Neural Network Generalization

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Overview

The field studies how neural networks move beyond fitting their training examples, including when overparameterized models can memorize data yet still perform well on new examples or shifted distributions. Theoretical approaches use Rademacher complexity, PAC-Bayes bounds, algorithmic stability, kernel and mean-field analyses, and high-dimensional statistics to relate model structure and training to test error. Empirical work examines implicit regularization from optimizers and architectures, representation learning, double descent, grokking, and robustness across settings such as images, language, graphs, and scientific models. A central open tension is that bounds and measures such as flatness do not reliably predict performance in every regime, which depends strongly on data structure, learned features, and distribution shift.

Latest papers

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311 papers35 papers in the last four weeks, up 35% on the four weeks before. 0.2% of all new papers.

Feb 23, 2026

Grokking Finite-Dimensional Algebra

This paper investigates the grokking phenomenon, which refers to the sudden transition from a long memorization to generalization observed during neural networks training, in the context of learning multiplication in finite-dimensional algebras (FDA). While prior work on grokking has focused mainly on group operations, we extend the analysis to more general algebraic structures, including non-associative, non-commutative, and non-unital algebras. We show that learning group operations is a special case of learning FDA, and that learning multiplication in FDA amounts to learning a bilinear product specified by the algebra's structure tensor. For algebras over the reals, we connect the learning problem to matrix factorization with an implicit low-rank bias, and for algebras over finite fields, we show that grokking emerges naturally as models must learn discrete representations of algebraic elements. This leads us to experimentally investigate the following core questions: (i) how do algebraic properties such as commutativity, associativity, and unitality influence both the emergence and timing of grokking, (ii) how structural properties of the structure tensor of the FDA, such as sparsity and rank, influence generalization, and (iii) to what extent generalization correlates with the model learning latent embeddings aligned with the algebra's representation. Our work provides a unified framework for grokking across algebraic structures and new insights into how mathematical structure governs neural network generalization dynamics.

Pascal Jr Tikeng Notsawo, Guillaume Dumas, Guillaume Rabusseau

Université de Montréal · Mila, Montréal · CHU Sainte-Justine Research Center, Montréal +1

GrokkingNeural Network GeneralizationNeural Network Training Dynamics
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Feb 22, 2026

Mapping Networks

The escalating parameter counts in modern deep learning models pose a fundamental challenge to efficient training and resolution of overfitting. We address this by introducing the Mapping Networks which replace the high dimensional weight space by a compact, trainable latent vector based on the hypothesis that the trained parameters of large networks reside on smooth, low-dimensional manifolds. Henceforth, the Mapping Theorem enforced by a dedicated Mapping Loss, shows the existence of a mapping from this latent space to the target weight space both theoretically and in practice. Mapping Networks significantly reduce overfitting and achieve comparable to better performance than target network across complex vision and sequence tasks, including Image Classification, Deepfake Detection etc, with 99.5%\mathbf{99.5\%}, i.e., around 500×500\times reduction in trainable parameters.

Lord Sen, Shyamapada Mukherjee

National Institute of Technology Rourkela, Odisha, India

Weight-Space LearningNeural Network CompressionNeural Network GeneralizationHypernetworks
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Feb 19, 2026

Early-Warning Signals of Grokking via Loss-Landscape Geometry

Grokking -- the abrupt transition from memorization to generalization after prolonged training -- has been linked to confinement on low-dimensional execution manifolds in modular arithmetic. Whether this mechanism extends beyond arithmetic remains open. We study two sequence-learning benchmarks: SCAN compositional generalization and Dyck-1 depth prediction. Across both tasks and a wide range of learning rates, the commutator defect -- a curvature measure derived from non-commuting gradient updates -- rises well before generalization, with lead times following a superlinear power law (alpha approximately 1.18 for SCAN, approximately 1.13 for Dyck), consistent with prior results on modular arithmetic. Weight-space PCA reveals that spectral concentration is not a universal precursor; the commutator defect is. Causal interventions demonstrate a mechanistic role: amplifying non-commutativity accelerates grokking (roughly 32% on SCAN, roughly 50% on Dyck), while suppressing orthogonal gradient flow delays or prevents it. The three task families form a spectrum of causal sensitivity -- modular arithmetic is rigid, Dyck is responsive, SCAN is intermediate -- yet suppression delays or prevents grokking in all cases, establishing necessity as a universal finding. These results identify the commutator defect as a robust, architecture-agnostic, causally implicated early-warning signal for delayed generalization in transformers.

Yongzhong Xu

GrokkingNeural Network GeneralizationLoss Landscape Geometry
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Feb 18, 2026

On the Mechanism and Dynamics of Modular Addition: Fourier Features, Lottery Ticket, and Grokking

We present a comprehensive analysis of how two-layer neural networks learn features to solve the modular addition task. Our work provides a full mechanistic interpretation of the learned model and a theoretical explanation of its training dynamics. While prior work has identified that individual neurons learn single-frequency Fourier features and phase alignment, it does not fully explain how these features combine into a global solution. We bridge this gap by formalizing a diversification condition that emerges during training when overparametrized, consisting of two parts: phase symmetry and frequency diversification. We prove that these properties allow the network to collectively approximate a flawed indicator function on the correct logic for the modular addition task. While individual neurons produce noisy signals, the phase symmetry enables a majority-voting scheme that cancels out noise, allowing the network to robustly identify the correct sum. Furthermore, we explain the emergence of these features under random initialization via a lottery ticket mechanism. Our gradient flow analysis proves that frequencies compete within each neuron, with the "winner" determined by its initial spectral magnitude and phase alignment. From a technical standpoint, we provide a rigorous characterization of the layer-wise phase coupling dynamics and formalize the competitive landscape using the ODE comparison lemma. Finally, we use these insights to demystify grokking, characterizing it as a three-stage process involving memorization followed by two generalization phases, driven by the competition between loss minimization and weight decay.

