Why Diffusion Models Don’t Memorize:  The Role of Implicit Dynamical Regularization in Training
Authors: Tony Bonnaire
Raphaël Urfin
Giulio Biroli
Marc Mezard
Venue: NeurIPS 2025 oral ()
Keywords: Diffusion Models
Deep Learning
Probabilistic Methods
Primary area: deep_learning
TL;DR: Implicit dynamical regularization during training gives diffusion models a generalization window that widens with the training set size, so stopping within this window prevents memorization.

Abstract: Diffusion models have achieved remarkable success across a wide range of generative tasks. A key challenge is understanding the mechanisms that prevent their memorization of training data and allow generalization. In this work, we investigate the role of the training dynamics in the transition from generalization to memorization. Through extensive experiments and theoretical analysis, we identify two distinct timescales: an early time $\tau_\mathrm{gen}$ at which models begin to generate high-quality samples, and a later time $\tau_\mathrm{mem}$ beyond which memorization emerges. Crucially, we find that $\tau_\mathrm{mem}$ increases linearly with the training set size $n$, while $\tau_\mathrm{gen}$ remains constant. This creates a growing window of training times with $n$ where models generalize effectively, despite showing strong memorization if training continues beyond it. It is only when $n$ becomes larger than a model-dependent threshold that overfitting disappears at infinite training times.
These findings reveal a form of implicit dynamical regularization in the training dynamics, which allow to avoid memorization even in highly overparameterized settings. Our results are supported by numerical experiments with standard U-Net architectures on realistic and synthetic  datasets, and by a theoretical analysis using a tractable random features model studied in the high-dimensional limit.

Why Diffusion Models Don't Memorize: The Role of Implicit Dynamical Regularization in Training
Tony Bonnaire
Raphaäl Urfint LPENS Université PSL, Paris raphael. urf in@phns. ens fr
LPENS Université PSL, Paris
tony www.abresppys.....moooooooo ens.
Giulio Biroli
Marc Mézard
LPENS
Department of Computing Sciences
Université PSL,, Paris
Bocconi University, Milano
giulio. www.liiphy...eeeeeooeeeoo entrrssens.ff.
marc maa..meezard@mmioccon.. .it
Abstract
Diffusion models have achieved remarkable success across a wide range of gener- ative tasks. A key challenge is understanding the mechanisms that prevent their memorization of training data and allow gencralization. In this work, we inves- tigate the role of the training dynamics in the transition from generalization memorization. Through extensive experiments and theoretical analysis, we identify distinct timescales: high-quality samples, and later time Tmem beyond which memorization emerges. Crucially, find that meom increases linearly with the training size while Tgen remains constant. This creates growing window of training times with n where models generalize effectively, despite showing strong memorization if train- ing continues beyond it. It is only when n becomes larger than a model-dependent threshold that overfitting disappears at infinite training times. These findings reveal a form of implicit dynamical regularization in the training dynamics, which allow avoid memorization even in highly overparameterized settings. Our results are supported by numerical experiments with standard U-Net architectures on realis- tie and synthetic datasets, and by a theoretical analysis using a tractable random features model studied in the high-ding dineensinnii limit.
early time Tgen which models begin to generate
Introduction
Diffusion Models [DMs, 44, 18, 49, 50] achieve state-of-the-art performance in wide variety of AI tasks such as the generation images [41], audios [58], videos [30], and scientific data [28, 36). This class of generative models, inspired by out-of-equilbrium thermodynamics [44], corresponds to a two-stage process: the first one, called forward, gradually adds noise to data, whereas the second one, called backward, generates new data by denoising Gaussian white noise samples. In DMs, the reverse process typically involves solving a stochastic differential equation (SDE) with a force field called score. However, it is also possible define deterministic transport through an ordinary differential equation (ODE), treating the score as a velocity field, an approach that is for instance followed in flow matching [29].
Understanding the generalization properties of score-based generative methods a central issue in machine learning, and particularly important question is how memorization of the training set
Equal contribution.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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11 mm (p)
Ana dtrming
Architectural Regularization
Suma (8)
Memorization fraction
Number of data n
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Inverse quality
s((x)
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Memorization regime
Dynamical Regularization
regime
gen
Timm
7pez
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Figure 1: Qualitative summary of our contributions. (Left) Illustration of the training dynamics of a diffusion model. Depending on the training time т, we identify three regimes measured by the inverse quality of the generated samples (blue curve) and their memorization fraction (red curve). The generalization regime extends over large window of training times which increases with the training set size On top, we show a one dimensional example of the learned score function during training (orange). The gray line gives the exact empirical score, at a given noise level, while the black dashed line corresponds to the true (population) score. (Right) Phase diagram in the (n, plane illustrating three regimes of diffusion models: Memorization when n is sufficiently small at fixed p. Architectural Regularization for n" (p) (which is model and dataset dependent, discussed in [13, 23]). and Dynamical Regularization, corresponding to a large intermediate generalization regime obtained when the training dynamics is stopped early, i.e. [Tee:: Tmem)
is avoided in practice. A model without regularization achieving training loss only learns the empirical score, and is bound to reproduce samples of the training dataset at the end of the backward process. This memorization regime [27, 5, 4] is empirically observed when the training set small and disappears when increases beyond a model-dependent threshold [22]. Understanding the mechanisms controlling this change of regimes from memorization to generalization is central challenge for both theory and applications, Model regularization and inductive biases imposed by the network architecture were shown to play role [23, 43], as well as dynamical regularization due to the finiteness of the learning rate [56]. However, the regime shift described above is consistently observed even in models where all these regularization mechanisms are present. This suggests that the core mechanism behind the transition from memorization to generalization lies elsewhere. In this work, we demonstrate first through numerical experiments, and then via the theoretical analysis of a simplified model that this transition is driven by an implicit dynamical bias towards generalizing solutions emerging in the training, which allows avoid the memorization phase.
