Self-Consuming Generative Models with Curated Data Provably Optimize Human Preferences

Authors: damien  ferbach
quentin  bertrand
avishek joey bose
gauthier  gidel
Venue: NeurIPS_cc_2024_Conference
Published: 2024-05-15 18:57:33

Abstract: The rapid progress in generative models has resulted in impressive leaps in generation quality, blurring the lines between synthetic and real data. Web-scale datasets are now prone to the inevitable contamination by synthetic data, directly impacting the training of future generated models.
    Already, some theoretical results on self-consuming generative models (a.k.a., iterative retraining) have emerged in the literature, showcasing that either model collapse or stability could be possible depending on the fraction of generated data used at each retraining step.
    However, in practice, synthetic data is often subject to human feedback and curated by users before being used and uploaded online. For instance, many interfaces of popular text-to-image generative models, such as Stable Diffusion or Midjourney, produce several variations of an image for a given query which can eventually be curated by the users.
    In this paper, we theoretically study the impact of data curation on iterated retraining of generative models and show that it can be seen as an implicit preference optimization mechanism. However, unlike standard preference optimization, the generative model does not have access to the reward function or negative samples needed for pairwise comparisons. Moreover, our study doesn't require access to the density function, only to samples. We prove that, if the data is curated according to a reward model, then the expected reward of the iterative retraining procedure is maximized. We further provide theoretical results on the stability of the retraining loop when using a positive fraction of real data at each step. Finally, we conduct illustrative experiments on both synthetic datasets and on CIFAR10 showing that such a procedure amplifies biases of the reward model.