Jianliang He, Leda Wang, Siyu Chen +1

Department of Statistics and Data Science, Yale University

Lottery Ticket HypothesisModular AdditionNeural Network Training DynamicsFrequency-Domain Feature LearningNeural Network GeneralizationGrokking
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Feb 18, 2026

Conjugate Learning Theory: Uncovering the Mechanisms of Trainability and Generalization in Deep Neural Networks

In this work, we propose a notion of practical learnability grounded in finite sample settings, and develop a conjugate learning theoretical framework based on convex conjugate duality to characterize this learnability property. Building on this foundation, we demonstrate that training deep neural networks (DNNs) with mini-batch stochastic gradient descent (SGD) achieves global optima of empirical risk by jointly controlling the extreme eigenvalues of a structure matrix and the gradient energy, and we establish a corresponding convergence theorem. We further elucidate the impact of batch size and model architecture (including depth, parameter count, sparsity, skip connections, and other characteristics) on non-convex optimization. Additionally, we derive a model-agnostic lower bound for the achievable empirical risk, theoretically demonstrating that data determines the fundamental limit of trainability. On the generalization front, we derive deterministic and probabilistic bounds on generalization error based on generalized conditional entropy measures. The former explicitly delineates the range of generalization error, while the latter characterizes the distribution of generalization error relative to the deterministic bounds under independent and identically distributed (i.i.d.) sampling conditions. Furthermore, these bounds explicitly quantify the influence of three key factors: (i) information loss induced by irreversibility in the model, (ii) the maximum attainable loss value, and (iii) the generalized conditional entropy of features with respect to labels. Moreover, they offer a unified theoretical lens for understanding the roles of regularization, irreversible transformations, and network depth in shaping the generalization behavior of deep neural networks. Extensive experiments validate all theoretical predictions, confirming the framework's correctness and consistency.

Binchuan Qi

Tongji University, Siping Street, 200092 Shanghai, China · Zhejiang Yuying College of Vocational Technology, Sihao Street, Hangzhou, 310018, Zhejiang, China

Neural Network GeneralizationNeural Network OptimizationInformation-Theoretic Generalization BoundsEmpirical Risk MinimizationStochastic Gradient Descent
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Feb 17, 2026

A unified theory of feature learning in RNNs and DNNs

Recurrent and deep neural networks (RNNs/DNNs) are cornerstone architectures in machine learning. Remarkably, RNNs differ from DNNs only by weight sharing, as can be shown through unrolling in time. How does this structural similarity fit with the distinct functional properties these networks exhibit? To address this question, we here develop a unified mean-field theory for RNNs and DNNs in terms of representational kernels, describing fully trained networks in the feature learning (μμP) regime. This theory casts training as Bayesian inference over sequences and patterns, directly revealing the functional implications induced by the RNNs' weight sharing. In DNN-typical tasks, we identify a phase transition when the learning signal overcomes the noise due to randomness in the weights: below this threshold, RNNs and DNNs behave identically; above it, only RNNs develop correlated representations across timesteps. For sequential tasks, the RNNs' weight sharing furthermore induces an inductive bias that aids generalization by interpolating unsupervised time steps. Overall, our theory offers a way to connect architectural structure to functional biases.

Jan P. Bauer, Kirsten Fischer, Moritz Helias +1

Gatsby Computational Neuroscience Unit, UCL, London, UK · IAS-6, Forschungszentrum Juelich, Juelich, Germany

Recurrent Neural NetworksRepresentation LearningNeural Network Generalization
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Feb 16, 2026

BEACONS: Bounded-Error, Algebraically-Composable Neural Solvers for Partial Differential Equations

The traditional limitations of neural networks in reliably generalizing beyond the convex hulls of their training data present a significant problem for computational physics, in which one often wishes to solve PDEs in regimes far beyond anything which can be experimentally or analytically validated. In this paper, we show how it is possible to circumvent these limitations by constructing formally-verified neural network solvers for PDEs, with rigorous convergence, stability, and conservation properties, whose correctness can therefore be guaranteed even in extrapolatory regimes. By using the method of characteristics to predict the analytical properties of PDE solutions a priori (even in regions arbitrarily far from the training domain), we show how it is possible to construct rigorous extrapolatory bounds on the worst-case L^inf errors of shallow neural network approximations. Then, by decomposing PDE solutions into compositions of simpler functions, we show how it is possible to compose these shallow neural networks together to form deep architectures, based on ideas from compositional deep learning, in which the large L^inf errors in the approximations have been suppressed. The resulting framework, called BEACONS (Bounded-Error, Algebraically-COmposable Neural Solvers), comprises both an automatic code-generator for the neural solvers themselves, as well as a bespoke automated theorem-proving system for producing machine-checkable certificates of correctness. We apply the framework to a variety of linear and non-linear PDEs, including the linear advection and inviscid Burgers' equations, as well as the full compressible Euler equations, in both 1D and 2D, and illustrate how BEACONS architectures are able to extrapolate solutions far beyond the training data in a reliable and bounded way. Various advantages of the approach over the classical PINN approach are discussed.

