Why Diffusion Models Don’t Memorize: The Role of Implicit Dynamical Regularization in Training

We first want to thank you for the very positive report as well as for your different questions that helped us to improve our work. You will find below an answer to the several points raised in your review.

Main comments

About larger parameters regime for Fig. 3

We have conducted additional experiments in the meantime. In particular, we added an intermediary point $W=48$ ($p \approx 9$M) and found that for CelebA resized to 32x32 the range of $p\in\left[10^6, 16\times 10^6\right]$ parameters show convergence on the training timescales we focus on ($8\times \tau_\mathrm{gen}$ at maximum) and therefore does not require much larger $p$. We plan to update Fig. 3 and the discussion accordingly in the final version of the manuscript. Exploring the diagram for much larger $p$ on this dataset would demand significantly larger $n$ and much more training time, which goes beyond our computational capacities without, we believe, strengthening our core results (see also the answer below).

About Fig. 3 (right) and the architectural regularization

We agree that although the layouts of Fig. 1 and Fig. 3 are similar, they plot related but distinct diagnostics, which may be confusing. Fig. 1 is a schematic diagram drawn in the $\tau\to\infty$ regime, illustrating the theoretical findings. On the other hand, Fig. 3 fixes $\tau = \ell \tau_\mathrm{gen}$ and empirically identifies the smallest $n(p, \tau)$ at which $f_\mathrm{mem}(\tau)=0$. Building upon your question, we were able to identify, for our range of $p$ and training time (2M steps), convergence of $n(p,\tau)$ as $\tau$ increases, therefore drawing the architectural regularization line $n^\star(p)$ hinted by Fig. 1. We thank you for this suggestion and we will definitely add this line to Fig. 3 to reinforce the parallel with Fig. 1 and validate the observations on realistic trained models and datasets.

About generalization to adaptive optimizers

In the numerical experiments on CelebA we use SGD with fixed momentum. To demonstrate that the observed phenomenon is not specific to SGD, we included in Appendix B.3. numerical experiments on a toy Gaussian Mixture dataset using Adam which exhibit the same behaviour. Since the submission, we also conducted additional experiments on CelebA at fixed $W=64$, and on random features, both trained with Adam, showing that $\tau_\mathrm{mem}$ also scales linearly with $n$ in these cases. We will be pleased to incorporate these new results in the appendix of the camera‐ready version of the manuscript, accompanied with a remark in the conclusion.

Is the phenomenon unique to diffusion models?

Thank you for raising this important question on the generality of our reuslts. We believe that our findings extend to other generative models. For instance, the same analysis can be done transparently for stochastic interpolants [1] and for Flow Matching [2]. Indeed, the learning of the score function (or velocity field) can be rephrased as a regression task (the denoising score-matching loss in the case of diffusion models). In the context of supervised learning, [3] observed a similar timescales for overfitting with an overparametrized two-layer neural network. More specifically, they show that if $n/p$ is kept fixed, then this timescale goes linearly with $p$, which is similar to our result. We will mention this point in the final version of the paper.

Making memorization scores more tangible

We show in Fig. 7 some generated images along with their nearest neighbor in the training set. We will make use of the additional page of the camera-ready version to include a similar (but better-looking) figure in the main text showing a random batch of generated sample making the memorization fractions more tangible (e.g. in Fig. 2).

About the assumptions of L239

We thank you for asking this clarification. It is indeed important to assess the role of these assumptions and simplifications. We will do it in the camera-ready version if the paper is accepted. There are two assumptions made in the analytical part: (i) the activation function $\sigma$ admits an expansion in Hermite polynomials and (ii) $\sigma$ is odd. Both are mainly technical and are not a limitation of the approach. Assumption (i) is equivalent to being in the functional space $L^2(e^{-x^2/2}dx)=\{f :\int \mathrm{d} x \lvert f(x)\rvert^2e^{-x^2/2}<+\infty\},$ and is verified by all the activation functions used in practice ($\tanh$, ReLU...). Regarding (ii), it is the same as in [4], and greatly simplifies computations. Moreover, the target score function we focus on also has this symmetry, therefore it's natural to consider such families of activation functions. It can be relaxed by requiring only that $\mu_0=\mathbb{E}_{z\sim\mathcal{N}(0,1)}[\sigma(z)]=0$, which simply amounts to a constant shift of the activation function $\tilde{\sigma}=\sigma-\mu_0$.