Contributions and theoretical picture. We investigate the dynamics of score learning using gradient descent, both numerically and analytically, and study the generation properties of the score depending the time which the training is stopped. The theoretical picture built from our results and combining several findings from the recent literature is illustrated in Fig. 1. The two main parameters are the size of the training set n and the expressivity of the class of score functions on which one trains the model, characterized by a number of parameters when both n and are large one can identify three main regimes. Given p. if is larger than (p) (which depends on the training set and on the class of scores), the score model is not expressive enough to represent the empirical score associated of the training set. In this regime, even with a very large training time T - co, memorization does not occur because the model is regularized by its architecture and the finite number of parameters. When < n² (p) the model is expressive enough to memorize, and two timescales emerge during training: one, Tgens is the minimum training time required to achieve high-quality data generation; the second, Tmem Tgen. signals when further training induces memorization, and causes the model to
data, and instead provides smooth interpolation, approximately independent

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increasingly reproduce the training samples (left panel). The first timescale, Tgen, is found independent of n, whereas the second, Tmem- grows upproximately linearly with n, thus opening large window of training times during which the model generalizes if early stopped when 7 E Tgen: Tmem] Our results shows that implicit dynamical regularization in training plays a erucial role in score-based generative models, substantially enlarging the generalization regime (see right panel of Fig. 1), and hence allowing to avoid memorization even in highly overparameterized settings. We find that the key mechanism behind the widening gap between Tgen and Tmem is the irregularity of the empirical score at low noise level and large n. In this regime the models used to approximate the score provide a smooth interpolation that remains stable for long period of training times and closely approximates the population score, a behavior likely rooted in the spectral bias of neural networks [38]. Only at very long training times do the dynamics converge to the low lying minimum corresponding to the empirical score, leading to memorization (as illustrated in the ID examples in the left panel of Fig. 1).
The theoretical picture described above is based on our numerical and analytical results, and builds up on previous works, in particular numerical analysis characterizing the memorization nemmrrzztion-gnneralzzatoooo transition [16, 57], analytical works on memorization of DMs [13, 23, 22], and studies on the spectral bias of deep neural networks [38]. Our numerical experiments use a class of scores based on a realistic U-Net [42] trained on downscaled images of the CelebA dataset [31]. By varying and p. we measure the evolution of the sample quality (through FID) and the fraction of memorization during learning, which support the theoretical scenario presented in Fig. 1. Additional experimental results on synthetic data are provided in Supplemental Material (SM, Sects. A and B). On the analytical side, we focus on class of scores constructed from random features and simplified models of data, following [13]. In this setting, the timescales of training dynamics correspond directly to the inverse eigenvalues of the random feature correlation matrix. Leveraging tools from random matrix theory, we compute the spectrum in the limit of large datasets, high-dimensional data, and overparameterized models. This analysis reveals, in fully tractable way, how the theoretical picture of Fig. I emerges within the random feature framework.
Related works.
The memorization transition in DMs has been the subject of several recent empirical investiga- tions [8, 45, 46) which have demonstrated that sate-of-the-ar image DMs - includiig Stable Diffusion and DALL-E - can reproduce non-negligible portion of their training data, indicat- ing a form of memorization. Several additional works [16, 57] examined how this phenomenon is influenced by factors such as data distribution, model configuration, and training procedure, and provide a strong basis for the numerical part of our work.
A series of theoretical studies in the high- sigh--imensional regime have analyzed the memorization- generalization transition during the generative dynamics under the empirical score assumption [5, 1, 52], showing how trajectories are attracted to the training samples. Within this high- dimensional framework, [9, 10, 55, 13] study the score learning for various model classes. In particular, [13] uses Random Feature Neural Network [39]. The authors compute the asymptotic training and test losses for t - oo and relate it to memorization. The theoretical part of our work generalizes this approach to study the role of training dynamics and early stopping in the memorization generalization
cemorizaiionnggneaalizzaio transition.
Recent works have also uncovered complementary sources of implicit regularization explaining how DMs avoid memorization. Architectural biases and limited network capacity were for instance shown to constrain memorization in [23, 22, 4], and finiteness of the learning rate prevents the model from learning the empirical score in [56]. Also related to our analysis. [26, 4] show the beneficial role of early stopping the training dynamics to enhance the generalization.
Finally, previous studies on supervised learning [38, 59], and more recently on DMs [54], have shown that deep neural networks display a rrequency-dependee learning bias towards low frequency functions. This fact plays an important role in the results we present since the empirical score contains a low frequency part that is close to the population score, and a training time to learn the high-frequency part and hence memorize, that we find to scale with has not been studied from this perspective the context of score-based generative methods.
learning speed, and hence
part that is dataset- dependent. To the best of our knowledge, the
*Code available github.com/tboomii//lyyy Difiuusioooiii--MMoe-eee
3

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Setting: generative diffusion and score learning. Standard DMs define a transport from target distribution Pg in Rd to a Gaussian white noise (o, Ia) through a forward process defined as an Ornstein-Uhlenbeck (OU) stochastic differential equation (SDE):
dx = -((xt) dB(t),
where dBB(t) square root of two times a Wiener process. Generation performed by time-reversing the SDE (1) using the score function sxx, () = Vx log (x(x),
dx = (x()) + 2s( t)] dt + dB()),
(2)
where (x) is the probability density time along the forward process, and the noise dB((t is also the square root of two times a Wiener process. As shown in the seminal works [21, 53), s(x, ()
can be obtained by minimizing the score matching loss
((x, 1) = arg min Ex- [IVA +
where 4, = 1- In practice, the optimization problem is restricted to a parametrized class of functions se (x(t), t) defined, for example, by a neural network with parameters 0. The expectation over x is replaced by the empirical average over the training set (n iid samples x" drawn from Po),
C, (xx )==1
1 Σ n
Be- [II Diso (x" (x)())) <11
where - + Vic. The loss in (4) can be minimized with standard optimizers, such as stochastic gradient descent ISGD, 40] Adam [25]. In practice, single model conditioned on the diffusion time is trained by integrating (4) over time [24]. The solution of the minimization of (4) is the so-called empirical score (e.g. [5, 27]), defined as Semp (x, t) Vx log Pemup( (x). with
pemp (x)
xx-x"e-1113
(5)
)2 d/2
This solution is known to inevitably recreate samples of the training set at the end of the generative process (i.e., it perfectly memorizes), unless n grows exponentially with the dimension d [5]. However, this is not the case in many practical applications where memorization is only observed for relatively small values of n, and disappears well before ri becomes exponentially large in d. The empirical minimization performed in practice, within a given class of models and given minimization procedure, does not drive the optimization to the global minimum of (4), but instead to a smoother estimate of the score that is independent of the training set with good generalization properties [22], as the global minimum of (3) would do. Understanding how it is possible, and in particular the role played by the training dynamics to avoid memorization, is the central aim of the present work.