Jonathan Gorard, Ammar Hakim, James Juno

Princeton University Princeton, NJ 08544, USA · Princeton Plasma Physics Laboratory Princeton, NJ 08540, USA

Neural PDE SolversPartial Differential EquationsNeural Network GeneralizationNeural Network VerificationNeural Network Approximation Theory
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Feb 9, 2026

On the Dynamics of Delayed Generalization in Linear Models

Delayed generalization, where training accuracy becomes nearly perfect substantially before test accuracy, has often been studied in the context of the depth and compositional structure of deep networks. In this paper, we ask whether delayed generalization can occur in simpler settings. We show that it can arise in a linear model trained with gradient descent and an exponential-tail loss on linearly separable data. We characterize the finite-time hyperplane dynamics into three phases: a population phase, a support vector dominated unlearning phase and a generalization phase. The convergence of the residual beyond the maximum-margin direction is sufficient for the unlearning phase to begin. In the support vector dominated phase, we characterize the hyperplane dynamics. The time scale in these dynamics causes delayed generalization. We validate our theory through controlled experiments varying population and support-vector distributions, and further verify the predicted bias behavior in a deep model with a single output neuron on MNIST. Our results provide a unified characterization of the asymptotic and finite-time dynamics underlying delayed generalization in linear models.

Atreya Vedantam, Nataraj Das, Chandrashekar Lakshminarayanan

Indian Institute of Technology Madras, Chennai.

Linear ClassificationNeural Network GeneralizationGradient Descent Dynamics
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Feb 1, 2026

Theoretical Analysis of Measure Consistency Regularization for Partially Observed Data

The problem of corrupted data, missing features, or missing modalities continues to plague the modern machine learning landscape. To address this issue, a class of regularization methods that enforce consistency between imputed and fully observed data has emerged as a promising approach for improving model generalization, particularly in partially observed settings. We refer to this class of methods as Measure Consistency Regularization (MCR). Despite its empirical success in various applications, such as image inpainting, data imputation and semi-supervised learning, a fundamental understanding of the theoretical underpinnings of MCR remains limited. This paper bridges this gap by offering theoretical insights into when MCR yields a more favorable finite-sample estimation-error upper bound, viewed through the lens of neural network distance. Under ideal interpolation and compatibility conditions, we show that the MCR estimation-error upper bound is no larger than vanilla supervised training and becomes strictly smaller when the mixed-sample alternative is favorable. We then extend the analysis to the non-ideal regime, where optimization and compatibility residuals can potentially offset this finite-sample advantage. Guided by these insights, we propose a novel practical diagnostic that leverages the duality gap and a calibrated drift estimator to infer the potential benefit of MCR training. We present detailed empirical evidence to support our theoretical claims and to show the effectiveness and accuracy of our practical diagnostic. We further provide simulations on real-world datasets to show the versatility of MCR under different model architectures designed for different data sources.

Yinsong Wang, Shahin Shahrampour

University of Nevada Reno, Reno, NV 89557 USA · Department of Mechanical and Industrial Engineering at Northeastern University, Boston, MA 02115 USA

Neural Network GeneralizationLearning with Missing DataStatistical Learning TheoryConsistency Regularization
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Jan 27, 2026

To Grok Grokking: Provable Grokking in Ridge Regression

We study grokking, the onset of generalization long after overfitting, in a classical ridge regression setting. We prove end-to-end grokking results for learning over-parameterized linear regression models using gradient descent with weight decay. Specifically, we prove that the following stages occur: (i) the model overfits the training data early during training; (ii) poor generalization persists long after overfitting has manifested; and (iii) the generalization error eventually becomes arbitrarily small. Moreover, we show, both theoretically and empirically, that grokking can be amplified or eliminated in a principled manner through proper hyperparameter tuning. To the best of our knowledge, these are the first rigorous quantitative bounds on the generalization delay (which we refer to as the "grokking time") in terms of training hyperparameters. Lastly, going beyond the linear setting, we empirically demonstrate that our quantitative bounds also capture the behavior of grokking on non-linear neural networks. Our results suggest that grokking is not an inherent failure mode of deep learning, but rather a consequence of specific training conditions, and thus does not require fundamental changes to the model architecture or learning algorithm to avoid.

