Generalization, memorization, and overfitting for diffusion models trained in the lazy high-dimensional regime

• URL: http://arxiv.org/abs/2608.23938v1
• Date: Tue, 25 Aug 2026 00:54:59 GMT
• Title: Generalization, memorization, and overfitting for diffusion models trained in the lazy high-dimensional regime
• Abstract summary: We develop a generative counterpart to the theory of benign overfitting and algorithmic regularization for neural networks.<n>We derive exact risk trajectories under gradient flow training.<n>We analyze how these estimators combine along the reverse-time SDE and characterize the distribution of the resulting samples.
• Score: 17.718748692214508
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Modern score-based generative models have achieved remarkable empirical success in high-dimensional tasks such as image, audio, and video synthesis. These models reduce distribution learning to a sequence of regression problems that, if solved exactly on finite data, would ultimately reproduce the training samples. Their ability to generalize must therefore arise from the implicit or explicit regularization during training. In this work, we develop a generative counterpart to the theory of benign overfitting and algorithmic regularization for overparameterized neural networks in the supervised lazy-training regime. We study denoising score matching in a vector-valued reproducing kernel Hilbert space with an inner-product kernel. In the proportional high-dimensional regime $n\asymp d$, we derive exact risk trajectories under gradient flow training. These trajectories exhibit three phases governed by qualitatively distinct estimators: a spectral estimator that generalizes, a pure-noise score with localized peaks that interpolate the training objective, and an empirical Bayes estimator that memorizes the data. We then analyze how these estimators combine along the reverse-time SDE and characterize the distribution of the resulting samples. The analysis reveals familiar mechanisms from supervised learning, including kernel linearization and self-induced regularization from the nonlinear part of the kernel, but also reveals a distinct phenomenology specific to generative modeling.

Related papers

• A Theoretical Analysis of Memory and Overfitting Phenomena in Stochastic Interpolation Models [28.505356058958125]
  We show that, in the continuous-time oracle setting, both deterministic estimation and generation processes recover training samples.<n>Under discretization, generated samples remain centered around training samples, with deviations controlled by the step size.
  arXiv  Detail & Related papers  (2026-06-07T10:14:07Z)
• On Interpolation Formulas Describing Neural Network Generalization [8.154667543705687]
  We introduce a gradient kernel that extends the deterministic version via a continuous-time diffusion.<n>It reveals that training samples contribute through loss-dependent weights and gradient alignment along the trajectory.<n>We visualize then link the generalization error to the null space of the integral operator induced by the gradient kernel.
  arXiv  Detail & Related papers  (2026-03-14T10:03:29Z)
• Inference-Time Scaling of Diffusion Language Models with Particle Gibbs Sampling [70.8832906871441]
  We study how to steer generation toward desired rewards without retraining the models.<n>Prior methods typically resample or filter within a single denoising trajectory, optimizing rewards step-by-step without trajectory-level refinement.<n>We introduce particle Gibbs sampling for diffusion language models (PG-DLM), a novel inference-time algorithm enabling trajectory-level refinement while preserving generation perplexity.
  arXiv  Detail & Related papers  (2025-07-11T08:00:47Z)
• An Analytical Characterization of Sloppiness in Neural Networks: Insights from Linear Models [18.99511760351873]
  Recent experiments have shown that training trajectories of multiple deep neural networks evolve on a remarkably low-dimensional "hyper-ribbon-like" manifold.<n>Inspired by the similarities in the training trajectories of deep networks and linear networks, we analytically characterize this phenomenon for the latter.<n>We show that the geometry of this low-dimensional manifold is controlled by (i) the decay rate of the eigenvalues of the input correlation matrix of the training data, (ii) the relative scale of the ground-truth output to the weights at the beginning of training, and (iii) the number of steps of gradient descent.
  arXiv  Detail & Related papers  (2025-05-13T19:20:19Z)
• Memorization and Regularization in Generative Diffusion Models [5.128303432235475]
  Diffusion models have emerged as a powerful framework for generative modeling.<n>The analysis highlights the need for regularization to avoid reproducing the analytically tractable minimizer.<n>Experiments are evaluated in the context of memorization, and directions for future development of regularization are highlighted.
  arXiv  Detail & Related papers  (2025-01-27T05:17:06Z)
• Scaling and renormalization in high-dimensional regression [72.59731158970894]
  We present a unifying perspective on recent results on ridge regression.<n>We use the basic tools of random matrix theory and free probability, aimed at readers with backgrounds in physics and deep learning.<n>Our results extend and provide a unifying perspective on earlier models of scaling laws.
  arXiv  Detail & Related papers  (2024-05-01T15:59:00Z)
• Towards Theoretical Understandings of Self-Consuming Generative Models [56.84592466204185]
  This paper tackles the emerging challenge of training generative models within a self-consuming loop. We construct a theoretical framework to rigorously evaluate how this training procedure impacts the data distributions learned by future models. We present results for kernel density estimation, delivering nuanced insights such as the impact of mixed data training on error propagation.
  arXiv  Detail & Related papers  (2024-02-19T02:08:09Z)
• Capturing dynamical correlations using implicit neural representations [85.66456606776552]
  We develop an artificial intelligence framework which combines a neural network trained to mimic simulated data from a model Hamiltonian with automatic differentiation to recover unknown parameters from experimental data. In doing so, we illustrate the ability to build and train a differentiable model only once, which then can be applied in real-time to multi-dimensional scattering data.
  arXiv  Detail & Related papers  (2023-04-08T07:55:36Z)
• Model, sample, and epoch-wise descents: exact solution of gradient flow in the random feature model [16.067228939231047]
  We analyze the whole temporal behavior of the generalization and training errors under gradient flow. We show that in the limit of large system size the full time-evolution path of both errors can be calculated analytically. Our techniques are based on Cauchy complex integral representations of the errors together with recent random matrix methods based on linear pencils.
  arXiv  Detail & Related papers  (2021-10-22T14:25:54Z)
• Automatic Recall Machines: Internal Replay, Continual Learning and the Brain [104.38824285741248]
  Replay in neural networks involves training on sequential data with memorized samples, which counteracts forgetting of previous behavior caused by non-stationarity. We present a method where these auxiliary samples are generated on the fly, given only the model that is being trained for the assessed objective. Instead the implicit memory of learned samples within the assessed model itself is exploited.
  arXiv  Detail & Related papers  (2020-06-22T15:07:06Z)
• Kernel and Rich Regimes in Overparametrized Models [69.40899443842443]
  We show that gradient descent on overparametrized multilayer networks can induce rich implicit biases that are not RKHS norms. We also demonstrate this transition empirically for more complex matrix factorization models and multilayer non-linear networks.
  arXiv  Detail & Related papers  (2020-02-20T15:43:02Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.