2 Generalization and memorization during training of diffusion models
Data & architecture. We conduct our experiments on the CelebA face dataset [31], which we convert to grayscale downsampled images of size d 32 32, and vary the training set size n from 128 up to 32768. Our score model has a U-Net architecture [42] with three resolution levels and base channel width of W with multipliers 1, and 3 respectively. All our networks are DDPMs [18] trained to predict the injected noise at diffusion time t using SGD with momentum at fixed batch size min 512). The models are all conditioned on i.e. a single model approximates the score at all times, and make use of a standard sinusoidal position embedding [51] that is added to the features of each resolution. More details about the numerical setup can be found SM (Sect. A).
Evaluation metrics. To study the transition from generalization to memorization during training, we monitor the loss (4) during training using a fixed diffusion time 0.01. At various numbers of SGD updates T, we compute the loss on all training examples (training loss) and on a held-out test set of 2048 images (test loss). To characterize the score obtained after a training time т, we assess the originality and quality of samples by generating 10K samples using a DDIM accelerated sampling [47]. We compute (i) the feéchet-Inception Distance [FID, 17] against 10K test samples which we use to identify the generalization time Tgeni and (ii) the fraction of memorized generated

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100
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1.2
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x-400
10 0.0
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Generea Ntightor
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Figure 2: Memorization transition as a function of the training set size n for U-Net score models on CelebA. (Left) FID (solid lines, left axis) and memorization fraction fmem (dashed lines, right axis) against training time T for various n. Inset: normalized memorization fraction (T)m fem (Tmax) with the rescaled time 7/n. (Middle) Training (solid lines) and test (dashed lines) loss with T for several n at fixed t = 0.01. Inset: both losses plotted against T/n. Error bars on the losses are imperceptible. (Right) Generated samples from the model trained with 1024 for 100K or r 1.62M steps, along with their nearest neighbors in the training set.
samples fmem (T) granting access to Tmem- the memorization time. Following previous numerical studies [57, 16]. generated sample x, is considered memorized if
X, a
12
Ex
k
(6)
x, a 2
where at and a are the nearest and second nearest neighbors of x, in the training set in the L2 sense. In what follows, we choose work with k = 1/3 [57, 16], but we checked that varying k to 1/2 or 1/4 does not impact the claims about the scaling. Error bars in the figures correspond to twice the standard deviation over 5 different test sets for FIDs, and noise realizations for Lirain and Lesst- For fmem. we report the 95% CIs on the mean evaluated with 1,000 bootstrap samples.
Role of training set size on the learning dynamics. At fixed model capacity (p 4 x 10 base width W = 32), we investigate how the training set size n impacts the previous metrics. In the left panel of Fig. 2, we first report the FID (solid lines) and fuem (r) (dashed lines) for various All trainings dynamics exhibit two phases. First, the FID quickly decreases to reach a minimum value on timescale Tgen ( 100K) that does not depend n. In the right panel, the generated samples 100K clearly differ from their nearest neighbors in the training set, indicating that the model generalizes correctly. Beyond this time, the FID remains flat. fuem (T) is zero until a later time Tmem after which it increases, clearly signaling the entrance into a memorization regime, as illustrated by the generated samples in the right-most panel of Fig. very close their nearest neighbors. Both the transition time Teem and the value of the final fraction fuem ( (Tmax) (with Tmax being one to four million SGD steps) vary with n. The inset plot shows the normalized memorization fraction fmem (T) (omemm(Tnnn (Tmax) against the rescaled time T/n, making all curves collapse and increase at around r/n zs 300, showing that Teem & n, and demonstrating the existence of a generalization window for 7 E [Tgen, Tmem] that widens linearly with n, as illustrated in the left panel of Fig. 1.
As highlighted in the introduction, memorization in DMs is ultimately driven by the overfitting of the empirical score Smem [x, f). The evolution of Lussin (T) and (T) at fixed / = 0.01 are shown in the middle panel of Fig. for n ranging from 512 to 32768. Initially, the two losses remain nearly indistinguishable, indicating that the learned score Se (x, t) does not depend on the training set. Beyond critical time, Cirain continues to decrease while Liest increases, leading generalization loss whose magnitude depends on n.. As increases, this critical time also increases and, eventually, the training and test loss gap shrinks: for 32768, the test loss remains close to the training loss, even after 11 million SGD steps. The inset shows the evolution of both losses with 7/ 7/n. demonstrating that the overfitting time scales linearly with the training set size n, just like Tiem identified in the left panel. Moreover, there is consistent lag between the overfitting time and Tinm at fixed n, reflecting the additional training required for the model overfit the empirical score sufficiently to reproduce the training samples, and therefore to impact the memorization fraction.
nonzero
5

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Figure 3: Effect of the number of parameters in the U-Net architecture the timescales of the training dynamics. (Left) FID (panels A, B) and normalized memorization fraction (panels C. D) for various n. and W during training. In panels B and D,
Swem mem time is rescaled such that all curves collapse. (Right) (n. phase diagram of generalization vs memorization for U-Nets trained on CelebA. Curves show, for 7 E {Tgen Tegen &rgen): J. the minimal dataset size npp) satisfying fmem = O. The shaded background indicates the memorization- generalization boundary for 7 Tgen-
Memorization is not due to data repetition. We must stress that this delayed memorization with n not due to the mere repetition of training samples, as a first intuition could suggest. In SM Sects. A and B, we show that full-batch updates still yield Tmem T all models have processed each sample equally often, larger n consistently postpone memorization. This confirms that memorization in DMs is driven by fundamental n-dependent change in the loss landscape not by a sample repetition during training.