Mingyue Xu, Gal Vardi, Itay Safran

Department of Computer Science, Purdue University, West Lafayette, IN, USA · Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Israel · Stein Faculty of Computer and Information Science, Ben-Gurion University of the Negev, Israel

Neural Network GeneralizationRidge RegressionGrokkingGradient Descent DynamicsGeneralization Bounds
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Jan 11, 2026

The Impact of Anisotropic Covariance Structure on the Training Dynamics and Generalization Error of Linear Networks

The success of deep neural networks largely depends on the statistical structure of the training data. While learning dynamics and generalization on isotropic data are well-established, the impact of pronounced anisotropy on these crucial aspects is not yet fully understood. We examine the impact of data anisotropy, represented by a spiked covariance structure, a canonical yet tractable model, on the learning dynamics and generalization error of a two-layer linear network in a linear regression setting. Our analysis reveals that the learning dynamics proceed in two distinct phases, governed initially by the input-output correlation and subsequently by other principal directions of the data structure. Furthermore, we derive an analytical expression for the generalization error, quantifying how the alignment of the spike structure of the data with the learning task improves performance. Our findings offer deep theoretical insights into how data anisotropy shapes the learning trajectory and final performance, providing a foundation for understanding complex interactions in more advanced network architectures.

Taishi Watanabe, Ryo Karakida, Jun-nosuke Teramae

Graduate School of Informatics, Kyoto University · Artificial Intelligence Research Center, AIST · RIKEN AIP

Neural Network GeneralizationNeural Network Training DynamicsTwo-Layer Neural NetworksDeep Linear Networks
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Nov 21, 2025

Addressing A Posteriori Performance Degradation in Neural Network Subgrid Stress Models

Neural network subgrid stress models often have a priori performance that is far better than the a posteriori performance, leading to neural network models that look very promising a priori completely failing in a posteriori Large Eddy Simulations (LES). This performance gap can be decreased by combining two different methods, training data augmentation and reducing input complexity to the neural network. Augmenting the training data with two different filters before training the neural networks has no performance degradation a priori as compared to a neural network trained with one filter. A posteriori, neural networks trained with two different filters are far more robust across two different LES codes with different numerical schemes. In addition, by ablating away the higher order terms input into the neural network, the a priori versus a posteriori performance changes become less apparent. When combined, neural networks that use both training data augmentation and a less complex set of inputs have a posteriori performance far more reflective of their a priori evaluation.

Andy Wu, Sanjiva K. Lele

Department of Aeronautics and Astronautics, Stanford University, Stanford, California, USA · Department of Mechanical Engineering, Stanford University, Stanford, California, USA

Neural Network GeneralizationData AugmentationComputational Fluid Dynamics
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Nov 16, 2025

On the Dimension-Free Approximation of Deep Neural Networks for Symmetric Korobov Functions

Deep neural networks have been widely used as universal approximators for functions with inherent physical structures, including permutation symmetry. In this paper, we construct symmetric deep neural networks to approximate symmetric Korobov functions and prove that both the convergence rate and the constant prefactor scale at most polynomially with respect to the ambient dimension. This represents a substantial improvement over prior approximation guarantees that suffer from the curse of dimensionality. Building on these approximation bounds, we further derive a generalization-error rate for learning symmetric Korobov functions whose leading factors likewise avoid the curse of dimensionality.

Yulong Lu, Tong Mao, Jinchao Xu +1

School of Mathematics, University of Minnesota, Minneapolis, 55455, MN, USA · Computer, Electrical and Mathematical Science and Engineering Division, King Abdullah University of Science and Technology, Thuwal 23955, Saudi Arabia · School of Mathematics, Georgia Institute of Technology, Atlanta, 30332, GA, USA

Neural Network GeneralizationNeural Network Approximation Theory
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Oct 20, 2025

DFNN: A Deep Fréchet Neural Network Framework for Learning Metric-Space-Valued Responses

Regression with non-Euclidean responses---e.g., probability distributions, networks, symmetric positive-definite matrices, and compositions---has become increasingly important in modern applications. In this paper, we propose deep Fréchet neural networks (DFNNs), an end-to-end deep learning framework for predicting non-Euclidean responses---which are considered as random objects in a metric space---from Euclidean predictors. Our method utilizes the representation-learning power of deep neural networks (DNNs) to the task of approximating conditional Fréchet means of the response given the predictors, the metric-space analogue of conditional expectations, by minimizing a Fréchet risk. The framework is highly flexible, accommodating diverse metrics and high-dimensional predictors. We establish a universal approximation theorem for DFNNs, advancing the state-of-the-art of neural network approximation theory to general metric-space-valued responses, without making model assumptions or relying on local smoothing. We further establish rigorous generalization guarantees for DFNNs and derive corresponding risk bounds, providing, to the best of our knowledge, the first such theoretical results for deep learning regression with metric-space-valued responses. Empirical studies on synthetic distributional and network-valued responses, as well as real-world applications to predicting compositional responses in an Aitchison simplex and spherical responses, demonstrate that DFNNs consistently outperform all existing methods.

Kyum Kim, Yaqing Chen, Paromita Dubey

Rutgers University · University of Southern California

Neural Network GeneralizationNeural Network Approximation Theory
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Oct 18, 2025

Memorizing Long-tail Data Can Help Generalization Through Composition

Deep learning has led researchers to rethink the relationship between memorization and generalization. In many settings, memorization does not hurt generalization due to implicit regularization and may help by memorizing long-tailed examples. In this paper, we consider the synergy between memorization and simple composition -- the ability to make correct prediction on a combination of long-tailed features. Theoretically, we show that for a linear setting, memorization together with composition can help the model make correct predictions on rare test examples that require a combination of long-tailed features, even if such combinations were never observed in the training data. Experiments on neural network architecture on simple data show that the theoretical insight extends beyond the linear setting, and we further observe that the composition capability of the model depends on its architecture.