Focusing on the case where the first layer of weights is frozen is a limitation. It allows us to study the memorization-generalization phenomenon in a well-defined and solvable setting, the celebrated Random Features model [5]. While simplistic, it has already served as theoretical framework to study several behaviors of deep neural network such as the double descent phenomenon [6,7]. We want to stress that our approach can be extended to more general settings. In fact, the Neural Tangent Kernel scaling limit of deep neural networks can be mapped to a specific Random Feature model [8,9]. Therefore, one can extend the approach we developed in this work to study deep neural networks models of the score in the NTK limit - the only difference being that the random matrix problem to solve (Theorem 3.1) is more involved but conceptually identical. After the completion of this work we started to work on this project and the associated results will be presented in a forthcoming publication.

We will clarify the nature and importance of each assumption in the final version of the paper.

Availability of the code

Thank you for raising this point. We will make all codes used for training our models and reproducing the main figures of the paper publicly available in a non-anonymized Github repository upon publication.

Smaller comments

1. $\tau$ is the training time, measured in the paper as the number of gradient updates. We will make this clearer in the final version of the paper.

2. We will make this sentence clearer to avoid any confusion.

3. We understand that the chosen wording might be misleading at first sight and we are committed to making it clearer at the very beginning of the next version of the paper.

4. Thank you, we now specify that the evolution of the FID is used to actually measure $\tau_\mathrm{gen}$ in the numerical experiments.

5. The choice $k=1/3$ comes from previous numerical studies [10, 11] to fit the visual appreciation of memorization on image datasets. We made sure that varying $k$ (e.g. to $1/2$ or $1/4$) does not affect the scaling behaviour of $\tau_\mathrm{mem}$.

6. We reworked the diagram in Fig. 3 (right) as you suggested to add markers indicating clearly the measured points. We also completed it with a two parameter values with $W=48$ and $W=8$. All available $W$ are used.

7. In Section 3, we rescale the Loss of Eq. (4) by the dimension $d$ of the data to have a well-defined large dimensional limit. We agree that the formulation of the sentence is ambiguous and we will clarify it in the camera-ready version of the paper.

8. The lines in Fig. 6 are empirical, obtained using a PyTorch implementation of full-batch gradient descent. Details of the numerical setup can be found in Appendix D. We will make this explicit in the figure caption by indicating that the curves are empirical.

9. In full generality, the score function $s(x,t)$ is a function of $x\in\mathbb{R}^d$ and the diffusion time $t$. In the analytical part of the article, we fix the diffusion time $t$ and study the learning of the score at this specific time $t$. In other words, we train a new model for the score at each diffusion time $t$. Usually, practitioners only use one model for the whole generative process which depends on the time $t$. However, in our setting, the learned parameters $A$ depend on the diffusion time. Thus the score function depends on time through its parameters $s_A(x)=s_{A(t)}(x)$. This setting was also used in some previous works [4,12] and greatly simplifies the analysis. We thank you for raising it and we will make the dependence of $A$ on time $t$ (and therefore of the score function $s_{A(t)}$) more explicit in the camera-ready version of the article to avoid any confusion.

10. Figure 5 is computed with parameters $\psi_p=64$ and $\psi_n=32$ which corresponds to the Regime I (overparametrized regime) of Thm.3.2. which is the regime of interest of the article. The first bulk in the inset, whose support lies approximately between $\lambda = 0$ and $\lambda = 0.016$, corresponds to the density $\rho_1$ in the theorem. The second bulk, supported between $\lambda = 15$ and $\lambda = 35$, corresponds to $\rho_2$. We thank the reviewer for pointing out the lack of precision in the legend of Figure 5, and we will revise it accordingly in the final version of the paper.

[1] Albergo, Boffi, Vanden-Eijnden, arxiv: 2303.08797, 2023.

[2] Lipman, Chen, Ben-Hamu, et al., ICLR, 2023.

[3] Montanari, Urbani, arxiv:2502.21269, 2025

[4] Georges, Veiga, Macris, arxiv:2502.00336, 2025.

[5] Rahimi, Recht, Neurips, 2007.

[6] Mei, Montanari, Communications on Pure and Applied Mathematics,2019.

[7] d’Ascoli, Refinetti, Biroli, et al., ICML, 2020.

[8] Jacot, Gabriel, Hongler, NeurIPS, 2018.