Main paper: 
 introduction: 
 Today state-of-the-art generative models can produce multi-modal generations virtually indistinguishable from human-created content, like text (Achiam et al., 2023) , images (Stability AI, 2023) , audio (Borsos et al., 2023) , or videos (Villegas et al., 2022; Brooks et al., 2024) . The democratization of these powerful models by open-sourcing their weights (Stability AI, 2023; Jiang et al., 2023; Touvron et al., 2023) or allowing public inference (Ramesh et al., 2021; Midjourney, 2023; Achiam et al., 2023) paves the way to creating synthetic content at an unprecedented scale-the results inevitably populate the Web. In particular, existing datasets are already composed of synthetic data such as JourneyDB (Pan et al., 2023) and SAC (Pressman et al., 2022) . Moreover, Alemohammad et al. (2024, Fig. 2) showed LAION-5B (Schuhmann et al., 2022) , a large-scale Web-crawled dataset used to train future generative models, already contains synthetically generated images.There is strong evidence that the synthetic data on the web has been, to a large extent, curated by human users. For instance, the LAION-Aesthetics datasets contains synthetically generated (Pan et al., 2023) . Samples from other diffusion models can be found in Figures 12a and 12b .images that have been curated using a reward model learned from human feedback on the Simulacra Aesthetic Captions dataset (Pressman et al., 2022) . Additionally, the JourneyDB dataset, contains human-picked images from the Midjourney (Midjourney 2023 ) discord server, that have been upscaled, i.e., images that have been requested in a higher resolution (see Figure 1 ). More generally, the user interface of the most popular state-of-the-art text-to-image models (e.g., Midjourney and Stable Diffusion 2.1 Huggingface implementation) proposes four alternatives for a single prompt for the user to pick their favorite.While the consequences of iterative retraining of generative models on synthetically generated data have raised a lot of attention in the community (Alemohammad et al., 2024; Shumailov et al., 2023; Bertrand et al., 2024; Dohmatob et al., 2024a) , previous works do not consider that generated data could be curated. This subtle nuance may be of major importance. Indeed, in numerous applications, augmenting the datasets with curated synthetically generated data is found to enhance the performance of the downstream task (Azizi et al., 2023; Wang et al., 2023; Gillman et al., 2024) and even generative models themselves (Hemmat et al., 2023; Gulcehre et al., 2023) , hinting that quality might not be the biggest problem. On the other hand, recent works Wyllie et al. (2024) ; Chen et al. (2024b) showed that this might lead to new issues, such as bias amplification. This is why, in this work, we aim to theoretically understand how the process of curation of synthetic data is connected with the reward model underlying human preferences and what distribution is learned by generative models trained on such curated synthetic data.Main contributions. We summarize our core contributions as the following:• We first focus on the self-consuming loop on (only) curated synthetic samples: we show that the expected reward gets maximized in Lemma 2.2 and that its variance vanishes. We further provide a convergence result in Theorem 2.1.• We additionally study the iterative retraining loop when real data is re-injected at each step: we first improve previous results of stability using raw synthetic samples by Bertrand et al. (2024) and show convergence in Kullback-Leibler (KL) divergence to the optimal distribution Theorem 2.2. When using curated synthetic samples, we show that the KL divergence with the optimal distribution remains bounded Theorem 2.4, as well as an improvement on the expected reward Theorem 2.3, enlightening connections with Reinforcement Learning from Human Feedback (RLHF).• We finally illustrate our theoretical results on synthetic datasets (mixtures of Gaussians and two moons) as well as natural images on CIFAR10 in Section 4. We highlight how curation based on the confidence of a classifier can lead to the emergence of biases (Wyllie et al., 2024) . 
iterative retraining with curated synthetic data: 
 We now study the fully synthetic self-consuming loop with curated samples. Unlike previous works that do not take curation into account (Alemohammad et al., 2024; Shumailov et al., 2023) and focused on stability of the process (Bertrand et al., 2024) , we show that retraining with curated samples both maximizes an underlying reward whose variance collapses, and converges to maximum reward regions. In Section 2.1 we first specify explicitly the distribution induced by a discrete choice model and highlight connections with RLHF. We additionally show that the expected reward increases and that its variance vanishes Lemma 2.2. Finally, inspired by stability results of Bertrand et al. (2024) , in Section 2.2 we extend our study to settings when real data is injected. More precisely, we improve previous results of stability of Bertrand et al. (2024) without curation and provide novel theoretical bounds when the synthetic data is curated.Notation and conventions. Lowercase letters p denote densities while uppercase letters P indicate their associated probabilities. We denote p data ∈ P(R d ) the real data distribution and for t ∈ N, we denote p t ∈ P(R d ) the model distribution at step t of the iterative retraining loop. Analogously, θ t is the corresponding parameters of the learned parametric model p t . We write p 0 to indicate the initial model learned using maximum likelihood on p data , this includes many modern generative model families such as diffusion models (Ho et al., 2020; Song et al., 2021) and flow-matching methods (Lipman et al., 2022; Tong et al., 2023b) .Discrete K-choice model. We propose using a discrete choice model for the curation phenomenon illustrated in Figure 1 , where users pick their preferred image that will be upscaled in the next dataset. Modeling human preferences is usually done via the Luce choice rule model (Shepard, 1957; Luce et al., 1963; Dumoulin et al., 2023) , where the human is modeled as a rational Bayesian subject. The probability that x 1 is preferred to x 2 is formulated using an underlying reward function r(x) and Plackett-Luce model (equivalently Bradley-Terry model) (Bradley and Terry 1952) . Under Luce's choice axiom (Luce, 1959) , it is possible to generalize this formula to K ≥ 1 samples as in Equation 2. For given samples x 1 , . . . , x K drawn from p t , the random curated sample denoted x is chosen according to this Plackett-Luce model x ∼ PL(x 1 , . . . , x K ) as in Equation 2 (Bradley and Terry, 1952; Luce, 1959; Plackett, 1975) . In particular, the curation procedure can be summarized as follows1) Sample x 1 ∼ p t , . . . , x K ∼ p t , independently, 2) Pick x ∼ PL(x 1 , . . . , x K ) , i.e., P(x = x k |x 1 , . . . , x K ) = e r(x k ) K j=1 e r(xj ) , 1 ≤ k ≤ K.((2)Self-consuming loop. After generating and curating a synthetic dataset according to Equations 1 and 2, the next generation of generative models is trained either solely on the distribution of curated samples (λ → ∞), or on a mixture of reference samples (that either comes from real data p data or a reference generative model p 0 ) and synthetic curated samples (λ < ∞) depending on the studied settingp t+1 = arg max p∈P 1 1 + λ • E x∼p ref log p(x) + λ 1 + λ • E x1,...,x K ∼pt x∼PL(x1,...,x K ) log p(x) .(EQUATIONwhere P is the set of achievable distributions with our model. This work aims to study the retraining dynamics of the distribution defined in Equation 3. First, in Section 2.1 we study the simplified dynamics of Equation 3 in the regime λ → ∞, i.e., when solely retraining on curated synthetic data and show convergence of the process but variance collapse. In Section 2.2 we study the exact dynamics given in Equation 3 and the impact on the stability of retraining on a mix of real data synthetic curated data.In this section, we study the dynamics of the density learned through iterative discrete K-choice curation in the fully-synthetic setting (i.e., λ → ∞): Equation 3 boils down toEQUATION)As a warm-up, we first consider the limit of K → ∞ in Lemma 2.1 and draw explicit connections with RLHF. This simplification yields a closed-form formula form for the solution of Equation 4and provides intuitions for the dynamics of learning on curated samples.Lemma 2.1. Let p t+1 be defined as in Equation 4. If P = P(R d ) is the set of probability distributions on R d , and if we assume that E y∼pt e r(y) < ∞, then we have for all x ∈ R d , x) .p t+1 (x) K→∞ ----→ p t (x) e r(x) E x∼pt e r((5)Dependency on K and connection to RLHF.. The proof of Lemma 2.1 relies on the fact that we can obtain a closed-form formula for the density p t+1 induced from discrete K-choice curation on p t (Equation 4). This is done in Appendix A.4.1 where we show that its density can be written xi) . ( 6)p t+1 (x) = p t (x) • H K pt (x) , with H K pt (x) := E x1,...,x K-1 ∼pt K • e r(x) e r(x) + K-1 i=1 e r(The latter directly implies that for all K ≥ 1, H K pt (x) ∈ (0, K). In particular, small values of K act as a regularization which prevents the density from blowing up too much in high rewards areas. On the other hand, the higher the number of samples used for curation, the more it can affect the induced distribution. In the limit K → ∞, Lemma 2.1 shows an interesting connection between iterative retraining on curated data and reward maximization via RLHF. Given a supervised-finetuned model distribution π SFT and a regularization parameter β, the goal of RLHF is to find a policy that maximizes a reward r(x) fitted on human preferences : (Go et al., 2023; Rafailov et al., 2024) .π RLHF = arg max π E x∼π [r(x)] -βD KL π||π SFT , which has a closed form formula , π RLHF (x) ∝ π SFT (x)e r(x) /βTherefore, in the limit K → ∞, Equation 5 shows that performing iterative retraining with human curation for t iterations is equivalent to performing RLHF with hyperparameter β = 1 t from the initial distribution π SFT := p 0 . The corresponding regularization parameter β is inversely proportional to the number of retraining steps. This connection is surprising since performing maximum likelihood on a curated distribution (Equation 3) is a priori different than directly maximizing a reward with Kullback-Leibler (KL) regularization.To prove that curation both increases the expected reward and reduces the variance, we will need an additional assumption that the reward is bounded at initialization. We decompose this assumption into three sub-assumptions of increasing thrength: C. Assum. 2.1.B and P(r(x) = r * ) > 0.In particular, Assumption 2.1 A and B are satisfied if we suppose that the reward bounded, which is reasonable if we suppose it is continuous given that the set of images [0, 1] d is compact. Note that assuming (a), we can always choose r * such that (b) is satisfied by picking the smallest value that almost surely bounds the reward at initialization. On the other hand (b) imposes that r * is the smallest value that a.s. upper-bounds the reward. In a nutshell, r * should be thought as the smallest number that upper-bounds the random variable p 0 with probability 1. For example, a Uniform distribution on the interval [0, 10], r * = 10 whereas for unbounded distributions such as N (0, 1), r * does not exist. In other words, r * = inf{r ∈ R, P 0 (r(x) ≤ r * ) = 1}. This shows that r * is uniquely defined which is an important point as we will show convergence of p t towards the level set r(x) = r * in Lemma 2.2 and Theorem 2.1.Lemma 2.2 states the reward expectation increases proportionally to the reward variance.Lemma 2.2. Let p t+1 be the distribution induced from a discrete choice model on p t (Equation 4). Suppose Assumption 2.1 B holds, then the expected reward increases proportionally to its variance at each retraining iteration:E pt+1 e r(x) ≥ E pt e r(x) + K -1 KVar pt e r(x) e r * . (7)Especially the expected reward converges to the maximum reward and its variance vanishes:E pt e r(x) t→∞ ---→ e r * and Var pt e r(x) t→∞ ---→ 0 .Discussion. Lemma 2.2 shows that the reward augmentation is directly proportional to the reward variance at each retraining step. In other words, the more heterogeneous the reward is, the more its expectation increases at the next step. Lemma 2.2 further shows that the expected reward converges towards the reward maximizers. We can additionally deduce that the variance is doomed to vanish. This is detailed in Appendix A.4.3 which additionally states that the reward variance decreases fast enough to have finite sum. Finally, we note that Lemma 2.2 helps us understand the fixed points of this process: due to the variance term in Equation 7, a fixed point of the retraining loop must put mass on a single level set of the reward function. The reciprocal is obviously true as detailed in the appendix (Lemma A.3).We can finally show a stronger result of convergence for the Kullback-Leibler divergence. We will need to assume that at initialization, the probability density puts a positive mass on the level set r(x) = r * . This corresponds to Assumption 2.1 C. Without this assumption, the probability density support would consecutively vanish towards the maximizer of the reward preventing KL convergence. Under assumption 2.1 C, we can denote p * the probability density at initialization restricted to the domain that maximizes the reward and renormalized:p * (x) := p0(x)1 r(x)=r * P0(r(x)=r * ) .Theorem 2.1. Let for all t ≥ 0, p t+1 be the distribution induced from a discrete choice model on p t (Equation 4) where P = P(R d ) is the set of probability distributions on R d . If p 0 satisfies Assumption 2.1 C, then we can define p * (x) := p0(x)1 r(x)=r * P0(r(x)=r * ) and the self-consuming loop on curated samples p t converges to p * :D KL (p * ||p t ) t→∞ ---→ 0.Theorem 2.1 proved in Appendix A.4.5 shows that the process of retraining with curation Equation 2 eventually converges to the highest level set of the reward reached at initialization. In particular, in the limit of a large number of retraining steps, the probability of all smaller rewards vanishes. This can have strong implications when retraining the next generation of generative models on a curated Web-scaled dataset: the learned distribution will lose diversity and collapse to the highest reward samples.After showing convergence but variance collapse of the self-consuming loop on curated synthetic samples, we now study the impact on the stability of injecting real data at each step. This setting is motivated by the recent work of Bertrand et al. (2024) that showed stability of the iterative retraining loop with real and synthetic data around a local maximizer θ * of the training distribution likelihood. This setting is furthermore relevant since Web-scrolled datasets will presumably keep containing a mixture of real data and human-curated synthetic data. In Section 2.2.1 we first improve previous results on retraining on mixed datasets which underlines the beneficial impact of real data on stability and in Section 2.2.2, we prove both stability and reward augmentation in the setting of mixed real and curated synthetic data.To motivate the impact of real data on the stability of the retraining loop with curation, we focus first on its impact without curation and improve previous results in that setting in Theorem 2.2.Setting. In this section only, following the approach of Bertrand et al. (2024) , we will not assume infinite capacity for our distribution (i.e., P = P(R d ) and hence adopt a parametric approach P = P Θ := {p θ | θ ∈ Θ}. Given the current generative model distribution p θt , p θt+1 must at the next iteration maximize the combined log-likelihood of real and generated data with hyperparameter λ, i.e., Equation 3 becomes:p θt+1 = arg max p θ ∈PΘ 1 1 + λ • E p data [log p θ (x)] + λ 1 + λ • E p θ t [log p θ (x)] .We finally denote p θ * = arg maxp θ ∈PΘ E p data [log p θ (x)] a maximizer of the data distribution loglikelihood. We also make the following assumption taken from Bertrand et al. (2024) proved stability of the retraining loop provided λ is sufficiently small. However, their proof is restricted to λ < 1 2 , preventing the use of a fraction of synthetic data λ 1+λ bigger than one-third which they left as future work. In Theorem 2.2, we extend their proof to any fraction of synthetic data provided the best model distribution is sufficiently close to p data in Wasserstein distance (Villani et al., 2009) i.e., W 1 (p θ * , p data ) ≤ ε < α L . Additionally, we express the result in distribution, while they expressed it in parameter space.Theorem 2.2. Under Assumption 2.2, if Lε < α and λ < α 2Lε , then there exists a neighborhood of the optimal distribution parameters θ * such that for any initial parameters θ 0 in that neighborhood, p θt converges to p θ * exponentially fast:D KL (p θ * ||p θt ) = Õ λ(α + εL) α + λ(α -εL) 2t .Interestingly when curating the synthetic samples we cannot expect stability around the optimal distribution (θ * in Theorem 2.2) since it is no longer a fixed point of the retraining loop. We will instead show a closeness result in KL divergence combined with an increasing property of the expectation of the reward, which bears close connections to RLHF. We therefore now study the setting described in Equation 3 where the synthetic samples are curated using a discrete K-choice model and real data is reused at each step (λ < ∞). In other words, we suppose that the retraining step uses a mixture of a reference distribution and a curated distribution asEQUATION)In Theorem 2.3, we prove that when retraining on a mixture of curated samples and samples from the reference distribution, the reward increases with respect to the reference distribution: E pt e r(x) ≥ E p ref e r(x) + λ (1 + λ) 3 (K -1)Var p ref e r(x)Ke r * .Discussion. A first interesting case is taking the reference distribution p ref equal to p data . In that case, we recover the fact that p data is not a fixed point of the retraining loop as soon as different reward values have non-zero probabilities to happen (we recover the result from Lemma A.3). In fact, Theorem 2.3 shows that such a process initialized at p data will increase the reward expectation.The second interesting case is taking p ref = p 0 the generative model at initialization. In that case, retraining on a mixture of samples from the initial model and curated samples from the current model improves the reward expectation with respect to initialization.After showing that such a retraining loop improves the expected reward, we can conversely show that this process does not deviate too much from p ref .Theorem 2.4. Let λ > 0 and p ref ∈ P(R d ) with a density with respect to Lebesgue measure. Consider the process (p t ) defined in Equation 8,with p 0 = p ref . Suppose that λ < 1 K-1 , then, for all t ≥ 1 D KL (p t ||p ref ) ≤ -log (1 -λ(K -1)) .Applying Theorem 2.4 with p ref = p data shows that retraining on a mixture of real and curated synthetic samples does not deviate too much from the data distribution. On the other hand, when setting p ref to be any initial model distribution, we see that reusing samples from the initial model stabilizes the retraining loop around initialization.Connection with RLHF. Theorem 2.3 and Theorem 2.4 together emphasize that retraining on a mixture of reference and filtered synthetic data bears important connections with RLHF. Indeed, the RLHF objective is composed of both a reward maximization term and a KL regularization between the current and initial model. In turn, Theorem 2.3 states that the expected reward increases and Theorem 2.4 shows that the KL divergence with respect to initialization remains bounded. The upper bound on the KL divergence further indicates that setting K small, i.e., using fewer samples for comparison acts as a regularizer, as previously noticed.3 Aligning models with human preferences. With the urgent and critical safety concerns of public deployment, the need to align models with human preferences has gained significant importance. RLHF is a popular reinforcement learning technique to align an already pretrained and finetuned model on human preferences (Christiano et al., 2017; Stiennon et al., 2020; Lee et al., 2021; Ouyang et al., 2022; Shin et al., 2023) . It consists of two steps: first fitting a reward r(x) on human preferences using a dataset of pairwise human comparisons and then, maximizing the expected reward over the model distribution. A Kullback-Leibler regularization to the initial model is further used during the maximization step to avoid reward hacking (Skalse et al., 2022; Chen et al., 2024a) or catastrophic forgetting (Korbak et al., 2022) . Variants of RLHF have recently been proposed such as Direct Preference Optimization (DPO) which maximizes the reward directly without modeling it (Rafailov et al., 2024) , Identity Preference Optimization (IPO) (Azar et al., 2024) or Kahneman-Tversky Optimization (KTO) (Ethayarajh et al., 2024) . 
