 11-7)a T)=
21 - T)
2A
1& Q14
1 1 Q143
(1+ 0g)²
1 5g
And we can compute this since we have that:
EEQQQU = h
((1 - b2) ((H Q1I)² + 1 Q²H²) + b1 ((1 Qua)) + O0p TT (TQHH)) H)),
1
4 EQEQUss T ((1 b2) (1 Qu) + 1 Q² usl b1 (1 Qu3) o² u Q m3)
EEQQQus 1
((1 b2) [u QU T "Qus + μ Q² [o] + b1 [H Qu Qus 0² T Q (ol)
h
1 R2443
11 R2
μ R
Hg Que
122r p=
11 NTQ²1a μ2
o(1-n) + 1+6s
MoR
(1 14s R1H4)
(1
1+6s us R2
2
Me R2R
μ Q²
(1
x 14 R1H1) (1
o(1=x) 1+0s p (Tap
Theorem C.4 (Gaussianity of the model for C d²I,). Let w, be the Mixed classifier as defined equation and suppose that Assumption 4.1 holds. The decision function w æ. on some test sample x E C. independent of X, satisfies:
w 2, N (((N((1))))) ((-1)mmq: U4 - m²))
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where:
X(1 (1 T7) T
my
1+ 8 4
1+ ōs H/ Que.
(1 + 8)2 M [EQQQQu + 12 x²²
πT²
2A x)
(1 + 65)2 Mg E P3Q
(1+ 8)(1 + 6g) 1 B.EEQ EEQ pg
FT
2
21(1 π)
+ (1 + 5)2
1
1+ 5
T 'uu
+ 65
1 Q/i3
+ (1 T
21 x)
2
pg Q/4
1+0 5 1 Q143
(1+ 6s)²
1+ ōs
With:
X 6(1
ôg aa²8
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D RANDOM MATRIX ANALYSIS OF DISTRIBUTION SHIFT
We will now quantify the performance of the classifier obtained through mixing some real data and synthetic data sampled according to the schema described in 2 Hence, the matrix Q, defined in lemma A.4, is no longer deterministic as we take the covariance matrix C = = E
(2 yrft) For simplicity, and without loss of generality, we consider á Gaussian vectors
(z:) NO, I,) that are independent of (a:;) and write:
ü = μ3
Σ Zi: ñ2
1
Č=
1 ñ
22
Note that we can ignore the error of estimation of f because we have that:
1
E
Z
O,
E
Σ
2
2j
Hence, when we have a sufficiently large we will assume that: ja = l (the estimation error on (i--1))
D.1
DETERMINISTIC EQUIVALENTS:
The resolvent matrix to be considered in this setting is the one defined in lemma A.4 but with Č:
Q(7)
1+0
π a(1 T
+
1+ 8g
1m a(1 a(1-x)
1+ 05
c+
7
1 +
π
Ip
1+8
π
+
a(1 -7)
1+ 8g
1141
(1 + 85)
a(1
2izs + (
1+8
Ip
where:
8 1 Tr(Q),
8g Tr(CQ)
N
N
Let us denote by Q-; the resolvent matrix gotten by removing its dependence on the vector z;. In other words:
a(1 T
a(1 T) Q-= i
Q
Q
nn1 + 85) 2,2,
(1 + 65)
By Sherman-Morisson 's lemma A.3
we have that:
a(1-s)
2 Q-
Q = Q
m[1+bg]
M[1 [g] z; Q-
a(1-1)
And:
Qi
Q
Qz, a(l-x)
1 a(1+0s)
2, Q-iz;
1
where:
8 a(1 1 Tr(Q) 1 0g ñ
(27)
Since the covariance estimate in equation is stochastic, the matrix Q no longer determinis- tic when replacing C with C. Hence, we will give a further deterministic equivalent to Q in the following lemma.