In other words, even if at fixed
Effect the model capacity. To study more precisely the role of the model capacity on the cewpiiaattionggneeallzaiii transition, we vary the number of parameters by changing the U-Nets base width W E {8, 16, 16,32, 48, 64), resulting a total p E (0.26, 1, 4, 9, 16) x 10 parameters. In the left panel of Fig. 3. we plot both the FID (top row) and the normalized memorization fraction (bottom row) as functions of r for several width W and training set sizes n. Panels A and C demonstrate that higher-capacity networks (larger W) achieve high-quality generation and begin to memorize earlier than smaller ones. Panels B and show that the two characteristic timescales simply scale as Tgen & W-1 and Tiee nW-1 In particular, this implies that, for W 8, the critical training set size ngm (p) at which Tmem = Tgen is approximately independent of p (at least on the limited values of p focused on). When n Tgm (p), the interval [Tgen: Timm] opens up, that early stopping within this window yields high quality samples without memorization. In the right panel of Fig.
display this boundary (solid line) in the (n. p) plane by fixing the training time Tgen- that we identify numerically using the collapse of all FIDs at around W Tgen R 3 x 10 (see panel B), and computing the smallest n such that fmem (7) 0. The resulting solid curve delineates two regimes: below the curve, memorization already starts at Tgeni above the curve, the models generalize perfectly under early stopping. We repeat this experiment for 3rgen and T 8Tgen showing saturation to larger and larger p as increases. Eventually, for + co, we expect these successive boundaries to converge to the architectural regularization threshold n (p), i.e. the point beyond which the network avoids memorization because it is not expressive enough, as found in [13) and highlighted in the right panel of Fig. 1. order to estimate n=*()) (p), we measure for a given the largest m²(7) yielding mmem x O. The curve nu(T) approaches n (p) for large T. We therefore estimate n² (p) by measuring the asymptotic values of nT)), which in practice is reached already at
7 ==
Tmax = 2M updates for the values of W we focus

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3 Training dynamics of a Random Features Network
Notations.
We use bold symbols for vectors and matrices. The L norm of a vector x is denoted by ||x|| = (E, x2)1//2 We write f = O'g) to mean that in the limit n. p - oc, there exists a constant C' such that | < Clgl-
Setting.
We study analytically a model introduced in [13], where the data lie in d dimensions. We parametrize the score with a Random Features Neural Network [RFNN, 39]
A VP
Wx
SA SA
=
a
vd
(7)
An RFNN, illustrated in Fig. 4 (left), is two-layer neural-network whose first layer weights (W E RRxd)) are drawn from a Gaussian distribution and remain frozen while the second layer weights (A E Rdxx)) are learned during training. This model has already served as theoretical framework for studying several behaviors of deep neural network such as the double descent phenomenon [32. 11]. is an clement-wise non-linear activation function. We consider training set of iid samples x" ~ Px for v= = 1,.... n and we focus on the high-dimensional limit d, p, n - oo with the ratios 1/ = p/dd, th n/d kept fixed. We study the training dynamics associated to the minimization of the empirical score matching loss defined in (4) at fixed diffusion time t. This is a simplification compared to practical methods, which use single model for all t. It has been already studied in previous theoretical works [9, 13]. The loss (4) is rescaled by a factor 1/d in order to ensure a finite limit large d. We also study the evolution of the test loss evaluated on test points and the distance to the exact score s(x) = V log Px.
Grest = - [I (xxA)) + + Excore = 7 [|SA (xx) - V log Px]l"]
(8)
where the expectations Ex.e are computed over loss, defined as fgen = frest Lerain. indicates the degree of overfitting in the model while the distance to the exact score Escore measures the quality of the generation it is an upper bound on the Kullback-Leibler divergence between the target and generated distributions [48, 7J. The weights A are updated via gradient descent
Px and È NNO,, Ia)- The generalization
A (++1) A (A)) 1/ Cerain (A (K)),
where 17 is the learning rate. In the high- nigh-dimensional limit, as the learning rate 7 0, and after rescaling time as r = /n/ , the discrete-time dynamics converges to the following continuous-time gradient flow:
( = dV A Cerain (A(T)) = --24, AU 2d, Dd A. A. VT
(10)
VP
with
12
(x) Vd
a (Wx)(() o1
Vd )7.
(x) vd
U= = Eg o
V=
Eg O
(11)
Assumptions. For our analytical results to hold, make the following mathematical assumptions which are standard when studying Random Features [37, 15, 20] namely (i) the activation function admits Hermite polynomial expansion (2x) Σ aa (e,(x); and (ii) the data distribution Ps has zero mean, sub-Gaussian tails and covariance Σ Er [xx"] with bounded spectrum. We assume that the empirical distribution of eigenvalues of Σ converges weakly in the high dimensional limit to a deterministic density (>(A) and that Tr /d converges a finite limit (for a more precise mathematical statement see SM Sect. C.3). Moreover, we make additional assumptions that are not essential to the proofs but which simplify the analysis: (iii) the activation function a verifies 440 E, [=)) 0 for standard Gaussian; and (iv) the second layer A is initialized with zero weights A(T = 0) = 0. In numerical applications, unless specified, we use (2)) = tanh(z) and Px (/ Ia).
7

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Wx
0,008
0.008
Wx
vd
125
100
0.004
0.00
130
0.3
835
0.002
00
0.002
0.000
0.000
W E 13xx
A E Rd
20 40 60
0
20
40 60
Figure 4: (Left) Illustration of an RFNN. (Middle/Right) Spectrum of U. Density p()) from
Theorem 3.1 in the overparameterizee Regime described in Theorem 3.2, with v/ 64, 8/ 8, == 0.01, and p>(A) 6(A - 1). The bulk of the spectrum (orange) is between À Rs 10 and X 45. The histogram shows the eigenvalues from single realization of U at d = 100. Inset: zoom near == (in blue) showing the first bulk P1 and the delta peak at = sy (Right) Same as (Middle), but with pΣ 50(A 0.5) _0(A 1.5). The first bulk in blue remains unchanged, as it depends only xx Tr(E)/ / in both cases, while the second bulk varies with Σ.
1.0
3.0
0.8
2.5
10
10
2.0
0.6
1.5
10
0.4
10
10
10
10
1.0
57mmm
0.2
0.5
0.0
10c
10a
10a
10a
10a
10a 10
16
7
f/
(A)
(B)
(C)
Figure 5: Evolution of the training and test losses for the RENN. (A) Distance the score
Escore against training time for vh = 4, 8, 16, 32.2/ 64, t 0.1 and d = 100. In the inset, the training time is rescaled by Fmem / Amiii (B) Training (solid) and test (dashed) losses for various vh. The inset shows both losses rescaled by Tieen (C) Heatmaps of Leen for 103 (top) and 1 (bottom) as function of tha and v.p. All the curves use Pytorch [35] gradient descent. More numerical details can be found in SM Sect. D.