Mo Zhou, Haoyang Ma, Rong Ge

University of Washington · Duke University

Neural Network GeneralizationNeural Network MemorizationLong-Tail LearningCompositional Generalization
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Oct 14, 2025

A Function-Centric Perspective on Flat and Sharp Minima

Flat minima are strongly associated with improved generalisation in deep neural networks. However, this connection has proven nuanced in recent studies, with both theoretical counterexamples and empirical exceptions emerging in the literature. In this paper, we revisit the role of sharpness in model performance and examine sharpness as a function-dependent property rather than as a direct indicator of poor generalisation. We conduct extensive empirical studies ranging from single-objective optimisation, synthetic non-linear binary classification tasks, to modern image classification tasks. In single-objective optimisation, we show that flatness and sharpness are relative to the function being learned: equally optimal solutions can exhibit markedly different local geometry. In synthetic non-linear binary classification tasks, we show that increasing decision-boundary tightness can increase sharpness even when models generalise perfectly, suggesting that sharpness is not reducible to memorisation alone. Finally, in large-scale experiments, we find that sharper minima often emerge when models are regularised (e.g., via weight decay, data augmentation, or SAM), and can coincide with better generalisation, calibration, robustness, and functional consistency. Our findings are consistent with a function-centric interpretation of minima geometry: flatness should not be treated as a universal proxy for generalisation, and sharper minima can reflect desirable inductive biases associated with improved predictive behaviour, calling for a function-centric reappraisal of minima geometry.

Israel Mason-Williams, Gabryel Mason-Williams, Helen Yannakoudakis

Department of Informatics, UKRI Safe and Trusted AI, London, United Kingdom · Queen Mary University of London, London, United Kingdom · Department of Informatics, King’s College London, London, United Kingdom

Flat MinimaLoss Landscape GeometryNeural Network Generalization
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Oct 9, 2025

Deep Neural Networks Inspired by Differential Equations

Deep learning has become a pivotal technology in fields such as computer vision, scientific computing, and dynamical systems, significantly advancing these disciplines. However, neural Networks persistently face challenges related to theoretical understanding, interpretability, and generalization. To address these issues, researchers are increasingly adopting a differential equations perspective to propose a unified theoretical framework and systematic design methodologies for neural networks. In this paper, we provide an extensive review of deep neural network architectures and dynamic modeling methods inspired by differential equations. We specifically examine deep neural network models and deterministic dynamical network constructs based on ordinary differential equations (ODEs), as well as regularization techniques and stochastic dynamical network models informed by stochastic differential equations (SDEs). We present numerical comparisons of these models to illustrate their characteristics and performance. Finally, we explore promising research directions in integrating differential equations with deep learning to offer new insights for developing intelligent computational methods that boast enhanced interpretability and generalization capabilities.

Yongshuai Liu, Lianfang Wang, Kuilin Qin +6

School of Mathematical Sciences, Beijing Normal University, Beijing, China · Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong

Neural Network GeneralizationDynamic Neural NetworksNeural ODEs
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Oct 2, 2025

PENEX: AdaBoost-Inspired Neural Network Regularization

AdaBoost sequentially fits so-called weak learners to minimize an exponential loss, which penalizes misclassified data points more severely than other loss functions like cross-entropy. Paradoxically, AdaBoost generalizes well in practice as the number of weak learners grows. In the present work, we introduce Penalized Exponential Loss (PENEX), a new formulation of the multi-class exponential loss that is theoretically grounded and, in contrast to the existing formulation, amenable to optimization via first-order methods, making it a practical objective for training neural networks. We demonstrate that PENEX effectively increases margins of data points, which can be translated into a generalization bound. Empirically, across computer vision and language tasks, PENEX improves neural network generalization in low-data regimes, matching and in some settings outperforming established regularizers at comparable computational cost. Our results highlight the potential of the exponential loss beyond its application in AdaBoost.

Klaus-Rudolf Kladny, Bernhard Schölkopf, Michael Muehlebach

MPI for Intelligent Systems, Tübingen · Tübingen AI Center · ELLIS Institute Tübingen

Neural Network Generalization
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Sep 28, 2025

Quantifying How Training Gradient Sparsity Affect Spiking Neural Network Accuracy And Robustness

Spiking Neural Networks (SNNs) have recently received increasing attention in both computational neuroscience and artificial intelligence owing to their potential for energy-efficient computation and reduced memory requirements. Despite these advantages, improving adversarial robustness in SNNs (particularly for vision-based applications) remains an emerging and relatively underexplored research problem. Recent work has suggested that encouraging sparse gradients can act as a regularization mechanism to improve resistance against adversarial perturbations. In this study, we report an unexpected observation: under certain architectural configurations, SNNs inherently exhibit sparse gradients and can attain state-of-the-art adversarial defense performance without requiring any explicit regularization strategy. Further investigation reveals an inherent trade-off between robustness and generalization. Specifically, increased gradient sparsity enhances resistance to adversarial attacks but may reduce the model's generalization capability, whereas denser gradients tend to improve generalization while simultaneously increasing susceptibility to adversarial perturbations. These findings provide new perspectives on the role of gradient sparsity in the training dynamics of SNNs.