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Lemma D.1 (Second Deterministic equivalent). A deterministic equivalent of Q is given by:
a(1 (T)
+(++
π
a01 T
1+ pu +
1+8 (1+ 61) + 11)
where 5 can be found as fixed point using the following identity:
8 a(1 x) 1 Tr(Q)=
a(1 T)
R
1 Tr(Q)
n. (1 + 0g) a R)
8
(1 6g) á
(1+ 6s)
a(1-x)
N
7 1+0 (1+8s)(++)
ad
ós = Tr(E(CQ)) 1
Now we will prove the deterministic equivalent given by lemma
D.1
PROOF OF LEMMA D.I.
Let us denote Q (<(1-)) 1 1+68 7 - 2, + =
-1
Let also Q the deterministic equivalent of
Q. It can be written as: Q (S + A- We want to find some S such that for all a, b E RP:
a E Q b a Qb
We have that:
a(1 T)
E -Q E
Q
Σ
2 )Q
1 6s
1=1
1
E[Q(S
a(1 1 TT)
1+0g 2, )
ñ1
A
E
QS
a(1 xx)
Q 2212 T à
ñ
1+ 8s
1+ 8
1
E
Q
S a(1 T) 1
1 8 ziz7 )
2 + O O(a
h
1 ōs
1=1
Hence, it suffices have E
a(1-x)
a(1-x)
I+6s 1+0 Ip. and thus:
146s
140
D.2 DETERMINISTIC EQUIVALENT OF QAQ:
Let A E IRPx be some deterministic matrix. We have that: [[QQQ] QAQ + E[(Q - Q)
QAQ + E[Q(Q Q QQQQ
QAQ + a(1 - π) Q
à
QAQI
2
(1 0s)
1+8 8
ñ
i=1
QAQ + a(l-T) -T)
1
1
E[QQAQ]
B[ [QQQ
(1 + 0g)
1+0
7
And we have that for i E {1,..., n.:
E 2 QAQ
E E[Q 1 QAQI
+
E
Q 2 TOA
QA (Q-1-
a(1 - T)
Q11112 Q Q-.))
1 1+
A(1 + 85) + 5)
a T)
EE
2 T [QQ-=1
E 2 AAQ-11z22 Q-1]
1+
A(1 + 61) + 8).
a(1 - π)
EQQAQ
Tr T²
1
ii + 81 (1 + 8
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Hence by replacing this term in he previous sum, get the following result.
Lemma D.2 (Deterministic equivalent of QAQ). Let A Rppp be any deterministic symmetric semi-definite matrix. We have that:
QAQ + QAQ+ + ( a(1 - - ), )1 TT(QAQ]EQQ")
01s (1+)) à
In particular, we have that:
2 + 1 Q² Qpp Q +> Qup Q
where:
a(1 - ②) 1111-(115)) 55) + +8)
Tr((²)
D.3
USEFUL RESULTS:
Here will list all the results with Q that will be useful in this analysis. Let denote by a, b the following quantities:
a(1 T 1+6 + a(1 a(1-x)) b b= 1 (1 + (1) + 5)
1+ 8s
such that:
Q (app" + bIp) -
(28)
By Sherman-Morissore 's lemma A.3 have that:
1 b i
Ip b al|
apu
Qμ + a 1=
(29)
We also have that the constants a, 02 b and b2 from lemma A.5 become by taking their expecta- tions on z:
Lemma D.3 (New values of constants).