Emergence of the two timescales during training. We first show in Fig. 5 that the behavior of training and test losses the RF model mirrors the one found realistic cases in Sect. 2, with separation of timescales Tgen and Tnem which increases with n. Equation (10) is linear in A and hence it can solved exactly (see SM). The timescales of the training dynamics are given by the inverse eigenvalues of the p matrix A, A Building on the Gaussian Equivalence Principle [GEP, 14, 15, 33] and the theory linear pencils [6], George et al. (2025) derive a coupled system of equations characterizing the Stieltjes transform of the eigenvalue density p()) of U for isotropic Gaussian data that lie in a D-dimensional subspace with D < d and D O()). We offer an alternative derivation presented in SM for general variance using the replica method [34] heuristic method from the statistical physics of disordered systems yielding the more compact formulation for obtaining the spectrum stated in Theorem 3.1. Before stating the theorem, introduce
b. = [uoo( oxx 'x0u + (A,)), a, = Eu (
ox u + VA,U)
u
(12)
e-lox
of = (( + -oi++// (iv)o(e oxu + VAww]] dee -202
(13)
5² Enjo(T,u)²] - 0.e 0² v² -
(14)

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where x² = 1 r, = -- A, = - 1) and the expectation is over the u, v, w random variables which are independent standard Gaussian AV 1).
Theorem 3.1. Ler (z) = } Tr((U - =Ip)=1, r(z) 1 Tr(E (U - =Ip)-1-E>" -W2 and
5() = Tr(W" (U =Ip) W), with & E C. Let
s(1) = 1/ + q'
(15)
1p Upale 2
f(r, 4)
(16)
ofe 2e
2p q
1 +
Then q(2), r()) and s(2) satisfy the following set of three equations:
/ dAz(())
1
(17)
8(q) + XF(r.9)*
/ (2
X 8(g) + Xr(r. g) Soal
r
(18)
Up - 2) +
1/p
8
0,
(19)
1 atc- 1ee-21pp vr +
9
q²
Vnn
The eigenvalue distribution of U, p(A), can then be obtained using the Sokhotski-Plemelii inversion formula p(A) = lim 1 Im q (e)
We now focus on the asymptotic regime thp: thn >> 1, typical for strongly wver-parameterized models trained on large data sets. In this limit, the spectrum of U can be described analytically by the following Theorem 3.2.
Theorem 3.2 (Informal). Let p denote the spectral density of U.
Regime (overparametrizedd); u,p > Un L.
p(A) =11- Vi
1+ Un
8(2 -s²) + Un
1
p1(A) P2(A).
Up
Regime II (underparamettrzeed) ein > Uip >> 1.
p()) (1- U
1
(A)+
(2(A).
where P1 is atomless measure with support
$t + vz (1 (1- /p///n
ot + 23 1+ //p/// Un
and P2 coincides with the asymptotic eigenvalue bulk density of the population covariance Ü Ex [U]; P2 is independent of tn and its support is on the scale Up. The eigenvectors associated with 5(A - >t) leave both training and test losses unchanged and are therefore irrelevant. In the limit vip Vin the supports of #1 and p2 are respectively on the scales v Vn/ and Upp i.e. they are well separated.
The proofs of both theorems are shown in SM (Sect. C). We recall that training timescales are directly related to eigenvalues À via the relation --1 = u Ae Amin- Theorem 3.2 therefore demonstrates the emergence of the two training timescales Twem and Tgen in the overparametrized regime of the RENN model. They are respectively associated to the measures P1 and P2. which are well separated in regime I, for v > vA, > 1, as shown in Fig. 4.
Generalization: The timescale Tgen on which the first relaxation takes place is associated to the formation of the generalization regime. related to the bulk P2 and or order 1/4, This regime only depends the population covariance Σ of the data and is independent of the specific realization

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of the dataset. On this timescale, which is of order one, both the training Lirann and test Leest losses decrease. The generalization loss Lgen Lusst Liain is zero, and Esooe tends to a value that we find to scale as (177")) with 0.59 numerically (see Fig. 5).
Memorization: The timescale 7mem. on which the second stage of the dynamics takes place, is associated to overfitting and memorization. It is related to the bulk P1. and scales as t Amminn where Amin is the left edge of ft. In the overparameterized regime p > n,, Tmem becomes large and of order /n/A, thus implying scaling of mmem with On this timescale, the training loss decreases while the test loss increases, converging to their respective asymptotic values as computed in [13]. Fig. 5 indeed shows that all training and test curves separate, correspondingly the generalization loss Eeea increases, at a time that scales with d Ammmin as shown in the inset.
As increases, the asymptotic (T - oc) generalization loss Egen decreases, indicating reduced overfitting. For n > n (p(p) p. although some overfitting remains (i.e., fgge > 0), the value of fegen is sensibly reduced, and the model is no longer expressive enough to memorize the training data, as shown in [13]. This regime corresponds to the Architectural Regularization phase in Fig. 1. We show in Fig. 5 (panel C) how the generalization loss Egen varies in the (n, p) plane depending on the time r at which training is stopped. In agreement with the above results, we find that the pnteraiizationmmmoizztiooo transition line depends on T and moves upward for larger values of T, similarly to the numerical results exposed in Fig. 3 and the illustration in Fig. 1.
4 Conclusions
We have shown that the training dynamies of neural network-based score functions display form of implicit regularization that prevents memorization even in highly overparameterized diffusion models. Specifically, we have identified two well-separated timescales in the learning: Tgen- at which models begins to generate high-quality, novel samples, and Tmem. beyond which they start to memorize the training data. The gap between these timescales grows with the size of the training set, leading to a broad window where early stopped models generate novel samples of high-quality We have demonstrated that this phenomenon happens in realistic settings, for controlled synthetic data, and in analytically tractable models. Although our analysis focuses on DMs, the underlying score-learning mechanism we uncover is common to all score-based generative models such as stochastic interpolants [3] or flow matching [29]; we therefore expect our results to generalize to this broader class.
I Limitations and future works.
While we derived our results under SGD optimization, most DMs are trained in practice with Adam [25]. In SM Sects. A.3 and D, we show that the two key timescales still arise using Adam, although with much fewer optimization steps. Studying how different optimizers shift these timescales would be valuable for practical usage.