experiments: 
 This section aims to empirically illustrate our previous theoretical results on how curation impacts the self-consuming loop. In Algorithm 1, we recall and detail the different steps performed in our experiments.Synthetic datasets. We first focus on two synthetic datasets: a mixture of Gaussians and the two moons dataset. For both datasets, we study the two settings of solely retraining on curated synthetic samples (λ = ∞) and mixed datasets (λ = 1). In Figure 4 , we iteratively retrain a denoising diffusion probabilistic model (DDPM, Ho et al. 2020 ) on a mixture of 8 Gaussians. The reward r(x) used for the discrete choice model is the clipped negative Euclidean distance to one of the centers of the Gaussians x * , i.e., r(x) := -γ max{0, x -x * -r min } where we choose γ = 10, r min = 1. Clipping the distance is used to ensure that the process does not collapse to a single point. Indeed applying Theorem 2.1, we know that the density will converge to a renormalized Gaussian distribution restricted to the ball centered at x * of radius r min . In Figure 5 , we plot the retraining curves on the two moons dataset: to compute the reward, we use an MLP classifier with 2 hidden layers of width 512 which yields probabilities q 0 (x), q 1 (x) for each class. The reward is then defined as : r(x) := γq 0 (x), γ > 0. Both Figure 4 and Figure 5 illustrate that retraining on solely curated samples induces collapse to regions that maximize the reward: respectively one mode of the MoG or one single moon. On the other hand, the use of real data results at the same time both in stability and higher density in high reward regions. Further experimental details are provided in Appendix A.5.Setting. We train a normalizing flow using optimal transport conditional flow matching (Lipman et al., 2022; Shaul et al., 2023; Tong et al., 2023b) with the torchcfm library Tong et al. (2023a Tong et al. ( , 2024)) . The initial model has been pretrained on the 50000 train images of the CIFAR-10 dataset (Krizhevsky et al., 2009) . At each iteration, we generate 5 • 10 4 samples using the current model from which we keep 2.5 • 10 3 samples filtered by discrete K-choice comparisons. The reward r(x) is computed using the class probabilities q 0 (x), . . . , q 9 (x) from a pretrained VGG11 classifier (Simonyan and Zisserman, 2014) with 92.39% test accuracy. Due to the expensive compute cost of retraining a generative model for multiple iterations (c.f. Appendix A.5.4), we plot only one run on each figure. To ensure the reproducibility of our results, we plot the retraining curves for 3 independent runs in Figure 11 in the appendix, illustrating that they have small variance.Using probability of one class as reward. As a first experiment, we filter samples following the probability of the classifier on a predefined class. We arbitrarily chose the class 0 corresponding to planes. The reward is then defined as r(x) = γ • q 0 (x), γ > 0. We plot the evolution of the class proportions as well as the averaged reward across 10 retraining steps in Figure 2 with γ = 5.Figure 2 shows collapse to the single plane class as the reward increases monotonically, illustrating Lemma 2.2.Using the confidence of the classifier as a reward: the emergence of bias. As a second experiment, we use the confidence of the classifier as a reward, i.e., r(x) = γ • max 0≤i≤9 q i (x), γ > 0.As written, the reward is therefore uncorrelated from the class but, remains implicitly correlated to it by the fact that the classifier statistics are class dependent. In Figure 3 we plot the evolution of the class proportions as well as the average reward. As expected by our theoretical results in Section 2, the average reward increases monotonically. On the other hand, we clearly see that the class proportions become more and more heterogeneous throughout the retraining loop. While confirming our theoretical study this plot therefore additionally shows that retraining on filtered samples increases bias, in a setting where the reward is implicitly correlated to diversity. Taking a step back, this has strong societal and ethical implications as it may imply that in a filtered internet biases may emerge or strengthen as we explain in Section 6.Reusing real samples: stability and reward augmentation. Finally, we illustrate our results from Section 2.2.1 by mixing real and filtered synthetic samples with hyperparameter λ = 1 2 . Figure 3 shows that the process remains stable as the proportion of classes remains approximately uniform Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?Answer: [Yes] Justification: We provide details on the model architectures, datasets and experimental hyperparameters in Section 4. 
conclusion and open questions: 
 We study the impact of data curation on the training of generative models in the self-consuming loop. We provide theoretical results demonstrating that the expected reward underlying the curation process increases and its variance collapses (Lemma 2.2) as well as a convergence result (Theorem 2.1) for the generative model. We additionally provide stability guarantees when reusing real data at each step (Theorem 2.3 and Theorem 2.4) establishing close connections with RLHF and preference optimization. Our work sheds light and theoretically grounds a novel phenomenon: increasing the proportion of curated synthetic data on the Web automatically optimizes preferences for future trained large models. A limitation is that we do not propose an algorithm to address emerging issues like bias amplification as we feel it goes beyond the scope of our paper and is a substantially complex field already intensively explored (Grover et al., 2019; Wyllie et al., 2024; Chen et al., 2024b) . We believe, however, that it should be a research priority and constitutes an interesting avenue for future work. Another interesting direction is to study the impact of the particular reward function underlying filtering (confidence, quality, diversity...) on the emerging bias amplification. 
broader impacts: 
 Training and aligning large generative models are prone to substantial ethical concerns regarding their alignment objective (Shen et al., 2023) , representational disparities of the training datasets (Clemmensen and Kjaersgaard, 2022) , or the presence of harmful images in the datasets (Birhane et al., 2021; Schramowski et al., 2023; Birhane et al., 2024) . Our work mostly focuses on the impact of the curation of synthetic datasets which itself heavily depends on the agent performing the curation and its underlying reward function. In particular the documentation of the Simulacra Aesthetic Captions dataset (Pressman et al., 2022) alerts that the human-based curation step is performed by a group of individuals that lacks diversity, mostly from Western, Educated, Industrialized, Rich, and Democratic (WEIRD) individuals (Henrich et al., 2010) . A similar bias is likely occurring in the JourneyDB (Pan et al., 2023) dataset and, more generally, in the synthetic data appearing on the web. However, our work mostly revolves around a theoretical analysis and raises awareness of the implicit alignment and potential bias amplification of self-consuming generative models. We therefore firmly believe that the potential benefits of this awareness outweigh the potential unforeseen negative consequences of this work.A Appendix / supplemental material A.1 Extended related work On retraining with data curation to align on preferences. In Gupta and Zou (2019) , the authors tackle the problem of generating synthetic DNA sequences using Generative Adversarial Networks (GAN, Goodfellow et al. (2014) ). They introduce an external function analyzer to rate synthetic samples from the generator and add the highest-scored ones into the discriminator training set. In Yao et al. (2022) , the authors propose a new GAN framework to incorporate users preferences in the training. They show state of the art results in generating the user-desired data distribution and theoretically prove the convergence of their method. The key difference to our work is that they aim to generate a diversity of samples that are desired by users and their focus is therefore not on the collapse of their method to a maximal reward set.On the impact of alignment on diversity. In Kirk et al. (2023) , the authors investigate how the different stages of alignment affect a models generalization capabilities and output diversity. They empirically show that the output diversity of the RLHF policy is decreased w.r.t. the supervised finetuned policy, which is consistent with our theoretical insights (e.g. lem. 2.2, thm 2.1). However there are major differences with our setting as their contribution is empirical and we investigate the impact of iteratively retraining a model several times on synthetic samples while they study a single training round using RLHF. In Perez et al. (2022) , the authors investigate how a language model can be used to generate prompts that lead to a harmful behavior of another language model (red teaming). This is useful to prevent such behavior before public deployment of a large language model. They find that LM-red teaming is a powerful tool that successfully unveils harmful behaviors of language models and help mitigate them. However, such a method can suffer from a diversity problem in the generated prompts if the red-teaming language model is itself biased. In Gao et al. (2023) , the authors study how over-optimizing an imperfect proxy of a reward model affects the average reward of the ground-truth reward. They uncover for both reinforcement learning based and best-of-n methods, functional forms of the underlying reward model scores as a function of the KL between the optimized policy and the initial policy that they confirm empirically. In Bai et al. (2022) , the authors propose to use iterative online learning for RLHF, where the preferences are updated on a weekly basis using human feedback. In particular at each iteration step they use their best RLHF policy to produce comparisons submitted to crowd-workers. The comparisons are then mixed with existing data and the process continues. The intuition for such methods is that diversity could be improved compared to RLHF since the dataset is generated using different states of the reward models. They additionally empirically show benefits of this practice on metrics such as Elo scores as evaluated by crowd-workers.On model collapse. Briesch et al. (2023) investigates experimentally the self-consuming loop specifically in the case of LLMs and evidences model collapse in that setting. Feng et al. (2024) show that pruning unwanted samples or selecting the best ones from multiple synthetic samples can prevent model collapse in the self-consuming loop.Iterative finetuning of language models and rejection-sampling. While there is already a large literature on RLHF to iteratively finetune LLMs, rejection-sampling is one way to optionally see finetuning as a sampling problem amenable to probabilistic inference. Indeed, recent works Zhao et al. (2024) ; Kong et al. (2024) frame iterative finetuning as drawing samplesusing rejection sampling, Twisted Sequentional Monte Carlo etcfrom the unnormalized posterior distribution p(x) ∝ e r(x) p 0 (x),where p 0 (x) is an initial generative model trained on real data. From this perspective, our framework studies the case where we curate data by using human reward and obtain x ∼ p(x) which are samples from the posterior, without access to the density. This allows us to then finetune p 0 (x) to approximate the posterior p(x), which in our notation is a step of iteratively finetuning on curated data.Accounting for the accumulation of data. In Gerstgrasser et al. (2024) , the authors show that model collapse is evitable, in the case where a model is retrained on the accumulation of all its previous iteraions (and not only data generated from the last iteration). We believe that their setting could be adapted in order to show that accumulating data provides additional stability to the retraining loop and avoids collapse of the reward variance. We leave this exciting research direction as future work. 
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 Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments? Answer: [No] Justification: Due to the expensive compute cost of retraining a generative model for several iterations we chose to not include error bars. However, we reported in Figure 11 three independent runs in one setting of our experiments and noticed small variance in the different runs.Guidelines:• The answer NA means that the paper does not include experiments.• The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.• The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions). • The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.) • The assumptions made should be given (e.g., Normally distributed errors).• It should be clear whether the error bar is the standard deviation or the standard error of the mean. • It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a 96% CI, if the hypothesis of Normality of errors is not verified. • For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates). • If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text. 
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 Appendix: Frameworks going beyond a single reward model are especially relevant in practical scenarios in LLM alignment. An interesting reference on this topic is the recent work Munos et al. (2023) which addresses such extension by learning a preference model of samples given a prompt P (x x|y) (as a function of the two variables x, x)instead of the Plackett-Luce reward model r(x) (less general when preferences are non-transitive) which they refer to as Nash Learning from Human Feedback.We now outline how to derive an extension to a mixture of reward to our setting: First, we can introduce a new latent variable z that describes the randomness in the reward used, which leads to the following expression of the curated distribution after one step of curation: (xi;u) In our setting, we were able to prove that the expected reward increases and the distribution converges to the maximum level set of a unique reward (Lem 2.2, Thm 2.1). However, in the presence of multiple rewards, it is not straightforward that the rewards have the same maximal level sets. Therefore this may yield interesting dynamics and the convergence of p t may differ. We believe such an extension of our results is outside of the scope of this work and think that it is a fascinating avenue for future work. For example, it may be interesting to study if a reward component in the mixture dominates, thereby dictating the convergence, e.g. if it gets large differences between two samples. In that case, the distribution may converge to only one maximal level set introducing a new model collapse behavior as the mixture of rewards would be dictated by a single reward.p t+1 (x) = p t (x) • H K pt (x) with H K pt (x) := E x1,...,x K-1 ∼pt,u K • e r(x;u) e r(x;u) + K-1 i=1 e rA.3 Comparison of our results against other results from the litterature A.3.1 On retraining from scratch vs iterative fine-tuning 1. Experiments. All retraining step in experiments on mixture of Gaussians and two moons are performed from scratch, whereas in the case of CIFAR dataset, due to the high compute cost of retraining the model from scratch (20 hours on an A100 GPU) we performed fine-tuning at each step. Fine-tuning is always performed on 10 6 images which corresponds to 20 epochs on the original dataset using batch size 128 (i.e. 7.8 * 10 3 gradient steps). We always use the same amount of images for fair comparison between different proportions of real data injected. In contrast, in Alemohammad et al. (2024) , the collapse is shown when the model is retrained from scratch at each iteration. In Shumailov et al. (2023) , the experiments are performed using retraining from scratch for Variational AutoEncoders and Gaussian Mixture Models and sequential fine-tuning for Large Language Models. In Bertrand et al. (2024) , toy experiments on two moons and mixture of Gaussians are performed by retraining from scratch while experiments on CIFAR10, FFHQ are performed using iterative fine-tuning. It is worth noting that stability using real data in the setting of iterative fine-tuning is easier to obtain than when retraining from scratch, since the model parameters are initialized around a good potential set of parameters. However, we point out that model collapse occurs also in the setting of iterative fine-tuning as shown in Figure 2 of Bertrand et al. (2024) (red curves).2. Theory. Finally regarding our theoretical results, only theorem 2.2 happens in the setting of iterative fine-tuning (since it uses the same setting as in Bertrand et al. (2024) ). However, all our other results and in particular theorem 2.1, 2.3, 2.4 do not explicitly assume a special learning algorithm in parameter space. Instead, we consider having a perfect learning model and consider the evolution of the expected reward for such a learning model. In that sense, it applies to learning from scratch with the additional assumption that the model attained perfectly fits the curated distribution.Alemohammad et al. ( 2024) studies the self-consuming loop in three different settings where the model is retrained a) only on synthetic data b) on a mixture of synthetic data and a fixed set of real data samples b) on synthetic data and a fresh set of real data samples at each step. In setting (a), the retraining loop collapses. In (b) it collapses too but with some delay related to the amount of fixed real data. In (c) the retraining loop does not degrade performances provided there is enough fresh real data at each step. Bertrand et al. (2024) proved stability in the setting (b) under some theoretical assumptions and in the iterative finetuning framework. Comparatively, our experiments on mixtures of Gaussians are performed using fresh real data at each step while the CIFAR experiments are performed in the fixed real data framework.A.4.1 Proof of Lemma 2.1 Lemma 2.1. Let p t+1 be defined as in Equation 4. If P = P(R d ) is the set of probability distributions on R d , and if we assume that E y∼pt e r(y) < ∞, then we have for all x ∈ R d , x) .p t+1 (x) K→∞ ----→ p t (x) e r(x) E x∼pt e r((5)Proof. First, by minimization of the cross-entropy, we know that for any distribution q, arg max p E x∼q [log(p(x))] = q. Therefore, if p t+1 is the solution of Equation 4, then we have directly that p t+1 has the law of x, where x is defined in Equations 1 and 2. We can now specify explicitly the distribution p t+1 . Let p t be the current distribution at time t. We first samplex 1 , • • • , x K i.i.d.