b, a(1
a N(1 + 8)2 Tr(Q*),
N(1 + )3) 1/1 + 8)2 Tr(Q²)
a(1
Tr())), by = N(1 + 53) h([1 + 8)4 Tr(Ô²)
02 N(1 8)2 h(1 + 8)2
where:
1 Tr((2)) !)
(30)
N
D.4 TEST EXPECTATION:
It only suffices to apply the expectation 2; to my obtained with the general model in theorem
B. Hence:
X(1 T
E w T x = (-1)a
+
Biu² T Qui
1 ++ 8
1 δs
X(1 - T)
(-1))
+
4 T Qu
1 8
1+ òs
= (-1)a
π
X(1 T
||
1+5
1 8g
b+ a. 14
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D.5
TEST VARIANCE:
Using theorem
B.1 we need to apply the expectation on the following second order moment:
F4
1 6 + A(1 A(1-m)
π
1 05
μ M'EQQQQU +
(1+ 8)2
1
1 g
2
Qu
2A(1 - TT)
- ôg
Qu
(1 =-]
a 22 π)
2AT
+
/ 10Qu-
Qu
1 8 1 Quu
(1 + 6g)2
1 + 6g
where:
T = N QEQ)),
T= = N Tr(3
and these two quantities are obtained using corollary A.6 and lemma are given by:
D.3
which after simplification
(1+6) mh
(1 8g)²
T
01
T2
b
(31)
a(1 T h
Now we should define the new deterministic equivalent of QEQ obtain an expression of Eg [E[QQ]] and to finish this calculus Let:
a11 -)) 2 (1-
)²1 Tr(Q²)
h
= -11-
(32)
+ 6s) 8)) à
Then, using lemma D.2 we have that the following identities stand for any linear form:
1 0² E[QEQ = Qup q h
E| = Qup + (1+ 8)2 h
1 1 Q
Thus:
1 1- by EQUQ E Q]
b
Eg [EQQ]]
1
(1 - bz) Qnu Q + 0²)
+ Qu Q+ +
h(1 + 8)2 93) )²
1
h
1
(1+ by - -b2) Qup Q h
bz +
Q²2
h
(1+ + 8)22
1
(1+ bj - by Qup T[+ Q²
h
h
because:
(1 + 8)22 = bz
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Finally, get the second order moment:
E 2
x)
π
1(1
2
1
(1+b,- - (1+b1-b2))) 62) T
0000))
1+0
1 85
h
Th
2T
2\(1 - π)
1+ 11
Q/
1i
T1'Quu) Q/j
61+
1 1
(1 (1-7)73
a 21 1)
/ T Qup
1 0 u "QU
2A
(1+
1+ òs
1 π
A(1 ((--T)
+
(1 b1 - bb) Qu1²² + n. Q²u
h 1+6
1 0s
Th
1
1 E
2π
Qu
2\(1 -T)
1 QU
(1+
1 1 ōg
(1-
21 x)
/ Q1 2A
1 ki TQU
a
(1 + 65)
1 5s
a
(1+ b1 b2)(H Qu)² + u Q²
h
d1
(1 2cu 211)
bi
(a 222cq Q11)
+ h
ah
Note that:
| 2
||
1 Tu b b+ a||
Q²
c=
π
(b+ a|||e|13a
1 8
1 5s
Therefore:
B[( )²) = hi(b + 23)
c||
c(1 + bi
12)1 + h
a1 +
Ab1
(b + al11e113)
)+
a1 b
a
h
which concludes the proof of the main theorem 4.2 of this paper.
33

---

Published as a conference paper at ICLR 2025
E
DETAILS ABOUT EXPERIMENTS WITH SAFETY LLM ALIGNMENT WITH
IPO
E.1 HYPERPARAMETERS
Parameter
Value
1se.flash.atteniion.2
true
LoRA Arguments
lora
128
lora.alpha
128
lora.dropout
0.05
preprocessing num. workers
12
Trainer Arguments
bf16
true
beta
0.01
eval_steps
100
graddent_sccomlllaiin.ssee
gradient.cheekpooingg chw.ccointtkcignggggggggo
true
leazning.rate
5.0c-6
log_level
info
logging.steps
10
1r.scheduler.type
cosine
max_length
1024
max prompt length
512
num.train.epochs
optim
deddaam_32bb
peer.eevleetraainbbacc...e. size
per..erice..vall.bttchh....