Nhan Trong Luu, Duong Trung Luu

College of Information and Communication Technology, Can Tho University, Vietnam · College of Engineering and Computer Science, VinUniversity, Hanoi, Vietnam · Center of Digital Transformation and Communication, Can Tho University, Vietnam

Neural Network GeneralizationNeural Network RobustnessAdversarial RobustnessSpiking Neural NetworksNeural Network Training Dynamics
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Jun 30, 2025

Neural Langevin Machine: a local asymmetric learning rule can be creative

Fixed points of recurrent neural networks can be leveraged to store and generate information. These fixed points are captured by the Boltzmann-Gibbs measure, which leads to neural Langevin dynamics that relax to those fixed points for generative learning of a real dataset. We call this type of generative model a neural Langevin machine, which derives an asymmetric and firing-rate-speed-adjusted learning rule requiring only local neural signals, thereby bearing biological relevance in terms of local predictive learning. An out-of-equilibrium regime of the generative process is revealed, together with a memorization-to-generalization transition with increasing training data size. The neuro-inspired machine can also realize a continuous exploration of the phase space for different kinds of generative images and can denoise a corrupted image as well.

Zhendong Yu, Weizhong Huang, Haiping Huang

PMI Lab, School of Physics, Sun Yat-sen University, Guangzhou 510275, People’s Republic of China · Guangdong Provincial Key Laboratory of Magnetoelectric Physics and Devices, Sun Yat-sen University, Guangzhou 510275, People’s Republic of China

Neural Network GeneralizationImage GenerationRecurrent Neural NetworksEnergy-Based ModelsGenerative ModelingLocal Learning
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Jun 23, 2025

GeNeRT: A Physics-Informed Approach to Intelligent Wireless Channel Modeling via Generalizable Neural Ray Tracing

Neural ray tracing (RT) has emerged as a promising paradigm for channel modeling by integrating physical propagation principles with neural networks. However, existing neural RT methods remain limited by strong spatial dependence and weak adherence to electromagnetic laws. We propose GeNeRT, a generalizable neural RT framework that improves generalization and accuracy through relative geometric features, scatterer semantics, and a Fresnel-inspired polarization-driven architecture. GeNeRT is trained through a three-stage strategy: polarization-specific module-wise pre-training captures general ray-surface interaction behavior; system-wise end-to-end training uses only receiver-side channel impulse responses to learn site-specific propagation characteristics; and measurement-based fine-tuning employs sparse measured multipath components (MPCs) to adapt polarization-related modules to real-world environments. Extensive outdoor simulations demonstrate robust intra-scenario transferability and inter-scenario zero-shot generalization. In an unseen scenario, GeNeRT achieves an overall error of −35.36-35.36 dB and an average-delay error of 4.91 ns, compared with −10.85-10.85 dB and 32.38 ns for the best baseline. With only 75 measured reflected MPCs, fine-tuning further reduces the overall error from −14.48-14.48 to −22.90-22.90 dB and the average-delay error from 6.28 to 3.58 ns. Ablation studies confirm the effectiveness of the proposed architecture and training strategy.

Kejia Bian, Meixia Tao, Shu Sun +2

School of Information Science and Electronic Engineering, Shanghai Jiao Tong University, Shanghai 200240, China · Shanghai Innovation Institute, Shanghai 200231, China

Neural Network GeneralizationWireless CommunicationsRadio Propagation ModelingPhysics-Informed ML
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Jun 18, 2025

Interpretability and Generalization Bounds for Learning Spatial Physics

While there are many applications of ML to scientific problems that look promising, visuals can be deceiving. Using numerical analysis techniques, we rigorously quantify the accuracy, convergence rates, and generalization bounds of certain ML models applied to linear differential equations for parameter discovery or solution finding. Beyond the quantity and discretization of data, we identify that the function space of the data is critical to the generalization of the model. A similar lack of generalization is empirically demonstrated for commonly used models, including physics-specific techniques. Counterintuitively, we find that different classes of models can exhibit opposing generalization behaviors. Based on our theoretical analysis, we also introduce a new mechanistic interpretability lens on scientific models whereby Green's function representations can be extracted from the weights of black-box models. Our results inform a new cross-validation technique for measuring generalization in physical systems, which can serve as a benchmark.

Alejandro Francisco Queiruga, Theo Gutman-Solo, Shuai Jiang

OpenAI, San Francisco, CA, USA · Google, Mountain View, CA, USA · Sandia National Laboratories, Albuquerque, NM, USA

Neural Network GeneralizationMechanistic InterpretabilityScientific MLPhysics-Informed MLGeneralization Bounds
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Jun 16, 2025

Random Matrix Theory for Deep Learning: Beyond Eigenvalues of Linear Models

Modern Machine Learning (ML) and Deep Neural Networks (DNNs) often operate on high-dimensional data and rely on overparameterized models, where classical low-dimensional intuitions break down. In particular, the proportional regime where the data dimension, sample size, and number of model parameters are all large and comparable, gives rise to novel and sometimes counterintuitive behaviors. This paper extends traditional Random Matrix Theory (RMT) beyond eigenvalue-based analysis of linear models to address the challenges posed by nonlinear ML models such as DNNs in this regime. We introduce the concept of High-dimensional Equivalent, which unifies and generalizes both Deterministic Equivalent and Linear Equivalent, to systematically address three technical challenges: high dimensionality, nonlinearity, and the need to analyze generic eigenspectral functionals. Leveraging this framework, we provide precise characterizations of the training and generalization performance of linear models, nonlinear shallow networks, and deep networks. Our results capture rich phenomena, including scaling laws, double descent, and nonlinear learning dynamics, offering a unified perspective on the theoretical understanding of deep learning in high dimensions.