∼ p t . and then independently sample an index i K following the Plackett-Luce model:EQUATION)By noting that the events {i K = i} K i=1 are disjoint, we can write the resulting density:p t+1 (x) = K i=1 yj ,j̸ =i p t (y 1 , • • • , y i-1 , x, y i+1 , • • • , y K )P(i K = i|x, y j , j = i) j̸ =i dy j .By independence since the K samples are drawn i.i.d. and since the Plackett-Luce formula is symmetric, all K terms in the sum are equal. This leads to rewriting:p t+1 (x) = K y1,••• ,y K-1 p t (y 1 , • • • , y K-1 , x)P(i K = K|y 1 , • • • , y K-1 , x)dy 1 • • • dy K-1 = p t (x)K y1,••• ,y K-1 e r(x) e r(x) + K-1 i=1 e r(yi) p t (y 1 ) • • • p t (y K-1 )dy 1 • • • dy K-1 = p t (x) • H K pt (x) where H K pt (x) = y1,••• ,y K-1 e r(x) e r(x) K + K-1 i=1 e r(y i ) K p t (y 1 ) • • • p t (y K-1 )dy 1 • • • dy K-1 (10)We now can study the limit K → ∞. Consider the random variable X = e r(x) as x ∼ p t . By assumption, E[X] < ∞. We can therefore apply the law of large numbers. Namely, ifEQUATIONFurthermore, for all x, y 1 , . . . , y K-1 , we have 0 ≤ e r(x) e r(x) + ∑ K-1 i=1 e r(y i ) ≤ 1 and e r(x) K K→∞ ----→ 0.Rewriting Equation 10:H K pt (x) = y1,••• ,y K-1 e r(x) e r(x) K + K-1 K ∑ K-1 i=1 e r(y i ) K-1 p t (y 1 ) • • • p t (y K-1 )dy 1 • • • dy K-1we get that: y) which directly impliesH K pt (x) K→∞ ----→ e r(x) E y∼pt e r(p t+1 (x) K→∞ ----→ p t (x)e r(x) E y∼pt e r(y)Without assuming that the reward is bounded, we can show using Jensen inequality that the reward expectation increases at each retraining step.Lemma A.1. When performing K-wise filtering, the expected reward increases, i.e., ∀t ≥ 0: x) .E pt+1 e r(x) ≥ E pt e r((12)Proof. Consider the random variableY = K-1 K ∑ K-1 i=1 e r(y i ) K-1 when y 1 , • • • , y K-1 i.i.d. ∼ p t .For a, b > 0, the function x → a b+x is convex on R * + . Hence by Jensen inequality, for any x:H K pt (x) = E Y e r(x) e r(x) K + Y ≥ e r(x) e r(x) K + E[Y ] = e r(x)e r(x) K + K-1 K E pt e r(x) Finally, we can write:E pt+1 e r(x) = e r(x) p t (x)H K pt (x)dx ≥ p t (x) e 2r(x) e r(x) K + K-1 K E y∼pt e r(y) dx ≥ E x∼pt e r(x) 2 Ex∼p t [e r(x) ] K + K-1 K E y∼pt e r(y) = E x∼pt e r(x)where we have used again Jensen inequality on the convex function x 2 x K +c on R * + wherec := K -1 K E y∼pt e r(y) > 0 A.4.3 Proof of Lemma 2.2Lemma 2.2. Let p t+1 be the distribution induced from a discrete choice model on p t (Equation 4). Suppose Assumption 2.1 B holds, then the expected reward increases proportionally to its variance at each retraining iteration:EQUATION)Especially the expected reward converges to the maximum reward and its variance vanishes:E pt e r(x) t→∞ ---→ e r * and Var pt e r(x) t→∞ ---→ 0 .Proof. By symmetry, we can write:KE pt+1 e r(x) = x1,••• ,x K K e 2r(x1) + • • • + e 2r(x K ) e r(x1) + • • • + e r(x K ) K k=1 p t (x k )dx k = x1,...,x K K j=1 e r(xj ) e r(x1) + • • • + e r(x K ) e r(x1) + • • • + e r(x K ) + e r(xj ) (K -1)e r(xj ) -i̸ =j e r(xi) e r(x1) + • • • + e r(x K ) K k=1 p t (x k )dx k = KE pt e r(x) + x1,...,x K i<j e r(xi) -e r(xj ) 2 e r(x1) + • • • + e r(x K ) K k=1 p t (x k )dx k ≤ KE pt e r(x) + i<j 2Var pt e r(x) Ke r * ≤ KE pt e r(x) + K(K -1) 2 2Var pt e r(x) Ke r * ≤ KE pt e r(x) + (K -1)Var pt e r(x) e r *This brings finally,E pt+1 e r(x) ≥ E pt+1 e r(x) + K -1 KVar pt e r(x) e r *We now prove that the expected reward converges and we will first show the following lemma:Lemma A.2. ∀ε ≥ 0, ∀t ≥ 0,P t+1 (r(x) ≥ r * -ε) ≥ P t (r(x) ≥ r * -ε) (13) Proof. Consider (x 1 , . . . , x K ) i.i.d.∼ p t and denote B ε := {x, r(x) ≥ r * -ε}. Then,P t (r(x) ≥ r * -ε) = 1 K E x1,...,x K K i=1 1 xi∈Bε .On the other hand,P t+1 (r(x) ≥ r * -ε) = E x1,...,x K K i=1 1 xi∈Bε e r(xi) K k=1 e r(x k )Proving Lemma A.2 is then equivalent, by permutation symmetries to showing that ∀k ≤ K, if r(x 1 ), . . . , r(x k ) ≥ r *ε and r(x k+1 ), . . . , r(xK ) < r * -ε, then k K ≤ k i=1 e r(x i ) ∑ K k=1 e r(x k ) .We can then write:k i=1 e r(xi) K k=1 e r(x k ) = k i=1 e r(xi) K k=1 e r(x k ) = kµ 1 kµ 1 + (K -k)µ 2 ≥ k K Where µ 2 := ∑ K i=k+1 e r(x i ) K-k ≤ ∑ k i=1 e r(x i ) k =: µ 1 Let ε > 0.By assumption on r * (Assumption 2.1 B), we know that there exists δ > 0 such that P 0 (r(x) ≥ r * -ε) ≥ δ and hence using Lemma A.2, ∀t ≥ 0, P t (r(x) ≥ r * -ε) ≥ δ. Therefore, while E pt e r(x) ≤ e r * -2ε, we know thatVar pt e r(x) ≥ ε 2 P t (r(x) ≥ r * -ε) ≥ ε 2 δ .Therefore, while E pt e r(x) ≤ e r * -2ε, we have using Lemma 2.2 thatE pt+1 e r(x) ≥ E pt+1 e r(x) + K -1 K ε 2 δ e r * . Since K-1 K ε 2 δe r * > 0, this can happen for only a finite number of steps and hence we know that there exists a time T ε ≥ 0 such that (remind that the expectation of the reward is increasing by Lemma 2.2):∀t ≥ T ε , E pt e r(x) > e r * -2ε .Since, the expected reward is obviously recursively bounded by e r * at any iteration t, we just have proved that it converges.We now prove that the variance has finite sum. Indeed, just notice that using Lemma 2.2 that ∀T ≥ 0:T t=0Var pt e r(x) ≤ e r * K K -1 E p T +1 e r(x) -E p0 e r(x)≤ K K -1 e 2r * .This proves thatVar pt e r(x) < ∞. Especially since the reward variance has finite sum and is positive, it converges to 0.Lemma A.3. A probability density p is a fixed point of Equation 10if and only if it puts all its mass on a single level set of the reward function. In other words, there exists r * ∈ R such that P(r(x) = r * ) = 1.Proof. Given the density p, denote P the corresponding probability function and F K (p) the curated distribution using Equations 1 and 2. When the reward r is p-a.s. bounded, this is a direct consequence of Lemma 2.2. When this is not the case, we know the existence of two disjoint interval I, J ⊂ R such that P(r(x) ∈ I) > 0 and P(r(x) ∈ J) > 0. From the proof of Lemma 2.2, we have seen that, taking p t := p: x) using that I, J have strictly positive mass and disjoint rewards. Therefore, p cannot be a fixed point. KE F K (p) e r(x) = KE p e r(x) + x1,...,x K i<j e r(xi) -e r(xj ) 2 e r(x1) + • • • + e r(x K ) K k=1 p(x k )dx k > KE p e r(D KL (p * ||p t ) = -log(P t (r(x) = r * )).We therefore only have to show that P t (r(x) = r * ) t→∞ ---→ 1.We will first show the following lemma:Lemma A.4. ∀ε ≥ 0, ∀t ≥ 0,P t+1 (r(x) = r * ) -P t (r(x) = r * ) ≥ P 0 (r(x) = r * ) * (P t+1 (r(x) ≥ r * -ε) -P t (r(x) ≥ r * -ε))Proof. We will actually show:P t+1 (r(x) = r * ) -P t (r(x) = r * ) ≥ P t (r(x) = r * ) * (P t+1 (r(x) ≥ r * -ε) -P t (r(x) ≥ r * -ε)) (14) from what we directly deduce Lemma A.4 by using Lemma A.2.To prove Equation 14, just notice that for any x, y, if r(x) ≥ r(y) then H K pt (x) ≥ H K pt (y) by increasing monotonicity of z → z z+c on R * + for a positive constant c > 0. Therefore we know the existence of a constant C such that ∀x, y, if r(x) = r * and r(y) ≤ r * , then H K pt (x) ≥ C ≥ H K pt (y). For example, take C = inf x s.t. r(x)=r * H K pt (x). Then we can write:P t+1 (r(x) = r * ) -P t (r(x) = r * ) = 1 r(x)=r * (p t+1 (x) -p t (x))dx = 1 r(x)=r * p t (x)(H K pt (x) -1)dx ≥ 1 r(x)=r * p t (x)(C -1)dx = P t (r(x) = r * )(C -1)and:P t+1 (r(x) ≥ r * -ε) -P t (r(x) ≥ r * -ε) = 1 r(x)≥r * -ε (p t+1 (x) -p t (x))dx = 1 r(x)≥r * -ε (H K pt (x) -1)dx ≤ 1 r(x)≥r * -ε p t (x)(C -1)dx = P t (r(x) ≥ r * -ε)(C -1) ≤ (C -1)where in the last step we have used C -1 ≥ 0 because P t+1 (r(x) ≥ r * -ε)-P t (r(x) ≥ r * -ε) ≥ 0 by Lemma A.2 and P t (r(x) ≥ r * -ε) ≤ 1.Combining the last two equations we get:P t+1 (r(x) = r * ) -P t (r(x) = r * ) ≥ P t (r(x) = r * ) * (P t+1 (r(x) ≥ r * -ε) -P t (r(x) ≥ r * -ε))We can now prove P t (r(x) = r * ) t→∞ ---→ 1. Let δ > 0, suppose that at time t,P t (r(x) = r * ) ≤ 1 -δ . Denote for ε > 0, U ε = {x ∈ R d |r * > r(x) ≥ r * -ε}. We know that ε>0 U ε = ∅. Therefore, ∃ε t > 0 such that P t (U ε t ) ≤ δ 4 .Furthermore, for any t ′ ≥ t, we know thatEQUATION) by convergence of the expectation (Lemma 2.2) and Markov property. We therefore know that ∃t ′ ≥ t such that P t ′ (r(x) ≤ r * -ε t ) ≥ 1 -δ 2 . By using the preceding Lemma A.4, we get: and P t (r(x) = r * ) hence increases by at least δ 4 . Therefore, the condition P t (r(x) = r * ) ≤ 1 -δ must become invalid at some point. Since we have shown this for any δ > 0, this shows that P t (r(x) = r * ) → 1. P t ′ (r(x) = r * ) -P t (r(x) = r * ) ≥ p 0 • (P t ′ (r(x) ≥ r * -ε t ) -P t (r(x) ≥ r * -ε t )) ≥ P 0 (r(x) = r * ) • ((1 - δ 2 ) -(1 -δ + δ 4 )) ≥ P 0 (r(x) = r * ) • δE pt e r(x) ≥ E p ref e r(x) + λ (1 + λ) 3 (K -1)Var p ref e r(x)Ke r * .Proof. We proceed by recursion. First, we know that ∀t ≥ 1, Var pt e r(x) ≥ 1 1+λ 2 Var p ref e r(x) . Furthermore it is straightforward using Lemma A.1 and a recursion that ∀t ≥ 0, E pt e r(x) ≥ E p ref e r(x) . This brings that∀t ≥ 1, E pt e r(x) ≥ 1 1+λ E p ref e r(x) + λ 1+λ E p ref e r(x) + λ (1+λ) 3 (K-1)Varp ref [e r(x) ] Ke r *which brings the result.We can actually show the following lower bound on the limit:Lemma A.5. Consider the process p t+1 (x) = 1 1+λ p ref (x)+ λ 1+λ p t (x)•H K pt (x) with p 0 = p ref . Then, lim inf t→∞ E pt e r(x) ≥ E p ref e r(x) + λ (1 + λ) 2 (K -1)Var p ref e r(x)Ke r * .Proof. Using the proof of Theorem 2.3 we can show the following more precise lower bound at each step: denote A := λ 1+λ and B = 1(1+λ) 2 (K-1)Varp ref [e r(x) ] Ke r *, then for all t ≥ 1:E pt e r(x) ≥ E p ref e r(x) + A t B + A t-1 B + • • • + AB .This directly bring that :lim inf t→∞ E pt e r(x) ≥ E p ref e r(x) + AB ∞ i=0 A = E p ref e r(x) + AB 1 -A = E p ref e r(x) + λ 1 + λ 1 (1 + λ) 2 (K -1)Var p ref e r(x) Ke r * 1 1 -λ 1+λ = E p ref e r(x) + λ (1 + λ) 2 (K -1)Var p ref e r(x) Ke r * A.4.7 Proof of Theorem 2.4Theorem 2.4. Let λ > 0 and p ref ∈ P(R d ) with a density with respect to Lebesgue measure. Consider the process (p t ) defined in Equation 8,with p 0 = p ref . Suppose that λ < 1 K-1 , then, all t ≥ 1 D KL (p t ||p ref ) ≤ -log (1 -λ(K -1)) .Proof. We know that ∀K ≥ 2, ∀x ∈ R , H K pt (x) ≤ K.We can then show by recursion that ∀t ≥ 1, ∀x, pt(x)p ref (x) ≤ 1 1-λ(K-1). Indeed, it is true at initialization and if true at time t, then at time t + 1:p t+1 (x) p ref (x) ≤ 1 1 + λ + λ 1 + λ 1 1 -λ(K -1) • K ≤ 1 1 -λ(K -1)We then just replace this bound in the expression of the D KL (p t ||p ref ):D KL (p t ||p ref ) = E pt log( p t (x) p ref (x) ≤ log 1 1 -λ(K -1).We study here the impact of retraining on a combination of all previous iterations and show that the process remains constant. This motivates and enlightens previous works that consider only retraining on the distribution at the last iteration. Let α 0 , α 1 , α 2 . . . a fixed non-negative sequence and consider a retraining process using maximum likelihood:θ t+1 = arg max θ t i=0 α i E p θ i log(p θ (x)). We will assume for this lemma that the solution of this optimization problem is unique. Otherwise the lemma remains valid but for a carefully chosen solution when there are multiple possibilities.Lemma A.6. Suppose we start with the first T iterations predefined, i.e., by fixing p 0 , • • • , p T -1 . Then starting t = T , the learned distribution is constant, i.e., ∀t ≥ T, p t = p T . Discussion. As an example, suppose that we take p 0 = p data and p 1 an initial generative model trained on p data . Then, Lemma A.6 states that starting t = 2, the learned distribution at each step will be constant equal to p 2 . In other words, we cannot expect the process to converge to a global maximizer of the data log-likelihood. More generally, Lemma A.6 shows that if the respective proportion of previous iterations remains constant throughout the retraining loop, the process remains constant and hence cannot converge towards the data distribution. These considerations have interesting links with previous work by Gerstgrasser et al. (2024) which experimentally showed that accumulating data with fixed relative ratios breaks the curse of recursion. However, note that the focus is different since they are in the finite sample setting while we study the infinite sample setting. Finally Lemma A.6 implies that to ensure convergence, we need to relatively decrease the proportion of previous iterations and comparatively increase the relative proportion of the data distribution or only use the distribution of the current iteration. This has been done in Bertrand et al. (2024) for parametrized generative models under some assumptions Proof. We prove the result by recursion starting t = T . By definition:θ T = arg max θ T -1 i=0 α i E p θ i log(p θ (x))Then suppose that for all j such that T ≤ j ≤ t, θ j = θ T . Then we can write:θ t+1 = arg max θ t i=0 α i log(p θ (x))But we know by cross-entropy minimization thatθ T = arg max θ E p θ T log(p θ (x)) = arg max θ t i=T E p θ i log(p θ (x)) .Furthermore, by definition,θ T = arg max θ T -1 i=0 α i E p θ i log(p θ (x)) .In particular it maximizes the sum of both previous terms and hence θ t+1 = θ T A.4.9 Additional lemma of convergence in parameters Lemma A.7. ∀λ ∈ R + , if λ < α 2Lε , then for θ 0 in a neighborhood of θ * , we have the following rate of convergence:EQUATION)Proof. We follow the same steps and notations as in Bertrand et al. (2024) . The main idea is to get another bound on the operator norm of the Jacobian at θ * : J G(θ * ) (their lemma E.1 (iii)). We begin with their intermediate result (lemma E.1 (ii)):J G(θ * ) = (I + λA -1 B) -1 λA -1 BHowever we will bound this term differently. First note that B -A ≤ Lε.From this, we deduce by sub-multiplicativity of the matrix norm that:A -1 B -I ≤ A -1 B -A ≤ Lε αand by triangular inequality:A -1 B = A -1 (B -A) + I ≤ A -1 B -A + 1 ≤ 1 + Lε α .Now we use the triangular inequality again to write:J G(θ * ) ≤ (I + λA -1 B) -1 λA -1 B .But,(I + λA -1 B) -1 = ((I + λI) + λ(A -1 B -I)) -1 = 1 1 + λ (I + λ 1 + λ (A -1 B -I)) -1 ≤ 1 1 + λ 1 1 -λ 1+λ A -1 B -I ≤ 1 1 + λ -λ Lε αwhere we have used that Lε α < 1. Finally,J G(θ * ) ≤ λ 1 1 + λ -λ Lε α (1 + Lε α )and a sufficient condition for having J G(θ * ) < 1 isλ 1 1 + λ -λ Lε α (1 + Lε α ) < 1 or equivalently, λ < α 2Lε .With this new bound λ < α 2Lε which ensures that the operator norm of the Jacobian is smaller than 1, i.e., J G(θ * ) < 1, we can unroll the remaining steps of their proof to get Equation 16A.4.10 Proof of Theorem 2.2 Theorem 2.2. Under Assumption 2.2, if Lε < α and λ < α 2Lε , then there exists a neighborhood of the optimal distribution parameters θ * such that for any initial parameters θ 0 in that neighborhood, p θt converges to p θ * exponentially fast: D KL (p θ * ||p θt ) = Õ λ(α + εL) α + λ(α -εL) 2t .∑ K j=1 e r(x j ) , 1 ≤ k ≤ K . // Luce's model xi ← xk D filtered = {x i } ⌊λ•n⌋ i=1// New filtered dataset 5 • 10 3 initial samples from the real data distribution and 5 • 10 3 generated samples filtered from 10 4 generated initial samples. When mixing, we use equal fractions of real and filtered samples. For the two moons we add a Gaussian noise with standard deviation 1.10 -1 .Only Synth. It is crucial to note the difference between collapse of the reward variance and collapse of the overall distribution variance. To highlight this difference, in appendix A.5.2 we show heat-maps of respectively a reward with four different modes, the density of the mixture of Gaussians p 0 , and the limit density of theorem 2.1 as defined as p * (x) := p0(x)1 r(x)=r * P0(r(x)=r * ) . In appendix A.5.2, we show that the reward variance collapses to 0 while the variance of the overall distribution density does not seem to collapse. of the precision than on Figure 8 or Figure 10 , which correlates with the intuition that confidence is linked to precision. Finally, notice that the three runs on Figure 9 have small variance, as we have already highlighted. Figure 11 : CIFAR-10. Evolution of the proportion of the classes and the average reward when filtering based on the confidence of a classifier for three independent runs. The curves have small variance which supports our results when only one run was reported due to the high compute costs of retraining a generative models multiple times.Experiments on synthetic data (mixture of Gaussians and two moons) ran on a single GPU in a few minutes. However, retraining with filtering on CIFAR10 was more costly. On a A100 GPU of 40GB RAM and using 4 workers with total 32 GB RAM, retraining for 20 iterations with generation of 50000 samples took about 22 hours.A.6 Examples of sets of four generated images on MidJourney and Stable Diffusion 2.1We show in Figure 12 two sets of four images generated respectively with Midjourney and Stable Diffusion. In both cases, users can choose their preferred image out of the 4 proposed and more specifically in the case of Midjourney, upscale it. Figure 12 : Two sets of four images were generated using two different generative models. For Midjourney (Figure 12a ), users can select which image to upscale. The upscaled images are then incorporated into the JourneyDB dataset (Pan et al., 2023) . For Stable Diffusion, users can choose the preferred generated image.Justification: We provide required assumptions in Assumption 2.1 and Assumption 2.2 and all proofs are explicited in Appendix A.Guidelines:• The answer NA means that the paper does not include theoretical results.• All the theorems, formulas, and proofs in the paper should be numbered and crossreferenced. • All assumptions should be clearly stated or referenced in the statement of any theorems. • The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition. • Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material. • Theorems and Lemmas that the proof relies upon should be properly referenced.• The answer NA means that the paper does not include experiments.• If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not. • If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable. • Depending on the contribution, reproducibility can be accomplished in various ways.For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general. releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g., in the case of a large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed. • While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. (d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility.In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results. • The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them. • The full details can be provided either with the code, in appendix, or as supplemental material. 