All experiments in Sect. 2 are conducted with unconditional DMs. We additionally verify in SM Sect. B, using a toy Gaussian mixture dataset and classifier free guidance [19], that the same scaling of Tmem with n holds in the conditional settings. Understanding precisely how the absolute timescales Tiem and Tgen depend on the conditioning remains an open question.
Our numerical experiments cover range of p between 1M and 16M Exploring a wider range essential map the full (n.p) phase diagram sketched in Fig. and understand the precise effect of expressivity on dynamical regularization.
Finally, our theoretical analysis rely on well-controlled data and score models that reproduce the core effects. Extending these analytical frameworks to richer data distributions (such as Gaussian mixtures or data from the hidden manifold model) and structured architectures would be valuable to further characterize the implicit dynamical regularization of training fcort-functions, In particular investigating how heavy-tailed data distribution [2] affect the picture described here could be valuable.
Although DMs trained on large and diverse datasets likely avoid the memorization regime we study here, some industrial models were shown to exhibit partial memorization [8, 45]. Our results provide practical guidelines (early-stopping, control the network capacity) train DMs robustly and hence avoid memorization, which can be especially helpful in data-scarce domains (e.g.. physical sciences).
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Acknowledgments and Disclosure of Funding
The authors thank Valentin De Bortoli for initial motivating discussions on memorization- generalization transitions. RU thanks Beatrice Achilli, Jérome Garnier-Brun Carlo Lucibello and Enrico Ventura for insightful discussions. RU is grateful to Boeconi University for its hospitality during his stay. during which part of this work was conducted. This work was performed using HPC resources from GENCI-IDRE (Grant (22--A0110016199) GB acknowledges support from the French government under the management of the Agence Nationale PR[AI]RIE- -SAII---SIII (ANR- 23-IACLL-0008). MM acknowledges the support of the PNRR-PE-AI FAIR project funded by the NextGeneration EU program. After completing this work, we became aware that A. Favero, A. Sclocchi, and M. Wyart [12] had also been investigating the memorization cammoriatioongennraaizaaioo transition from similar perspective.
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Reviews:
[{"review_id": "T28FrBAmcE", "reviewer": "Reviewer_f3Fk", "summary": "This work investigates how diffusion models transition from generalization to memorization. The authors identify two training timescales: an early time $\\tau_{gen}$ when models begin generating new samples, and a later time $\\tau_{mem}$, when the model starts to memorize training data. The authors claim that the desired generalization behavior of diffusion models occur when the training time is between $\\tau_{gen}$ and $\\tau_{mem}$. The authors provide experimental evidence and theoretical analysis in the setting of a random feature network.", "questions": "Is there a way the findings and the insights of this paper may guide us to change something with the current diffusion model practice?", "limitations": "yes", "rating": 5, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 4, "originality": 4, "strengths_and_weaknesses": "Strength\n - The perspective that the model achieves generalization and then eventually memorization as the training progresses is quite compelling. To the best of my knowledge, this is a novel perspective and the experimental evidence seems sound.\n - The characterization that $\\tau_{gen}$ is roughly independent of $N$ while $\\tau_{mem}$ is proportional to $N$ is a non-obvious and interesting observation.\n - The theoretical analysis with random feature network is convincing.\n\nWeakness\n - Although I really appreciate the conceptual insight that this work offers, I believe that all diffusion models in practice never reach $\\tau_{mem}$ or are within the architectural regularization regime. Therefore, the present theory does not inform us as to what to do differently.", "was_revised": true, "final_justification": "The main finding of this work is novel and interesting. With the solid experimental and theoretical evidence, the paper provides clear value.", "rebuttal": "We thank you for the very supportive report.\n\n> **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?**\n\nWe 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.\nIn 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.\n\n---\n\n[1] Carlini, Hayes, Nasir, et al., Extracting Training Data from Diffusion Models, USENIX, 2023.\n\n[2] Somepalli, Singla, Goldblum, et al., Understanding and Mitigating Copying in Diffusion Models, NeurIPS, 2023."}, {"review_id": "gKsvuYHWN6", "reviewer": "Reviewer_hfaQ", "summary": "This paper investigates the generalization ability of diffusion model, and identifies two crucial time points: an early time when models begin to generate high-quality samples (generation time), and a later time when memorization emerges (memorization time). The generation time is a constant and the memorization time scales with the data size. The findings reveal a form of implicit dynamical regularization, and backed up with both theoretical and empirical investigations.", "questions": "It seems the results are restrictive to SGD. Is there a similar phenomena for other gradient based training algorithms with momentum terms?\n\nAre the results applicable to conditional diffusion models?\n\nIs the result applicable on datasets with long-tail distributions? How does the identified implicit dynamic regularization effect affect the learning from samples in long-tail?", "limitations": "yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "The paper is well-written in genaral and easy to follow. This work contributes to the theoretical understanding of generalization performance of diffusion models. I didn't go through the proofs, but the result is sound. The theoreical investigation is based on a random feature simplification. Could the authors provide more elaboration on how that's closely related to the more practical models or what's used in experiments?\n\nMore discussions on how such findings help design better training strategies for diffusion models would enhance the significance of this work.", "was_revised": true, "final_justification": "Most of my concerns are addressed, I therefore increase my score and recommend for acceptance.", "rebuttal": "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.\n\n> **Discussions about the practical impact of our work**\n\nSeveral 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.\n\n> **Generalization to other optimization algorithms with momentum terms**\n\nIn 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.\n\n> **Are the results applicable to conditional diffusion models?**\n\nThank 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.\n\nTo 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.\n\n> **Is the result applicable on datasets with long-tail distributions?**\n\nThank 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.\n\nA 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. \n\n----\n\n[1] Carlini, Hayes, Nasir, et al., Extracting Training Data from Diffusion Models, *USENIX*, 2023.