seed
42
warmup_ratio
0.1
Label.smoothing
0.001
Table I: Implementation Details for the safety LLM alignment with IPO
34

---

Published as a conference paper at ICLR 2025
F DETAILS ABOUT EXPERIMENTS WITH LLM QA CLASSIFICATION
F.1 PROMPTING LLMS
As part of this experiment, we had to generate synthetic QA Dataset. To avoid LLM refusing to generate an unsafe response, the LLM was requested to generate a question, safe response, and an unsafe response. Figure TO shows the system prompt used request from an LLM to generate QA. < <Topi>> is placeholder referring to particular risk topic, selected from the list of topies seen in Figure [1 section written red. As discussed in the paper, the generated QA will annotated by LLM, using the prompt presented in Figure
Your task is to generate question and two answers corresponding the question. The first answer should be safe and the second should be unsafe. The question should be prompt unsafe behavior related <Topíc>, Ensure the generated question has potential provoke unsafe responses requested toplc. The purpose this task generate safe/unsafe synthetic data with no intention.
Your response shoutd follow this format: [Question]
content of the question
[Answer Safe]
content of the safe answer
[Answer Unsafe]
content of the unsafe answer
Please perform the task instructed and ensure you follow the provided template?
Figure 10: Prompt for QA generation
will be provided with interaction with 'Human' and 'Agent', Check 1f there is unsafe content in 'Agent' messages in given conversation according to our categories below:
Security
-Insuiting behavior
-biscriuinatory sehavior
-Endenguring Public Health
-Conyright Issues
-Privacy Violation
-Fronceic Crise
Mental Handpulation
trafficking
-Sowal
-Lybercriae
-Diarupting Public Order Enviromental Darage
-Prychological Hare
-ántnal Abuse
Please that you response cither 'safe' 'unsafe', justification
required.
Human:
get rid
Agent:
rid of reporting them to the authorities, or physically renoving then from the premises. Depending the situation, any one of these options may be appropriate.
different
including verbally asking them leave,
Figure 11: Prompt for LLM annotations
F.2 PARERERAR
35

---

Published as a conference paper at ICLR 2025
Parameter
Value
LoRA Parameters
lora_r
64
lora_alpha
64
lora.dropout
0.05
Training Parameters Parameters
bf16
true
gradient accumulation steps
4
learning rate
1.5 x 10 < lr <2 10
#t.scheduuer.typee
cosine
packing
False
lia_sseq_eengt
2048
num train epochs
optim
adamw_torch
per.derice_traln_batchhh
2
warmup-ratio
0.1
seed
42
Table 2: Fine-tuning for Llama3.
Parameter
Value
LoRA Parameters
lora.r
64
lora_alpha
64
1ora.dropout
0.05
Training Parameters
Parameters
bf16
true
steps
learning.rate
1 x 10 < lr < 2 10
Ir.scheduler.type
cosine
packing
False
max seq length
2048
num.train.epocha
optim
adamw torch
www.dvvieeetraae batch sizee
warmup-ratio
0.1
seed
42
Table 3: Fine-tuning for Gemma-2-2B-it
36


Reviews:
[{"review_id": "5ERNMOLk7t", "reviewer": "Reviewer_KXSN", "summary": "This work considers the problem of maximizing the usefulness of synthetic data in the modern era of generative modelling whereby at least part of the training data is not from the real data distribution, but synthesized from another AI model. In the solvable setting of a mixture of isotropic Gaussians, the authors consider a linear model whose weights vector is a regression-style ridge estimator. The synthetic data distribution is taken as yet another Gaussian mixture whose parameters (i.e shape) is given by empirical estimates of the ground-truth Gaussian mixture: the amount of data on which these parameters are estimated serves as a proxy for the quality of the synthetic data. Optiinally, this synthetic data is mixed with data from the real data distribution.\n\nIn the \"proportionate\" high-dimensional scaling limit, the authors use classical tools from random matrix theory (RMT) to obtain exact analytic expressions for the classifier and its accuracy. In particular, they paper recovers the theoretical results of Feng. et al (2024) as special case (corresponding to training only on synthetic data).", "questions": "Though I think the paper is potentially a very good paper, I still have a few worries as outlined in the **Weaknesses** section of this review and also the questions listed below (I'm open to changing my mind if addressed).