Decision: This paper performs a theoretical analysis of the setting in which a generative model is trained on a curated dataset of its own samples.  It shows that the implicit reward that guides such curation can successfully be learned by the generative model, thus motivating the soundness of the practice.  The reviewers were unanimously in favor of acceptance, with the paper's strengths being that it performs a novel, rigorous theoretical analysis of a topic of great modern importance.  Moreover, the results are experimentally supported in interesting experiments (Reviewer cUrK).  The reviewers primary concerns were if the analysis extends to a mixture of reward functions, relationship to prior work, and on some of the underlying conditions / assumptions of the results.  I believe the authors have either successfully addressed  these concerns (e.g. the underlying conditions) or rebutted their significance (e.g. mix of rewards is a nice extension but not essential, some of the related works are concurrent or specialized to specific generative model classes).

Reviews: review: paper_summary: paper_summary: Self-consuming generative models are known to have collapse or stability problems, and the curation of synthetic data is often ignored. This paper theoretically studies the impact of data curation and proves that it optimizes the expected reward.
strength_weakness: strength_weakness: 1. The theoretical results are applied in a simplified setting, focusing on distribution and reward on $x$ only, instead of considering a joint distribution of $x$ given $y$ mimicking text-to-image generation. The theoretical results apply only to learning the distribution directly, without considering finite samples and optimization, and there is only one reward function. In recent works combining RLHF and diffusion models [1], multiple rewards are considered. How would the results hold with a (random) weighted sum of multiple rewards?
2. On the connection with model collapse: The authors motivate the results from previous literature on synthetic data leading to model collapse but do not discuss how and whether curation with a reward model will avoid model collapse. The reviewer thinks that improvements can occur when inconsistencies between text and image and implausible images are discarded through curation. However, other problems persist in model collapse, such as the inherent lack of diversity in synthetic data [2], and existing results show that human interaction with GPTs still produces data lacking diversity [3]. In this case, curation does not address the loss of diversity in synthetic data. The authors should discuss this clearly, especially since the paper is motivated by model collapse and spends considerable time discussing it.
3. Could the authors explain the contribution of Section 2.2.1? Comparing Sections 2.2.1 and 2.2.2, there seems to be no direct comparison to be made.