\n\n[2] Somepalli, Singla, Goldblum, et al., Understanding and Mitigating Copying in Diffusion Models, *NeurIPS*, 2023.\n\n[3] Wen, Liu, Chen, et al., Detecting, explaining and mitigating memorization in Diffusion Models, *ICLR*, 2024.\n\n[4] Chen, Yu, Xu, Towards Memorization-Free Diffusion Models, arxiv:2404.00922.\n\n[5] Ho, Salimans, Classifier-Free Diffusion Guidance, *NeurIPS*, 2021.\n\n[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."}, {"review_id": "C4RWgFNYon", "reviewer": "Reviewer_xUfF", "summary": "This article investigates two important timescales for score-based diffusion models: an early generalization time $\\tau_{\\mathrm{gen}}$, before which high-quality generation occurs, and a later memorization time $\\tau_{\\mathrm{mem}}$, after which the generative ability of the diffusion model becomes weakened or limited as the dynamics begin to dominate. The transition regime between these two timescales, related to the so-called dynamical regularization effect, is shown to scale linearly with the training set size $n$, i.e., $\\mathcal{O}(n)$. Furthermore, an architectural regularization phase is identified and linked to the expressivity of the underlying neural network.", "questions": "Question:\nIs there an explicit estimate for the two timescales $\\tau_{\\mathrm{gen}}$ and $\\tau_{\\mathrm{mem}}$? How do these timescales depend on the spectrum of $\\mathbf{U}$ as described in Theorems 3.1 and 3.2?", "limitations": "Yes.", "rating": 5, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 4, "originality": 4, "strengths_and_weaknesses": "Strengths:\nThe article is clearly written and well organized. It provides a valuable framework for studying one of the most important aspects of diffusion models. The theoretical analysis leverages connections to random matrix theory and spectral analysis, offering a powerful mathematical tool for researchers in the field.\n\nWeakness:\nThe diffusion model considered in the article is linear, with constant drift and diffusion coefficients (see equation (2)), and the analysis focuses only on one-layer neural network learning (see equation (8)). Although this setup serves well to demonstrate the phenomenon as a toy model, it is limited in capturing nonlinear dynamical effects and in analyzing deep neural networks. Admittedly, extending the analysis to such settings would be a highly challenging task.", "was_revised": true, "final_justification": "The paper is well written and makes strong theoretical contributions. The authors provided clear and satisfactory responses to reviewer concerns.", "rebuttal": "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.\n\n> **Constant drift and diffusion coefficients in the diffusion model and one-layer neural network for the analytical part.**\n\nWe 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.  \nThe 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\n$$\n\\mathrm{d} \\boldsymbol{\\mathrm{x}} = -\\frac{1}{2} \\beta(t) \\boldsymbol{\\mathrm{x}} \\mathrm{d}t + \\sqrt{\\beta(t)} \\mathrm{d}\\boldsymbol{B},\n$$\nwhere $\\beta(t)$ 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. \n\nWe 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, \nthe \"Neural Tangent Kernel\" scaling limit of deep neural networks can be mapped to a specific Random Feature model [1,2]. \nTherefore, 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. \nAfter the completion of this work we started to work on this project. Results will be presented in a forthcoming publication. \n\nWe will clarify all these aspects in the final version of the paper to highlight the broad applicability of our results.\n\n> **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?**\n\nAs 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:\n$$\n\\tau_{1} = \\frac{\\psi_p}{\\Delta_t \\min[\\mathrm{Supp}(\\rho_{1})]}\\ \\ ; \\ \\ \\tau_{2} = \\frac{\\psi_p}{\\Delta_t \\min[\\mathrm{Supp}(\\rho_{2})]}\n$$\n\nFurthermore, 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:\n$$\n\\tau_{\\mathrm{gen}} = \\tau_1 \\quad \\text{and} \\quad \\tau_{\\mathrm{mem}} = \\tau_2.\n$$\n\n----\n\n[1] Jacot, Franck, Clément, Neural tangent kernel: Convergence and generalization in neural networks, NeurIPS, 2018.\n\n[2] Chizat, Oyallon, Bach, On lazy training in differentiable programming, NeurIPS, 2019."}, {"review_id": "SoocEMtcRl", "reviewer": "Reviewer_piNC", "summary": "This work examines the role of early stopping in preventing memorization in diffusion models. They observe that when the number of samples $n$ are increased, the so called window of generalization, the stopping times at which generalization turns into memorization, scales with $n$. They verify this in a toy setting with Celeb A data, and verify this scaling theoretically for random Gaussian data in the random feature model for the score function, deriving results in the high-dimensional limit using the replica method based on previous work.", "questions": "Questions\n\n1. Is the phenomenon you observe unique to training diffusion models? Or does it also occur when you simply want to train a model to memorize given input data? It would be great to explicitly comment on this in the manuscript.\n2. Fig. 3 right: Generally, this plot is rather confusing, as the visualization suggests that one should to match the two lines to the architectural and dynamical regularization in the theoretical intro Figure 1. But it seems that since you do not have n large / p small enough, the architectural bias curve does not show up. Can you clarify, and edit the figure so that it is not suggesting direct correspondence between theory and experiment lines? (or correct my understanding, if this is the intended case!)\n3. Paragraph L.189: It generally seems that experiments on larger p/n would help support your results, you just state your expectation and refer it to future work, yet it seems that it would underline a crucial point about architectural regularization you make in the introduction and that would not be a substantially new direction of study.\n4. Optionally, it could be nice to make the memorization scores more tangible by showing images of the closest two samples used in the metric for a random subset of training samples - long with the values of the scores - similar to the FID in the Appendix.\n5. Can you clarify whether the assumptions you make in L.239 are mainly technical or due to fundamental limitations? (Similarly for the frozen first layer weights?)\n6. Is the code available only on request, or will it be published?\n\nSome small comments\n1. L.45 seems like the first time where you introduce $\\tau$ - for clarity it could be useful to distinguish it from the time of the diffusion process $t$ and specifiy whether you mean training budget in an online setting, or epochs of the optimization process.\n2. L.41 implicit bias — it would help to spell out what this bias is biasing towards (generalizing instead of memorizing solutions?)\n3. Personally I find the term “dynamical regularization” slightly misleading, as my first thought was that the regularization would be dynamical, i.e. adaptive. On second thought it had to do with the diffusion time dimensions. Both interpretations were wrong. It could be helpful to say very early in the introduction that early stopping is an example, which both indicates that you mean training dynamics and rules out the diffusion itself.\n4. L.152 it could be useful to mention explicitly that the FID is used to measure generalization.