[9] Chizat, Oyallon, Bach. NeurIPS, 2019.

[10] Yoon, Choi, Kwon, et al., ICML, 2023.

[11] Gu, Du, Pang, et al., Transactions on Machine Learning Research, 2025.

[12] Cui, Krzakala, Vanden-Eijnden, et al., ICLR, 2025.

We thank you for the very supportive report.

I believe that all diffusion models in practice never reach or are within the architectural regularization regime. Is there a way the findings and the insights of this paper may guide us to change something with the current diffusion model practice?

We agree that models trained on massive and richly diverse datasets operate above the architectural regularization threshold, and are therefore not concerned by the presented theoretical findings. However, even some models industrially trained on massive datasets like LAION were found to exhibit complete or partial memorization [1, 2]. There are also domains where training data are scarce - for instance many physical science applications like cosmology or climate science - hence often falling in a low-data and high-capacity regime. In all these cases, our work offers some concrete guidelines and possible fixes (early-stopping and/or controlling the network capacity) that could help to robustly train the diffusion model to avoid memorization without needing more domain-specific inductive biases. We would be happy to discuss these aspects in the conclusion of the camera-ready version of the paper.

[1] Carlini, Hayes, Nasir, et al., Extracting Training Data from Diffusion Models, USENIX, 2023.

[2] Somepalli, Singla, Goldblum, et al., Understanding and Mitigating Copying in Diffusion Models, NeurIPS, 2023.

We first want to thank you for the very positive report concerning the soundness and contributions of our work. You will find below an answer to the points raised in your review.

Constant drift and diffusion coefficients in the diffusion model and one-layer neural network for the analytical part.

We are not completely sure we fully understood the comment, so we will present an extended response and apologize in advance if part of it is not what you referred to.
The exposition we make of diffusion models in the introduction is indeed kept minimalist, but adding time-dependence to the drift and/or diffusion terms of Eq. 2 in fact amounts to a time reparameterization of the diffusion process. This is therefore not a limitation. All of our experiments in fact use the standard DDPM formalism with

where $\beta(t)$ is a predefined noise schedule. In Appendix A1, we show that the two formulations are indeed equivalent under a proper reparameterization of the time $t$. Hence, also the theoretical results hold for a non-constant drift and diffusion coefficient.

We do consider a linear score in the theoretical analysis. The motivation is to study the memorization-generalization phenomenon in a well-defined and solvable setting. As you stated, extending the analysis to more general settings would be a highly challenging task. We do hope that our work will trigger new research along these lines. Finally, it is true that we consider a one layer network to model the score, but our approach can be extended to more general settings. In fact, the "Neural Tangent Kernel" scaling limit of deep neural networks can be mapped to a specific Random Feature model [1,2]. Therefore, one can extend the approach we developed in this work to study deep neural networks models of the score in the NTK limit -- the only difference being that the random matrix problem to solve (Theorem 3.1) is more involved but conceptually identical. After the completion of this work we started to work on this project. Results will be presented in a forthcoming publication.

We will clarify all these aspects in the final version of the paper to highlight the broad applicability of our results.

Is there an explicit estimate for the two timescales $\tau_\mathrm{gen}$ and $\tau_\mathrm{mem}$? How do these timescales depend on the spectrum of $\boldsymbol{U}$ as described in Theorems 3.1 and 3.2?

As stated in Line 247 of the main text and proved in Proposition C.1 of the SM, the timescales of the training dynamics are given by the inverse eigenvalues of the matrix $\Delta_t \mathbf{U} / \psi_p$, where $\mathbf{U}$ is defined in Eq. (12). In Theorem 3.2, we show that the spectrum of $\mathbf{U}$ exhibits two well-separated bulks, with normalized densities denoted by $\rho_1$ and $\rho_2$. This structure allows us to define two characteristic timescales:

Furthermore, Proposition C.2 of the SM shows that on the fast timescale $\tau_2$, both the training and test losses decrease without overfitting, whereas on the longer timescale $\tau_1$, the generalization loss begins to increase. This behavior supports the identification:

[1] Jacot, Franck, Clément, Neural tangent kernel: Convergence and generalization in neural networks, NeurIPS, 2018.

[2] Chizat, Oyallon, Bach, On lazy training in differentiable programming, NeurIPS, 2019.