### Reference

[1] Liang, Youwei, et al. "Rich human feedback for text-to-image generation." *Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition*. 2024.

[2] Guo, Yanzhu, et al. "The Curious Decline of Linguistic Diversity: Training Language Models on Synthetic Text." *Findings of the Association for Computational Linguistics: NAACL 2024*. 2024.

[3] Padmakumar, Vishakh, and He He. "Does Writing with Language Models Reduce Content Diversity?." *The Twelfth International Conference on Learning Representations*.
questions: questions: See weakness.

One related work: [4] propose to use a correction function on the synthetic data to mitigate the model collapse. 

[4] Gillman, Nate, et al. "Self-Correcting Self-Consuming Loops for Generative Model Training." Forty-first International Conference on Machine Learning.
limitations: limitations: The limitations and societal impacts are well discussed.
paper_summary: paper_summary: This paper explores the scenario where generative models are iteratively trained on self-generated data curated by human users with some implicit reward. The key idea is that each iteration of training on the self-generated data reweights the previous distribution based on the implicit reward, which converges to reward maximization as the iteration approaches infinity. The paper also studies the scenario where the curated self-generative data is mixed with natural data and analyzes its implication on the stability of iterative training. Experiments on synthetic data and CIFAR 10 validate the insights from the theoretical analysis.
strength_weakness: strength_weakness: 1. Human preference can be heterogeneous, so in some cases, Eq. (2) does not hold. It would be interesting to see if the theoretical analysis can be extended to a mixture of rewards.
2. This paper didn't provide experiments on realistic datasets and large models such as LAION and Stable Diffusion (SD). Thus, it's hard to map the theoretical insights to how we should train the next generation of SD. It would be really interesting if the experiments on CIFAR were reproduced on SD to see what realistic biases are picked up, how much replay is needed to mitigate that, etc.
3. Conceptually, retraining with data curation is related to iterative finetuining in language models (e.g., rejection-sampling-based SFT). Would like to see more discussion on that.
questions: questions: See Weaknesses.
limitations: limitations: I think the authors adequately addressed the limitations.
paper_summary: paper_summary: This paper extends Bertrand et al. 2024’s analysis of model collapse to study settings where data is filtered (based on a particular preference model) before being used for training the next iteration of generative models.
strength_weakness: strength_weakness: - I do not think that Equation (3) faithfully models reality. Specifically, it assumes that the $t+1$-th model is fit to a mixture of (1) real data and (2) preference-filtered data from the $t$-th model. But realistically, synthetic data should amass over time, as I believe is the case with the datasets that the authors mention in their introduction, e.g., LAION-5B. This is a point made by "Is Model Collapse Inevitable? Breaking the Curse of Recursion by Accumulating Real and Synthetic Data." and I agree with them.