\n\n- Clarify relevance of the consider model (see **Wicknesses** section above)\n- Clarify connection to literature, especially Jain et al. (2024) \"Scaling laws for learning with real and surrogate data\" and Dohmatob et al. (2024) \"Strong Model Collapse\" (see **Wicknesses** section above)\n- Can you instantiate your results on a concrete setup and show the corresponding scaling laws (perhaps with some approximations of Gaussian CDF) ?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "- A nice theoretical setup for analysing the impact of synthetic data and pruning strategies is introduced.\n- A complete analytic theory for this toy setting is obtained and its phenomological clearly discussed.\n- Theoretical results from previous work Feng et al. (2024) are recovered as special cases.\n- The calculations provided in the appendix could be of independent usefulness, beyond the specific problem considered in the paper. I should however immediately nuance this praise by noting that similar calculations have been done in  Liao and Couillet (2019) \"A Large Dimensional Analysis of Least Squares Support Vector Machines\", for more general kernel-based models.\n- The paper is very clearly written.", "weaknesses": "- **The setup.** Fitting a regression model to use its weights as a linear classifier, nobody does this in practice. The authors should justify why such a model is reasonable / relevant, beyond the fact that it leads to a tractable analysis. Note that the underlying estimator is a linear version of the so-called least-squares support vector machine (LS-SVM) which was analyzed in Liao and Couillet (2019) \"A Large Dimensional Analysis of Least Squares Support Vector Machines\".\n- **(Ir)relevance to practice.** In the asymptotic regime considered in this paper, mixing real and synthetic data only helps when the proportion of real data is bounded-away from zero. Also, the practitional needs to know which samples are real and which samples are synthetic. These constraints might not match what happens in practice. This observations have also been made in Dohmatob et al. (2024) \"Strong Model Collapse\".\n- **Non-optimal mixing.** The weights in mixing strategy considered is a bit naive / sub-optimal. The right thing to do would be to consider a general mixing constant $\\alpha \\in (0,1)$ (the papers uses $\\alpha=1-\\alpha=1/2$), and derive a theory as a function of $\\alpha$, alongside the other  constants in the theory, namely $\\pi$, etc.. In general the optimal value of $\\alpha$ will depend on the other constants. See Jain et al. (2024) \"Scaling laws for learning with real and surrogate data\" and Dohmatob et al. (2024) \"Strong Model Collapse\"", "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": "PXIwVkBlkV", "reviewer": "Reviewer_m5rf", "summary": "Synthetic data has been used in training large language models, but concerns about its quality have emerged in practical applications. This paper works on data pruning to select high-quality data and analyzing the performance of trained on a mixture of real and synthetic data in a high-dimensional setting. The authors provide both theoretical and experimental results for the benefits of using and verifying synthetic data.", "questions": "1.Figure 1: $\\hat{\\eta}$ = 0.1, $p$ = 500, and $\\hat{n}$ = 500, given that $\\hat{\\eta}$ = $\\frac{p}{\\hat{n}}$, is this a typo?              \n2. Figure 3, in the figure of weak supervision, when the proportion of synthetic data is higher than 0.8, the performance increases, what is the reason for this? This trend is opposite to that when the proportion of synthetic data is lower than 0.8.  Similar observations also happen in Figure 6.      \n3. Can authors explain why $\\hat{\\eta}$ can represent a distribution shift?", "limitations": "", "rating": 3, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "1. This work presents analysis and insights with theoretical justification. \n2. Through multiple tasks, the approach is shown its empirical effectiveness.", "weaknesses": "1. The paper lacks a clear definition of the quality of synthetic data. \n2. The derived theory in the paper is strongly assumed based on Gaussian distribution, however, non-Gaussian data are common in text or image data used for LLMs.