Zhenyu Liao, Michael W. Mahoney

School of Electronic Information and Communications, Huazhong University of Science and Technology, Wuhan, China · ICSI, LBNL, and Department of Statistics, University of California, Berkeley, USA

Random Matrix TheoryNeural Network GeneralizationNeural Network Training DynamicsNeural Scaling LawsHigh-Dimensional Asymptotics
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Jun 1, 2025

Generalization in VAE and Diffusion Models: A Unified Information-Theoretic Analysis

Despite the empirical success of Diffusion Models (DMs) and Variational Autoencoders (VAEs), their generalization performance remains theoretically underexplored, especially lacking a full consideration of the shared encoder-generator structure. Leveraging recent information-theoretic tools, we propose a unified theoretical framework that provides guarantees for the generalization of both the encoder and generator by treating them as randomized mappings. This framework further enables (1) a refined analysis for VAEs, accounting for the generator's generalization, which was previously overlooked; (2) illustrating an explicit trade-off in generalization terms for DMs that depends on the diffusion time TT; and (3) providing computable bounds for DMs based solely on the training data, allowing the selection of the optimal TT and the integration of such bounds into the optimization process to improve model performance. Empirical results on both synthetic and real datasets illustrate the validity of the proposed theory.

Qi Chen, Jierui Zhu, Florian Shkurti

Department of Computer Science, University of Toronto · Data Science Institute · Vector Institute +2

Variational AutoencodersNeural Network GeneralizationInformation-Theoretic Generalization BoundsDiffusion ModelsGeneralization Bounds
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Apr 23, 2025

The Dynamics of Generalization in Deep Learning

We derive a differential equation that governs the evolution of the generalization gap when a model is trained by gradient descent-based methods. This differential equation is driven by two key quantities, a contraction factor that brings together trajectories corresponding to slightly different datasets, and a perturbation factor that accounts for them training on different datasets. The coupled decay of contraction and perturbation guarantees a controlled accumulation of generalization gap during training. We analyze this differential equation to show that the generalization gap is given by a quadratic form that consists of an ``effective Gram matrix'' that depends upon the training trajectory and a certain residual of the predictor at initialization. Our framework is applicable to general deep networks and smooth loss functions. In numerical experiments on different neural network architectures, datasets and sample sizes, we show that this quadratic form accurately captures the actual generalization gap. We also show how to instantiate our framework in a number of examples via analytical calculations. For example, for high-dimensional linear regression, our framework matches existing calculations of generalization gap in the literature exactly in under-parameterized, over-parameterized and critical regimes.

Rubing Yang, Pratik Chaudhari

Department of Applied Mathematics and Computational Sciences, University of Pennsylvania Philadelphia, PA, 19104, USA · Department of Electrical and Systems Engineering, University of Pennsylvania Philadelphia, PA, 19104, USA

Neural Network GeneralizationNeural Network Training DynamicsStatistical Learning TheoryGradient Descent Dynamics
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Apr 21, 2025

Deep learning with missing data

In the context of multivariate nonparametric regression with missing covariates, we propose Pattern Embedded Neural Networks (PENNs), which can be applied in conjunction with any existing imputation technique. In addition to a neural network trained on the imputed data, PENNs pass the vectors of observation indicators through a second neural network to provide a compact representation. The outputs are then combined in a third neural network to produce final predictions. Our main theoretical result exploits an assumption that the observation patterns can be partitioned into cells on which the Bayes regression function behaves similarly, and belongs to a compositional Hölder class. It provides a finite-sample excess risk bound that holds for an arbitrary missingness mechanism, and in combination with a complementary minimax lower bound, demonstrates that our PENN estimator attains in typical cases the minimax rate of convergence as if the cells of the partition were known in advance, up to a poly-logarithmic factor in the sample size. Numerical experiments on simulated, semi-synthetic and real data confirm that the PENN estimator consistently improves, often dramatically, on standard neural networks without pattern embedding. Code to reproduce our experiments, as well as a tutorial on how to apply our method, is publicly available.