- I might be missing something, but intuitively, Theorem 2.1 seems wrong, specifically the claim “eventually converges to the highest level set of the reward reached **at initialization**”. For the simplest possible counter example, suppose p_0 is a discrete Uniform distribution on integers 0 to 10 and the reward function is $r(x) = x$. Defining $r_* := 10$, we see that Assumption 2.1 is satisfied, including the variant where $\epsilon \geq 0$. It is hard to imagine that the iterative loop would concentrate on $10$ rather than something greater than $10$. The intuition is that sampling and filtering from the first model iteration might shift the distribution towards regions where it is possible to sample higher rewards.

- I think the experiments corresponding to Theorem 2.1(Figures 4 and 5) could be improved to better connect with the maths. Specifically: (1) Provide a heatmap of the level set of $r_*$ to show what $p_*$ is. (2) As I understand, the theorem doesn’t say that the distribution’s variance collapses, but rather, that the _reward variance_ collapses. In the figures, the reward functions are unimodal, and so we see the distributions converge towards unimodal behavior. You should modify the reward functions to be multimodal to demonstrate this distinction between the distribution's variance collapsing versus the reward's variance collapsing. One way to do this might be to define the reward function as 4 of the 8 MoGs (i.e. the reward is the max of the set of negative distances to the 4 chosen centroids). Then, we should see the model concentrate on those 4 chosen centroids.

- I feel like I don’t understand Theorems 2.3 or 2.4. In Theorem 2.3, the lower bound on the right hand side does not appear to depend on the model fitting iteration $t$, as best as I could tell, nor does the upper bound in Theorem 2.4. Here, I’m expecting the answer to depend heavily on the model fitting iteration. I read the adjacent discussion but didn't receive any clarity. I'm consequently not sure how to evaluate the significance of Section 2.2. Perhaps the authors could clarify?

- I think there are many additional papers you might want to cite. Some may be concurrent with yours (I intentionally did not search for a preprint in order to preserve double blind reviewing), and if so, that’s fine. Here are some suggestions on several different topics:

**On Model Collapse:**

Beyond Model Collapse: Scaling Up with Synthesized Data Requires Reinforcement. https://arxiv.org/abs/2406.07515

Is Model Collapse Inevitable? Breaking the Curse of Recursion by Accumulating Real and Synthetic Data.
https://arxiv.org/abs/2404.01413

**On mode collapse in RLHF:**

Understanding the Effects of RLHF on LLM Generalisation and Diversity. https://openreview.net/forum?id=PXD3FAVHJT

A Distributional Approach to Controlled Text Generation.
https://openreview.net/forum?id=jWkw45-9AbL

Red Teaming Language Models with Language Models
https://arxiv.org/abs/2202.03286

Improving alignment of dialogue agents via targeted human judgements
https://arxiv.org/abs/2209.14375

Aligning Language Models with Preferences through f-divergence Minimization
https://arxiv.org/abs/2302.08215

**On filtering data using reward models - often known by multiple names in the RLHF literature including “Best of N” or “rejection sampling” or “reranking” in the RLHF literature:**

Scaling Laws for Reward Model Overoptimization
https://arxiv.org/abs/2210.10760

Training a Helpful and Harmless Assistant with Reinforcement Learning from Human Feedback
https://arxiv.org/abs/2204.05862

There are many more
questions: questions: N/A
limitations: limitations: N/A
paper_summary: paper_summary: This paper studies the impact of data curation on iterated retraining of generative models. Theoretical results are derived for the convergence state of the retraining loop when using a fraction of curated synthetic data or a mixture of real data and curated synthetic data at each step. Empirical experiments on both synthetic datasets and CIFAR-10 demonstrate that the proposed approach can bias the generative model to generate samples with higher reward.
strength_weakness: strength_weakness: The writing of this paper needs further improvement. Some notations are unsuitable, and certain mathematical notations are introduced without explanation. Some literature references are missing.  
1. In Eq.2, $\mathcal{BT}\left(x_1, \ldots, x_K\right)$, where "BT" refers to the Bradley-Terry model, is typically used to model pairwise preferences, whereas the Plackett-Luce (PL) model is proposed for preferences involving more than two items.  
Eq.5: If $p_{t+1}(x)$ is not a normalized density function, the same applies to $p_{t+1}(x)$ in Eq.6.  
2. There are already studies on deep generative models (GANs) that investigate the convergence state of iterative retraining on curated data from the perspective of preference [1, 2]. It should include a discussion of these works.
3. The theoretical proofs especially the part in Section 2.2 are heavily inherited from the previous work [3], which somehow makes the theoretical contributions marginal.

[1] Gupta, A., Zou, J. Feedback GAN for DNA optimizes protein functions. Nat Mach Intell. 2019.  
[2] Yao, Y., Pan, Y., et. al. Differential-Critic GAN: Generating What You Want by a Cue of Preferences. IEEE Transactions on Neural Networks and Learning Systems, 2022.  
[3] Bertrand, Q., Bose, et. al. On the Stability of Iterative Retraining of Generative Models on their own Data. In The Twelfth International Conference on Learning Representations.
questions: questions: 1. Eq.3: The term $p_{ref}$ is not explained. What distinguishes $p_{ref}$ from $p_{data}$?  
2. In Theorem 2.2, deriving $\lambda < 0.5$ based on the assumption $L \varepsilon < \alpha$ and $\lambda < \frac{\infty}{2 D \varepsilon}$ appears to conflict with the claim in line 181.  
3. The explanation of $r_{\ast}$ in lines 134-137 is somewhat complex. Adding some equations would clarify its meaning.
limitations: limitations: Yes.
paper_summary: paper_summary: The authors investigate the properties of self-consuming loops that arise in the training of generative models. In particular, they investigate the impact that data curation has on the iterative retraining performance of these models. The paper contains theoretical and empirical analysis of how model performance is affected by various data curation assumptions (only synthetic examples, or real data being injected at each step, or human-preference curated synthetic samples, etc).
strength_weakness: strength_weakness: Clarity: in my opinion, the presentation of statements of the theorems in the paper needs improvement in order to be useful to the community. Consider for example Theorem 2.1. The assumptions for this theorem are distributed in the section preceding it, which makes it much harder to understand and contextualize that theorem. I would strongly suggest summarizing the assumptions and key notations in the statement of that theorem (and likewise for all the other results.) An excellent model for this would be the paper that the authors cited the most, Bertrand et al's "On the Stability of ..." . That paper's Theorem 1 (from the latest arXiv version) begins--"Given theta^\star as defined in Equation (7) that follows assumptions 1 and 2....". Given that a large part of this paper's contribution is its theorems, I would be likely to raise my score, and champion this paper, if I could first see an updated draft which makes explicit every assumption in the statement of each theorem.

See also the limitations section.
questions: questions: I'm not sure I understand how the experiment supports the theory here, please help clarify this for me--at line 250, the authors say "applying theorem 2.1, the density will converge to a renormalized Gaussian distribution restricted to the ball centered at x_* of radius r_min". But this isn't what I see in Figure 4, it looks like there are a bunch of different Gaussian balls with different densities, corresponding to different intensities. Should these gaussian balls all be the same density, or did I misunderstand something?

Some misc points that didn't affect my score/judgment of the paper, but the authors should consider fixing in the next draft:
- improper formatting in equation 8 (spacing/parentheses)
- awkward/grammatically incorrect wording in the sentence immediately after equation (7)
limitations: limitations: The two main things preventing me from giving a higher score are the following two limitations. I'm looking forward to hearing the author's rebuttal to these points (and hopefully seeing an updated draft, if possible).

1. Presentation of statements of theorems (what are the specific hypotheses? eg take a look at the latest version of the bertrand et al paper on the arXiv for a good way to do this, since that paper is structured in a similar way. They number their assumptions and make them more clear. See "Weaknesses" section for more details.) This is the primary limitation, from my perspective, as I think the usefulness of the paper is very limited by unclear theorem statements.

2. Proper contextualization of these results relative to the literature--namely, there's a difference between the work in Bertrand et al and Alemohammad et al, but the authors seems to be comparing them in an "apples to apples" way. Namely, the former work considers the case of iterative fine-tuning, whereas the latter considers the case of retraining from scratch. In the former case, it is strictly easier to avoid model collapse, since the model parameter update is "local", and in the latter case, the updates are "global." Specifically, on line 218 it says: "Alemohammad et al. (2024); Shumailov et al. (2023) first evidenced catastrophic degradation of the generated data in the fully synthetic loop. Bertrand et al. (2024) mitigate these conclusions in the setting where the model is retrained on a mixture of synthetic and real data and they show the stability of the process around the data distribution." I think this is a false statement, because Alemohammad et al considered re-training from scratch at each iteration, which wasn't considered in the Bertrand et al paper--but please correct me if I'm misunderstanding. And in that same vein, I think it is important to properly contextualize the present paper in the presence of that dichotomy--does this paper consider iterative fine-tuning, or iterative re-training from scratch? That is important information for the readers/the literature.

Comments: title: Response to the Reviewer's Comment
Response to the Reviewer's Comment [Part 3]
Response to the Reviewer's Comment [Parts 1 and 2]
Response to rebuttal
Response to Authors' Rebuttal [Part 3]
Response to Authors' Rebuttal [Part 2]
Response to Authors' Rebuttal [Part 1]
Thanks for the responses.
comment: We are grateful to the reviewer for their valuable feedback and for increasing their score. We will include in the updated manuscript the new presentation of the theorem statements with their assumptions, along with the contextualization of our results in the literature. We present below how we aim to update the statements of Assumption 2.2 and theorems 2.2, 2.3, and 2.4.

**Assumption 2.2:** For $\theta$ close enough to $\theta_*$, the mapping $x \mapsto \nabla^2_\theta\log p_\theta(x)$ is $L$-Lipschitz and the mapping $\theta \mapsto E_{p_{data}}\left[\log p_\theta(x) \right]$ is continuously twice differentiable with $E_{p_{data}}\left[\nabla^2_\theta\log p_\theta(x)\right] \preceq -\alpha I \prec 0$. Further suppose $W_1 (p_{\theta_*}, p_{data})\leq \epsilon$, i.e. $p_{\theta_*}$ is close to the data distribution $p_{\text{data}}$.

