A general statistical framework for studying asymptotically optimal noise-level allocation in diffusion training and an idealized independent-learner regime, intended to model temporal specialization in neural networks are developed.
Abstract
How should a diffusion model decide which noise levels to train on, and how much? Despite the importance of this choice, current noise schedules are based largely on heuristics or empirical tuning. Here, we develop a general statistical framework for studying asymptotically optimal noise-level allocation in diffusion training. Our first main result concerns the fully coupled regime, where information can spread between different time points. Under convexity or Polyak-Lojasiewicz-type assumptions, we show that the optimized training schedule admits an atomic minimizer, concentrated on finitely many noise levels. Our second main result specializes this framework to an idealized independent-learner regime, intended to model temporal specialization in neural networks. Under an additional feature-noise decoupling condition, a random-matrix analysis leads to an information-theoretic proxy: the decoupled sampling density is proportional to the square root of the generative entropy rate, the rate at which conditional entropy grows along the forward process. We test these predictions in controlled settings where the coupled objective can be optimized directly, including Dirac mixtures, low-dimensional manifolds, and MNIST. In these settings, the optimized schedules are consistently finite-support, while the smooth entropic proxy closely tracks the atomic optimum in neural-network models and breaks down mainly in the fully coupled parametric case, as the theory suggests. We then evaluate the entropic schedule in larger-scale experiments, where full schedule optimization is currently intractable. The results indicate that square-root entropy scheduling can substantially improve training efficiency on discrete domains and remains competitive with standard EDM-style heuristics on continuous images.
How diffusion models circumvent the curse of dimensionality to learn complex distributions over high dimensional spaces from a finite training set, instead of memorizing it, remains a fundamental mystery. To address this, we introduce analytically tractable Bayesian information restricted diffusion (BIRD) models, in which each pixel observes restricted information about noisy data. A BIRD model time-reverses diffusion by inferring which past training sample produced its current restricted observation using the Bayesian posterior. This model class generalizes existing analytical diffusion models that use spatially local information restriction. We show that spatially local BIRD models closely approximate trained diffusion models \textit{early in training}, across different architectures such as UNets and DiTs. Under minimal assumptions on the data distribution, we identify an information-theoretic phase boundary between memorization and generalization in the joint space of amount of training data, time in the reverse generative process, and amount of information restriction: a BIRD model memorizes when the mutual information between its restricted noisy observations and the training data exceeds the log number of training points, and it generalizes otherwise. Experiments across a range of datasets confirm our theoretically predicted location for the transition. We find that generation proceeds near the edge of memorization: both spatially local BIRD models and early-training diffusion models track the memorization-generalization phase boundary by increasingly restricting information over time. Overall, our results reveal a fundamental role for information restriction in generative AI to circumvent the curse of dimensionality.
Henry S. Hunt, M. Kamb, Surya Ganguli· arXiv.org· 1 citation
This work establishes a minimax lower bound under the score-entropy loss, and proposes an MLE-based thresholding estimator that matches this lower bound up to constant and polylogarithmic factors that depend on neighboring density ratios.
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.
Hugo Latourelle-Vigeant, Sinho Chewi, Aram-Alexandre Pooladian et al.· 0 citations
While autoregressive models optimize the exact data likelihood via the chain rule, diffusion models are typically trained with denoising objectives. We develop conservation laws based on generalized extrinsic information transfer (GEXIT) functions for a broad class of memoryless noise processes, showing that the data--model cross-entropy (CE) can be characterized exactly as an integral of local information-theoretic derivatives along the noise path. This yields a unified characterization of the likelihood for discrete and continuous diffusion, with the Gaussian case reducing to the well-known mutual information--minimum mean-square error (I-MMSE) relationship. An immediate implication is a locality property: one can compute the information-theoretic derivatives using only the marginal posteriors along the noise path. As a result, training reduces to learning the marginal posteriors by minimizing the negative log-likelihood. While the conservation law implies that the entropy does not depend on the noise path, finite-capacity denoisers approximate the posteriors with varying accuracy across noise types, leading to differences in performance. We validate these predictions on synthetic Markov sources and standard benchmarks, including text8 and CIFAR-10.
GRAS is simple yet effective: across regulatory DNA and protein design it attains the best training-free reward, outperforming prior training-free methods and matching or surpassing a reward-fine-tuned model, and it remains effective even for non-differentiable rewards.
Using a finite energy message-passing algorithm, it is demonstrated numerically that thermal noise enables effective generalization in the regime of constraint densities where both recovering the teacher and finding a zero temperature solution are computationally hard.
Enrico M. Malatesta, A. Passalacqua, Riccardo Zecchina· arXiv.org· 0 citations
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