Thank you for your answer and for maintaing your recommendation about the acceptance of our manuscript. We are grateful for your feedback and pleased that our rebuttal addressed your questions.

We thank you for your remarks and questions that, we believe, will help improve the quality of the paper. Please find below the answers to your interrogations.

Discussions about the practical impact of our work

Several recent work show that even industrial models [1, 2] trained on millions of data exhibit partial of complete memorization of some training samples. While our work is proposing a theoretical understanding of the memorization phenomenon, we believe it also provides interesting guidelines for practitioners on some ways of fixing this issue: either by reducing the total number of parameters in the network as long as it does not harm generation's quality, or by stopping the training earlier. Our work also shows that some intuitive and simple metrics can be used to monitor the amount of memorization during training (memorized fraction or train vs test loss at fixed diffusion times). We will be happy to discuss more how our results could better guide practice in the final version of the paper.

Generalization to other optimization algorithms with momentum terms

In all the experiments on CelebA reported in Sect. II, we in fact use stochastic gradient descent (SGD) with fixed momentum $\beta=0.95$. To demonstrate that the observed phenomenon is not specific to SGD, we included in Appendix B.3. numerical experiments on a toy Gaussian Mixture dataset using Adam which exhibit the same behaviour. Moreover, since the submission, we also conducted additional experiments on CelebA at fixed $W=64$, and on random features, both trained with Adam, showing that $\tau_\mathrm{mem}$ also scales linearly with $n$ in these cases. We will be pleased to incorporate these new results in the appendix of the camera‐ready version of the manuscript, accompanied with a remark in the conclusion.

Are the results applicable to conditional diffusion models?

Thank you for raising this interesting point. We know memorization is still observed in models trained conditionally, as for instance shown in [2, 3, 4]. Importantly, our results do not rely on the model being unconditional: any conditioning variable (class, text embeddings or other information) usually enters the model as extra input at the training level. In consequence, our observations are expected to hold when training conditional scores, in particular for each conditioning, we do expect $\tau_\mathrm{mem}$ to increase with $n$. It could happen that $\tau_\mathrm{gen}$ and $\tau_\mathrm{mem}$ depend on the class. In this case, before entering a full memorization (generalization) phase, one could be in a regime of $n$ and $p$ in which some classes are in the generalization (memorization) phase and others do not.

To validate our results on conditional diffusion models, we trained DDPMs with classifier free-guidance [5] on a mixture of two Gaussian (same data and model as in the appendix) and measured the associated memorization fraction during training for several dataset sizes $n$. We show that, even under guidance, $\tau_\mathrm{mem}$ scales linearly with $n$. We will add this result to the appendix and explicitly mention it in the main text of the camera-ready version of the paper as we think it is a valuable addition showing the generality of our results.

Is the result applicable on datasets with long-tail distributions?

Thank you for this question. Since the submission of this work, we have extended our analysis to compute the spectrum of $\mathbf{U}$ for Gaussian data with arbitrary covariance matrix $\Sigma$, including the cases where the density of eigenvalues of $\Sigma$ has power-law tails. We will add this computation in the final version of the paper. The case corresponding to non-Gaussian data, whose distribution has power law tails is more challenging. We have done preliminary numerical experiments on the RF model applied to datasets with long-tail distribution as in [6]. Our findings confirm that the increase of $\tau_\mathrm{mem}$ with $n$ is present also in this case.

A thorough understanding of how heavy-tailed data distributions affect the implicit regularization dynamics would require a more detailed analysis, which goes beyond the current scope of the paper. However, we agree with you that this is an interesting direction that we would like to pursue in a future work.

[1] Carlini, Hayes, Nasir, et al., Extracting Training Data from Diffusion Models, USENIX, 2023.

[2] Somepalli, Singla, Goldblum, et al., Understanding and Mitigating Copying in Diffusion Models, NeurIPS, 2023.

[3] Wen, Liu, Chen, et al., Detecting, explaining and mitigating memorization in Diffusion Models, ICLR, 2024.

[4] Chen, Yu, Xu, Towards Memorization-Free Diffusion Models, arxiv:2404.00922.

[5] Ho, Salimans, Classifier-Free Diffusion Guidance, NeurIPS, 2021.

[6] Adomaityte, Defilippis, Loureiro, et al., High-dimensional robust regression under heavy-tailed data: asymptotics and universality, Journal of Statistical Mechanics: Theory and Experiment, 2024.