**Theorem 2.2:** Under Assumption 2.2, if $L \epsilon < \alpha$ and $\lambda<\frac{\alpha}{2L\epsilon}$, then there exists a neighborhood of the optimal distribution parameters $\theta_*$ such that for any initial parameters $\theta_0$ in that neighborhood, $p_{\theta_t}$ converges to $p_{\theta_*}$ exponentially fast:

$$ KL(p_{\theta_*}||p_{\theta_t}) = \tilde{\mathcal{O}}\left(\left(\frac{\lambda(\alpha+\epsilon L)}{\alpha+\lambda(\alpha-\epsilon L)}\right)^{2t}\right)$$


**Theorem 2.3:** Let $\lambda > 0$ and consider the process $(p_{t})$ defined in eq. 8, with $p_0 = p_{ref}$. If $p_{ref}$ satisfies Assumption 2.1 B, then for all $t\geq1$:

$$E_{p_t}\left[e^{r(x)}\right] \geq E_{p_{ref}}\left[e^{r(x)}\right] + \frac{\lambda}{(1+\lambda)^3}\frac{(K-1)Var_{p_{ref}}\left[e^{r(x)}\right]}{Ke^{r_*}}$$

**Theorem 2.4:** Let $\lambda > 0$ and $p_{ref}\in \mathcal{P}(\mathbb{R}^d)$ with a density w.r.t. Lebesgue measure. Consider the process $(p_{t})$ defined in Equation 8, with $p_0 = p_{ref}$. Suppose that $\lambda < \frac{1}{K-1}$, then, for all $t\geq1$:

$$KL(p_t||p_{ref})\leq -\log\left({1-\lambda(K-1)}\right)$$

We hope this answers the reviewer’s remaining concerns and are happy to provide clarification to any additional questions the reviewer may have.
We are grateful to the reviewer for this interesting discussion on whether our assumptions realistically reflect the practical setting when today’s large models are retrained on web-scaled dataset. 

We agree together with the reviewer that the accumulation of data is arguably a realistic feature of web-scaled datasets such as LAION-5B. Especially, it is expected that new next-generation web-crawling datasets will incorporate synthetic images from previous years generative models together with the current state-of-the-art generation of generative models. We did not incorporate such feature in our setting as it would complicate the statements and the notations. The main focus of our work was on a new type of collapse centered on human preferences which is different from the previous literature. However, we believe such an extension is relatively easy and we present now how to address it:

Let $\\{\lambda_k^t\\}$ for $0 \leq k \leq t$ be a family of positive numbers such that forall k, $\lambda_k^t$ is decreasing in t and normalized such that $\forall t\geq 0, \sum_{k=0}^t\lambda_k^t=1$. Consider that the data accumulates with proportions given by the family $\\{\lambda_k^t\\}$ for $0\leq k \leq t$. In that case, equation 6 which states $p_{t+1}(x) = p_t(x)\cdot H^K_{p_t}(x)$ becomes $p_{t+1}(x) = \sum_{k=0}^t \lambda_k^t p_k(x)\cdot H^K_{p_k}(x)$. 

We believe that similarly to our theory, it is straightforward that $E_{p_t}[e^{r(x)}]$ is increasing. Further assume that $\forall k, \lambda_k^t\overset{t\rightarrow \infty}{\rightarrow} 0$. In that case, for all $k$ the contribution of each $p_k$ in the retraining at iteration $t$ of $p_t$ decreases to $0$. We additionally believe that **in that case, the expected reward will converge to $r_*$ the maximal reward at initialization and that $p_t$ will collapse to maximal reward regions**.
We think that this case is realistic, as *the proportion of data on the web generated by any model is doomed to vanish as more models are trained and deployed*. In that setting, accounting for the accumulation of data would therefore not prevent the collapse of the self-consuming model to maximal reward regions. This constitutes an interesting extension and we will add to the updated manuscript precise statements, along with proofs if the reviewer believes it strengthens our work.

Finally, we mention that the work [1] is concurrent with ours since it was first posted on Arxiv on 1st April 2024. This was an additional reason why we did not incorporate their setting in our study, as we were aware of such extension only late in this project.
We are grateful to the reviewer for their time and engaging with us during this rebuttal. We answer below the reviewer’s additional questions.
## On whether the support of $p_{t+1}$ is included in the support of $p_t$
We acknowledge the reviewer's comments regarding the fact that the support of $p_{t+1}$ is included in the support of $p_t$. We understand the two points provided by the reviewer as
>1) Depending on the choice of realizable distributions $\mathcal{P}$, the support might be the entire space of outcomes e.g., if $\mathcal{P}$ is the set of Gaussian distributions.

Note that having that the support is the entire space of outcome is not a problem at all for our theory. Our only requirement is that **the reward is bounded over the support of $p_0$** (at initialization) by $r_*$ (Assumption 2.1) which implies that the reward is bounded for all timesteps (since the support of $p_t$ is included in the support of $p_0$).
There exist many functions with unbounded support that are bounded (e.g. the sigmoid function). 
We believe that it is reasonable to assume that the intrinsic human reward is bounded. 

>2) In general, probabilistic models are often capable of placing mass/density on sets that were not in their training data (because they generalize)

Our setting prevents this scenario from happening: we are working in a setting were we neglect the errors due to the finiteness of **model’s capacity** and **training data**. In other words, we work in a setting where we have:

i.  **infinite capacity regime**:  *the set $\mathcal{P}$ of achievable distributions is the entire set of probability distributions* (lines 112-113 in lemma 2.1). In that case, equation 6 holds and hence the support of $p_{t+1}$ is included in the support of $p_t$. 

ii. **infinite training data regime**: This comes from the fact that we are assuming that the next distribution minimizes the population likelihood (and not the empirical one) in equations 3 and 4 ($p_{t+1}$ is defined using an expectation on the distribution $p_t$ and $p_{data}$).

Finally, i), ii) together prevent the points 1) and 2) from happening in our setting. 
Note that we mentioned i) explicitly in the statement of Lemma 2.1 “If $\mathcal P$ is the set of probability distributions on $\mathbb{R}^d$” and on Line 174.
The point ii) was indicated by the fact that we were using expectations and not finite sum in our paper.

We acknowledge that we should have made i) and ii) appear more clearly in our paper to avoid any confusion. **We will clarify and highlight our setting in the updated manuscript**. Current generative models are getting larger and larger, and are trained on more and more data with more than billions of parameters and datapoints. That is why, we believe it is reasonable to neglect the errors due to the finiteness of **model’s capacity** and **training data** in our theory and that our results satisfyingly capture the reward maximization phenomenon induced by human curation.

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edit: we updated this comment to enhance clarity of the response
I would like to thank the authors for the thoughtful reply to my review.

In particular, I appreciate the additional contextualization of works [1] and [2]; I think that the literature would greatly benefit from having that explanation from the section "On fresh real vs fixed real data". I also appreciate the more clear statement of Theorem 2.1.

Would it be possible to share here the other exact statements as well--namely, how the authors would propose to update the other exact statements of the other theorems, 2.2, 2.3, 2.4--in another response?

**I have updated my score to above the acceptance threshold, with the expectation that the authors will update the camera-ready version's main result statements with more clearly stated hypotheses and clearly referenced terms, similar to the rebuttal above.** Although my concern was mainly with Theorem 2.1 (it contained the least context out of all the other theorems), in my opinion, Theorems 2.2, 2.3, 2.4 should each be improved via clearer context, and I would want to see the updated statements if possible.

I strongly believe that having these statements stated that clearly would better allow the community to benefit from the authors' work.
> We thank the reviewer for referring us to the paper [1] and pointing out this interesting extension. We believe, together with the reviewer, that the setting of [1] could be adapted to show that accumulating data provides additional stability to the retraining loop and avoids collapse. However, we believe such a study is out of the scope of our work whose aim was to introduce and theoretically develop a new research question of model collapse from the view of preferences. We will, therefore, update our draft to provide additional clarification of this aspect and mention it as an exciting future direction. 

I feel like the authors and I miscommunicated here. The point I was trying to raise is: what are realistic assumptions to make about how model-data feedback loops should be modeled? I wasn't so much interested in that other paper as much as I was interested in whether this paper faithfully captures the settings we care about ("I do not think that Equation (3) faithfully models reality. Specifically, it assumes that the -th model is fit to a mixture of (1) real data and (2) preference-filtered data from the -th model. But realistically, synthetic data should amass over time, as I believe is the case with the datasets that the authors mention in their introduction, e.g., LAION-5B.")

My thinking is that the assumptions of this paper are not especially realistic because synthetic data should increase over time and the total amount of data should increase over time too.

**TLDR: I think your assumptions are not realistic and I think this harms the significance & relevance of your paper.**

If I missed the response to this point by the authors, I apologize and I would appreciate being pointed in the correct direction. Thank you!
> We understand the reviewer's natural concern. The crucial point is that the curated distribution at time $t+1$, $p_{t+1}$ is issued from learning a modified distribution from $p_t$ constructed by sampling from $p_t$ and curating samples using preferences. This implies that if a set has probability $0$ for $p_t$, it will never be sampled and therefore never be preferred over other samples. This means that this set will also have probability $0$ for $p_{t+1}$.

I'm not sure I buy this argument. I agree that if a set has probability $0$ for $p_t$, it will never be sampled and therefore will never be preferred over other samples. But you lose me for two reasons:

1. Depending on the choice of realizable distributions $\mathcal{P}$, the support might be the entire space of outcomes e.g., if $\mathcal{P}$ is the set of Gaussian distributions. In this case, no set would have probability $0$ for $p_t$. Your explanation then seems to hinge on a condition "if" that might not be applicable.

2. Even if the conditional statement is true i.e. there is some set with mass/density 0 under $p_t$, why is $p_{t+1}$ prohibited from extending its support to this set? In general, probabilistic models are often capable of placing mass/density on sets that were not in their training data.

Could the authors please clarify?
Thank you to the authors for their response!

> Improvement of the experiments on MoGs

> We provided in the additional pdf for figures a heat map of the level set of the reward when using centroïds (Fig 1a).

This is wonderful. Thank you for running these additional experiments - I really like them, and I think they'll improve the paper (in my opinion; if you disagree, you don't need to include them).

> Note that [2,3] are concurrent work according to NeurIPS guidelines:

Yes, then it's fine to not cite them. I tried to allude to that above ("Some may be concurrent with yours") but apparently didn't finish the sentence. I appreciate your care to the other citations.
I appreciated the author's discussion on the multi-reward setting and on the difference between the collapse of distribution and the collapse of reward variance. I have increased my score.
I appreciate the authors' clarification. I increased my score to "weak accept".

About $p_t$. Usually, the probability is normalized to 1. If not, it should add the clarification.