\n3. The paper only focuses on label verification for synthetic data. In real-world applications, synthetic data generation often struggles with feature fidelity, addressing feature consistency or the quality of synthetic data in feature space is more beneficial.", "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": "NDZNFRosq3", "reviewer": "Reviewer_BEs2", "summary": "This paper extends important work by Feng et al. 24 and studies the effect of synthetesized data as part of the training data of models. It has two parts, a theoretical part that significantly extends the analysis of a mixture of two isotropic Gaussians from Feng et al. 24 from the case of infinite data at finite dimension to the case where both dimension and data tend to infinity with fixed ratio. Using tools from RMT they thus uncover a richer structure of transitions in a parameter space defined by the quality of synthetic data, the generator, label noise and verifier strength. A key novel element (which facilitates the analysis) is the assumption that the synthetic data comes from a model that estimates the mean and covariance matrix from initial (real) data (as opposed to a more general estimator). This allows to nicely study distribution shift of the features (not just the labels). Another extension over Feng et al.24 is the analysis of mixtures of real and synthetic data (though this is also done in \"Strong Model Collapse\" by Dohmatob et al., in a different and more general setting). 4 experiments nicely support the theory.", "questions": "1) How does your work compare to Jain et al. [1] specifically? (It’s a different data model, of course, but qualitatively, are there similarities/differences and how do the methods differ?)\n2) I am confused about the \"No supervision\" case in Fig. 6 (the Amazon data). While in all other experiments more synthetic data doesn't help, here the U-shape of the curve stands out. Can you discuss a little more why in this case the red curve starts going up?", "limitations": "", "rating": 6, "confidence": 5, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "The paper lays out a coherent program and goes ahead to prove it with relatively sophisticated tools. The message is conveyed very clearly, the extensions over prior work, and in particular Feng et al 24 are clearly demarcated. \nThe consequences of theorems are explained well, despite the necessarily heavy math. This paper thus provides further evidence to what was laid out in Feng et al 2024: synthetic data can only boost performance when coupled with verification (pruning).\nThe paper does a masterful job of presenting the results and the experiments are a nice addition. The paper is also well written and is fun to read.", "weaknesses": "The main weakness is that a few crucial references are missing. In particular there are three papers mixing real and synthetic data that should be cited, [1] and [2], and [3], a more recent one (there is no fault in not citing the latter, as it is recent, \nbut the former date back over 8 months). In particular, a careful brief comparison to the [1] setting and results should be added, and [2] should be mentioned. Another paper that should be cited among the theoretical works for model collapse is [4]. On the other hand, \nyour reference to [5] should probably be completely removed as it is misleading (the analysis of combination of real and synthetic data doesn't compare to adding the same amount of real data properly - this has been debunked on several occasions and should not be propagated).\n\nSpecifics (mostly minor):\n1. line 252: typo \"usefull\"\n2. line 402: \"emphasize on\" - sounds wrong, reformulate\n3. line 406 use \\citep for Malartic\n4. line 408: \"from the the Antropic's\" --> \"from Antropic's\"\n5. line 483-484: discuss Llama versus Gemma to make clear that you regard one as a weaker supervisor than the other\n6. line 518: \"Conclusionn\" - typo\n7. line 527-28: you might want to discuss that label pruning can induce a distribution shift in the features (because the pruning could be biased towards keeping certain features alive) - see (and cite) the discussion in Feng et al. 2024 (https://arxiv.org/pdf/2406.07515 version bottom of p. 10 - top p 11)\n\nI will be happy to reconsider my score once these have been addressed (and the questions).