Tianyi Ma, Tengyao Wang, Richard J. Samworth

Statistical Laboratory, University of Cambridge · Department of Statistics, London School of Economics

Neural Network GeneralizationLearning with Missing DataNonparametric Regression
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Mar 19, 2025

Unified Enhancement of the Generalization and Robustness of Language Models via Bi-Stage Optimization

Neural network language models (LMs) are confronted with significant challenges in generalization and robustness. Currently, many studies focus on improving either generalization or robustness in isolation, without methods addressing both aspects simultaneously, which presents a significant challenge in developing LMs that are both robust and generalized. In this paper, we propose a bi-stage optimization framework to uniformly enhance both the generalization and robustness of LMs, termed UEGR. Specifically, during the forward propagation stage, we enrich the output probability distributions of adversarial samples by adaptive dropout to generate diverse sub models, and incorporate JS divergence and adversarial losses of these output distributions to reinforce output stability. During backward propagation stage, we compute parameter saliency scores and selectively update only the most critical parameters to minimize unnecessary deviations and consolidate the model's resilience. Theoretical analysis shows that our framework includes gradient regularization to limit the model's sensitivity to input perturbations and selective parameter updates to flatten the loss landscape, thus improving both generalization and robustness. The experimental results show that our method significantly improves the generalization and robustness of LMs compared to other existing methods across 13 publicly available language datasets, achieving state-of-the-art (SOTA) performance.

Yudao Sun, Juan Yin, Juan Zhao +3

Department of New Networks, Peng Cheng Laboratory, Shenzhen 518055, China · Department of Industrial Systems Engineering and Management, National University of Singapore, Singapore 117576 · School of Data Science and Artificial Intelligence, Chang’an University, Shaanxi, 710064, China

Adversarial TrainingNeural Network GeneralizationNeural Network RobustnessAdversarial Robustness of Language ModelsLanguage Model Robustness
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Mar 3, 2025

The Mathematical Theory of Double Descent as Shrinkage, not Overfitting: Matching Bounds for Path-Regularized Multilayer Neural Networks

We give a nonasymptotic generalization theory for multilayer fully connected neural networks trained with path regularization, for general Lipschitz losses and without assuming the loss bounded, the architecture sparse, or any width or depth limit. Approximation error is modelled explicitly, and along the way we settle an open problem of E et al. on approximation rates in generalized Barron spaces. We treat the underparameterized and overparameterized regimes separately, which is what allows the resulting curve to be non-monotone in the width. We then prove lower bounds on the error itself, valid at every target and at every local minimizer, and combine them with the upper bounds into a theorem asserting that the error - not merely our bound on it - descends, rises and descends again as the width grows, with a peak exceeding the final plateau by a factor diverging in the sample size. Having both sides of the bound also identifies what drives the ascent, and it is not overfitting: once the regularization exceeds the dual norm of the noise, which the prescribed schedule does at every width, a slow-rate oracle inequality controls the error with no variance term at all. The rise is instead the shrinkage demanded by the regularization that the underparameterized regime requires, and we show it cannot be removed by any width-independent regularization under which the theory's guarantees hold. This mechanism is a property of the estimator analyzed here; we do not claim it accounts for the double descent reported for unregularized training.

Hao Yu

Department of Automation, Tsinghua University, Beijing, 100086, China

Double DescentNeural Network GeneralizationNeural Network Approximation TheoryGeneralization Bounds
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Feb 4, 2025

Networks with Finite VC Dimension: Pro and Contra

Approximation and learning of classifiers of large data sets by neural networks in terms of high-dimensional geometry and statistical learning theory are investigated. The influence of the VC dimension of sets of input-output functions of networks on approximation capabilities is compared with its influence on consistency in learning from samples of data. It is shown that, whereas finite VC dimension is desirable for uniform convergence of empirical errors, it may not be desirable for approximation of functions drawn from a probability distribution modeling the likelihood that they occur in a given type of application. Based on the concentration-of-measure properties of high dimensional geometry, it is proven that both errors in approximation and empirical errors behave almost deterministically for networks implementing sets of input-output functions with finite VC dimensions in processing large data sets. Practical limitations of the universal approximation property, the trade-offs between the accuracy of approximation and consistency in learning from data, and the influence of depth of networks with ReLU units on their accuracy and consistency are discussed.

Vera Kurkova, Marcello Sanguineti

Institute of Computer Science of the Czech Academy of Sciences Pod Vodárenskou věží, 2 - 18207 Prague, Czech Republic · DIBRIS University of Genova Via Opera Pia, 13 - 16145 Genova, Italy

Neural Network GeneralizationNeural Network Approximation TheoryReLU Neural Networks
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Feb 2, 2025

Boosting Adversarial Robustness and Generalization with Dictionary Structure

This work investigates a novel approach to boost adversarial robustness and generalization by incorporating structural prior into the design of deep learning models. Specifically, our study surprisingly reveals that existing dictionary learning-inspired convolutional neural networks (CNNs) are robust against random noise but remain highly vulnerable to adversarial attacks. To address this, we propose Elastic Dictionary Learning Networks (EDLNets), a novel ResNet architecture that significantly enhances adversarial robustness and generalization. Extensive and reliable experiments demonstrate consistent improvements in adversarial robustness across multiple datasets, backbone architectures, and threat models. To the best of our knowledge, this is the first work to discover and validate that dictionary structure can reliably enhance deep learning robustness under strong adaptive attacks, unveiling a promising direction for future research.

Zhichao Hou, Weizhi Gao, Hamid Krim +2

Department of Computer Science, North Carolina State University, Raleigh, USA.

Neural Network GeneralizationAdversarial RobustnessConvolutional Neural NetworksDictionary Learning
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