\n5. Do you have an understanding on how sensitive your results are to your choice of k=1/3 (6)\n6. Fig. 3 right: Can you add markers to the phase diagram, so that is clear what the measured datapoints were? Which W is used here? All of them? \n7. L.227 From context clear but not easy to parse: Do you rescale time or the loss?\n8. In the caption of Fig. 6 could you clarify whether the lines are theoretical or empirical.\n9. L.229 it might be worth clarifying that the score is not a function which depends on time, but that the time is a property of the input, i.e. write $s(\\mathbf x_t)$.\n10. Can you clarify in Figure 5 whether we are in Regime I or II, and what can be identified as $\\rho_1$ or $\\rho_2$? Or if it is an misunderstanding, why this distribution is not related to the forms in Thm. 3.2?", "limitations": "They are openly adressed by the authors in a separate section and discussed in weaknesses in this review.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths\n1. The exposition of the problem is well-motivated and the manuscript is clearly written.\n2. The metrics selected for memorization and generalization in diffusion models are intuitive and easy to compute and interprete.\n3. The theoretical results are communicated clearly and extend existing results.\n\nWeakness\n1. As mentioned by the authors, larger parameter regimes for the models would be desirable, and they propose this as future work. However, it seems important to the empirical verification of the phase diagram in Fig. 3 to verify these regimes as well. It would be useful to understand if there are deeper reasons why these experiments were not included.\n2. It is not entirely clear how much of this analysis is specific to the score function in diffusion models - or if the phenomenon of first generalizing then memorizing is something that is more generally observed in random feature models. \n3. In the context of the score function, getting the score correct at different diffusion times $t$ is weighted the same, but errors could accucmulate on sampling new items. Yet, in the experiments with Celeb, the final generated samples are compared. It is unclear to me in how far these measurements are comparable. \n4. It is unclear whether the theory that holds for GD is inidicative of adaptive optimizers. For the empirical results on Celeb this could be tested for at least a smaller fraction of hyperparameter values to obtain an idication.", "was_revised": true, "final_justification": "All questions I had were adressed and with the extra page, I believe that the authors will be able to incorporate them in the given space.", "rebuttal": "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.\n\n## Main comments\n\n> **About larger parameters regime for Fig. 3**\n\nWe 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).\n\n> **About Fig. 3 (right) and the architectural regularization**\n\nWe 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.\n\n> **About generalization to adaptive optimizers**\n\nIn 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.\n\n> **Is the phenomenon unique to diffusion models?**\n\nThank 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.\n\n> **Making memorization scores more tangible**\n\nWe 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).\n\n> **About the assumptions of L239**\n\nWe 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 $$L^2(e^{-x^2/2}dx)=\\{f :\\int \\mathrm{d} x  \\lvert f(x)\\rvert^2e^{-x^2/2}<+\\infty\\},$$\nand 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 $\\tilde{\\sigma}=\\sigma-\\mu_0$.\n\nFocusing 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].\nWe 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]. \nTherefore, 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. \nAfter the completion of this work we started to work on this project and the associated results will be presented in a forthcoming publication.\n\nWe will clarify the nature and importance of each assumption in the final version of the paper.\n\n> **Availability of the code**\n\nThank 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.\n\n## Smaller comments\n\n1. $\\tau$ 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.\n\n2. We will make this sentence clearer to avoid any confusion.\n\n3. 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.\n\n4. Thank you, we now specify that the evolution of the FID is used to actually measure $\\tau_\\mathrm{gen}$ in the numerical experiments.\n\n5. 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 $\\tau_\\mathrm{mem}$.\n\n6. 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.\n\n7. 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.\n\n8. 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.\n\n9. 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.\n\n\n10. 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.\n----\n\n[1] Albergo, Boffi, Vanden-Eijnden, arxiv: 2303.08797, 2023.\n\n[2] Lipman, Chen, Ben-Hamu, et al., *ICLR*, 2023.\n\n[3] Montanari, Urbani, arxiv:2502.21269, 2025\n\n[4] Georges, Veiga, Macris, arxiv:2502.00336, 2025.\n\n[5] Rahimi, Recht, *Neurips*, 2007.\n\n[6] Mei, Montanari, *Communications on Pure and Applied Mathematics*,2019.\n\n[7] d’Ascoli, Refinetti, Biroli, et al., *ICML*, 2020.\n\n[8] Jacot, Gabriel, Hongler, *NeurIPS*, 2018.\n\n[9] Chizat, Oyallon, Bach. *NeurIPS*, 2019.\n\n[10] Yoon, Choi, Kwon, et al., *ICML*, 2023.\n\n[11] Gu, Du, Pang, et al., *Transactions on Machine Learning Research*, 2025.\n\n[12] Cui, Krzakala, Vanden-Eijnden, et al., *ICLR*, 2025."}]

Author rebuttal:

Meta-review: This paper makes a significant and cohesive contribution to the theoretical and empirical understanding of generalization and memorization in diffusion models, which are well acknowledged by the reviewers (in both pre-rebuttal and post-rebuttal phases). Empirically, through rigorous and well-designed experiments on the CelebA dataset, the authors clearly demonstrate the emergence of two distinct training timescales: an early timescale $\tau_{\text{gen}}$ for achieving high-quality generation, which remains constant with dataset size $n$, and a later memorization timescale $\tau_{\text{mem}}$, which scales linearly with $n$. This creates an expanding window for effective generalization, a finding that is robustly validated across varying model capacities and supported by comprehensive metrics including FID, memorization fraction, and train/test loss dynamics. The empirical findings benefit further study of the memorization effect of the diffusion model.

Theoretically, the authors provide a complementary and equally rigorous analysis using a tractable random features model, deriving the spectral properties of the feature correlation matrix in the high-dimensional limit. By leveraging tools from random matrix theory, they formally link the eigenvalue distribution to the separation of timescales, confirming the linear scaling of $\tau_{\text{mem}}$ with $n$ and elucidating the role of model and data complexity ratios ($\psi_p$, $\psi_n$). The synergy between clear empirical demonstrations and a solid theoretical foundation offers profound insights into the mechanisms of implicit dynamical regularization. 

Overall, the paper is exceptionally well-written and easy to follow, the rebuttal thoroughly addressed all reviewer concerns, and the findings are of fundamental importance to the machine learning community. Thus, I would recommend an oral presentation.
Decision: Accept (oral)