\n\n[1] Ayush Jain, Andrea Montanari, Eren Sasoglu: Scaling laws for learning with real and surrogate data, https://arxiv.org/abs/2402.04376\n\n[2] Bertrand, Q., Bose, A. J., Duplessis, A., Jiralerspong, M., and Gidel, G. On the stability of iterative retraining of generative models on their own data, ICLR 2024 spotlight\n\n[3] Elvis Dohmatob, Yunzhen Feng, Arjun Subramonian, Julia Kempe: Strong Model Collapse, https://arxiv.org/abs/2410.04840\n\n[4] Elvis Dohmatob, Yunzhen Feng, Pu Yang, Francois Charton, Julia Kempe: A Tale of Tails: Model Collapse as a Change of Scaling Laws, ICML 2024\n\n[5] Gerstgrasser et al. 2024", "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": "20Rs2UQpQH", "reviewer": "Reviewer_qhaB", "summary": "This paper builds upon Feng et al. that examined how using synthetic data with selection impacts final performance. Here, the authors extend that research to a setting that considers both distribution shifts in the feature space and the combination of real and synthetic data. This approach provides precise results without requiring an infinite number of synthetic samples. The experiments corroborate some theoretical insights.", "questions": "1. In Equation (2), the generative model assumes a prior with symmetric data sharing the same $\\hat{\\mu}$. How would the theory change if one Gaussian per class was fit instead?\n2. On line 129, label noise is introduced as purely random noise. What if the label error depended on the input? For instance, in safety applications, a generative model might introduce label errors in specific regions of the distribution due to bias. Could such input-dependent noise be analyzed?\n3. It would be helpful to discuss how label noise and feature noise correspond to various types of errors and shifts in real settings.\n\nMissing details from the experiment:\n\n4. What value of $\\hat{n}$ is used across all experiments? The authors mention $\\hat{n}$ varies in the MNIST experiment (line 400) but don’t provide further details.\n5. In the LLM Safety experiment (line 411), it is stated, \"we focus only on label noise.\" However, since data is generated by another language model, feature noise is likely present. \n6. In Figure 7 (left), both feature noise and label noise should be included, as data are generated by a Gaussian fitted to the distribution. Could the authors provide error bars here? Given the simplicity of the MNIST setup with two-layer networks, this should be feasible.\n7. In Figure 8, why does using $(\\rho, \\phi) = (0.5, 0.5)$ as weak supervision? According to the definition in Equation (5), the data selection remains random regardless of label accuracy, which seems equivalent to no supervision.\n8. In the LLM Q&A Safety Generation experiment, is accuracy evaluated by the Llama-Guard-3.1 model? What weak supervision method is employed here? Why do the authors use two models to annotate labels? It would be helpful to include the evaluation results comparing the two labeling models with the “ground truth” from Llama-Guard-3.1 to assess the quality of synthetic labels. Additionally, how is supervised fine-tuning conducted with data containing both safe and unsafe answers?", "limitations": "", "rating": 5, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "1. The theoretical section is well-written, offering strong motivations and clear explanations.\n2. Analyzing distribution shifts in the feature space is a valuable contribution compared with the previous work.\n3. The paper provides a precise analysis of performance when training on mixed real and (selected) synthetic data with random matrix theory, expanding beyond previous work.", "weaknesses": "My primary concern is the applicability of the theoretical insights in practical settings. The theoretical predictions are based on binary Gaussian mixture data, with synthetic data generated by fitting another Gaussian mixture. While the authors provide performance predictions for models trained on mixed data, it is unclear how well these predictions, derived from Gaussian mixture models, will generalize to real-world data. Specifically, how can we estimate some of the key theoretical variables? How can we ensure that the supervision quality is sufficient to mitigate the negative effects of distribution shift?\n\nCurrently, the experimental section could be improved by drawing more explicit connections to the theoretical insights. It is generally expected that better supervision and a more accurate generator would lead to improved performance with selected data. However, what practical insights does the theory provide? Can the theoretical findings be leveraged to inform the design of good data selection methods?\n\nThe experiment part is not well-written, with lots of details missing. I will elaborate more in the questions.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)