Neural Network Generalization

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21 papers in the last four weeks, up 600% on the four weeks before. 0.2% of all new papers.

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Latest papers 204

Oct 8, 2026quant-ph

Toward Joint Optimization of Circuit Depth and Training Data Size in Adaptively Grown Quantum Classifiers

Building a quantum model involves a tradeoff: how complex the circuit should be, and how much training data it needs. Caro et al. show that models with fewer trainable gates need less training data to generalize well. Q-FLAIR shows that a quantum feature-map circuit can be grown gate-by-gate, stopping once further growth stops improving the training loss. We ask whether these two results combine into a predictable scaling law. Does Q-FLAIR's own stopping rule pick larger or smaller circuits as training data grows? Does the resulting generalization behavior track Caro et al.'s bound? We reimplement Q-FLAIR's growth mechanism faithfully, including its analytic reconstruction and exact stopping rule. We run it on full-resolution (784-pixel) MNIST 3-vs-5 classification, at five training-set sizes from N = 2000 to 10000. We then fine-tune each resulting circuit, so we can measure Caro et al.'s notion of active gates, K. We find no predictable relationship between training-set size and the circuit size Q-FLAIR converges to. Circuit size and test accuracy both vary non-monotonically with N, and seed-to-seed variance is nearly as large as any trend across N. The empirical generalization gap never exceeds Caro et al.'s bound in 14 of 15 runs, so the bound holds as a valid guarantee in those runs. But the gap correlates only weakly with the bound's value (r = 0.12). This shows that K does not explain most of the variation we observe. Why a valid guarantee can coexist with such weak predictive power remains an open question, and answering it may be necessary before circuit depth and training data size can be jointly optimized in practice.

Saeefa Rubaiyet Nowmi, Md Mahmuduzzaman Kamol, Mohammad Saidur Rahman

Old Dominion University Norfolk, VA, USA · University of Texas at El Paso El Paso, TX, USA

Quantum Circuit Trainability Neural Network Generalization Quantum Feature Maps

Oct 8, 2026cs.CV

HAND: A Biologically-Inspired Activation Function that Improves Generalisation and Sample Efficiency in Image Classification

DNNs exhibit robustness and generalisation issues not seen in humans. They are also far less data-efficient learners, requiring considerably more training samples to accurately classify novel exemplars. Inductive bias could help with these issues by providing in-built mechanisms to improve generalisation, and hence, reduce reliance on learning from data. We incorporate a biologically-inspired inductive bias into a new activation function, HAND (Homeostasis, Accelerating Nonlinearity, and Divisive-nomalisation), and show its effectiveness with CNNs trained on image classification. Using HAND a ConvNeXt-tiny required 25 training epochs to reach the same accuracy on ImageNet1k as the unmodified model achieved after 200 epochs. Consistent with the effects of an inductive bias, the performance gap reduced with training time and increased data augmentation. When the volume of training data was reduced and unevenly distributed between classes (Long-tailed ImageNet) the improvements in accuracy were even larger and did not reduce with increased training time. Generalisation performance with the common-corruptions data, and the ability to reject samples from unknown classes, were unaffected or improved by HAND. Results generalised across CNN architectures and training data-sets. HAND can, therefore, reduce the required training time and/or the required volume and variety of training data, helping to improve sample efficiency.

Michael W. Spratling, Heiko H. Schütt

University of Luxembourg, Department of Behavioural and Cognitive Sciences, L-4366 Esch-sur-Alzette, Luxembourg.

Neural Network Activation Functions Neural Network Generalization Data-Efficient Learning Image Classification Convolutional Neural Networks

Oct 8, 2026cs.CV

Dissecting Representation Structure in Vision Transformers: A Rigorous Architectural Study

Representation structure is crucial for understanding Vision Transformer (ViT) architectures and their generalization behavior. However, prior studies neither isolate nor analyze module-level features nor investigate how their interactions contribute to performance estimation. In this work, we conduct the first rigorous analysis of feature information across diverse architectural scales, empirically uncover the relationship between ViT representation and generalization behavior, and leverage these insights to guide efficient ViT design. Our contributions are fivefold: Across diverse architectural scales, 1) We identify feature collapse at initialization, which leads to redundancy, and propose a reduction scheme to mitigate this issue. 2) We quantify feature information using entropy and the minimum eigenvalue, demonstrating that these metrics serve as reliable indicators for generalization prediction. 3) We show that feature in the token space provides a more faithful representation than those in embedding space. 4) We discover an unexpected finding: features produced by linear submodules within ViT layers are critical for the prediction of generalization performance. 5) Our proposed proxy improves the correlation ranking by 18-48% over prior baselines and can effectively identify ViT architectures that achieve higher accuracy at lower or comparable computational cost.

Kim-Cuc Nguyen, Ngai-Man Cheung

Temasek Laboratories Singapore University of Technology and Design Singapore

Vision Transformer Neural Network Generalization Efficient ViTs

Oct 8, 2026cs.LG

Do Flatter Minima Drive Better Generalization? An Algorithmic Separation in Grokking

Flat loss landscapes have long been linked to better generalization in neural networks. However, its role as a causal mechanism for generalization is less established. Grokking provides an unique testbed to understand this distinction: models are prone to fit observed data using non-generalizing structure and remain in that regime for prolonged periods, transitioning to generalization only under particular training conditions. In this work, we study whether flat loss landscapes can act as a driving mechanism in this transition. While recent work has argued for flatness as a necessary geometric condition for this transition, we find that biasing training toward flatter solutions using sharpness-aware minimization (SAM) is insufficient to reliably induce this transition, despite producing flatter solutions. However, when SAM is paired with mechanisms that drive generalization such as weight decay, an interesting property emerges: SAM can accelerate the transition to generalizing solutions by up to 4x at the epoch-level. We theoretically untangle this relationship between SAM and weight decay using a minimal interpolating two-layer ReLU model with both memorizing and generalizing solutions. We show that even in this simple setup, flatness alone cannot distinguish a memorizing solution from a generalizing one, while weight decay favors generalizing solutions. However, under a local stability analysis, there exists a window where a memorizing interpolant is locally stable under gradient descent but unstable under SAM in the low-norm regime, which can explain SAM's ability to accelerate this transition. Overall, our results provide a more interpretable account of the role of flatness in driving generalization, especially in settings where models are vulnerable to minimizing loss through learning non-generalizing structure.

Mohnish Harwani

Purdue University

Neural Network Generalization Flat Minima Grokking Sharpness-Aware Minimization

Oct 1, 2026cs.LG

Generalization in Neural Networks Through the Lens of Magnitude Potential

Explaining generalization and training dynamics in neural networks remains a challenge, and various approaches have been developed to study different aspects of these phenomena. In this paper, we introduce the idea of {\em magnitude potential} -- a quantity based on the theory of metric magnitude -- that reflects how well an arbitrary point is represented by a given set. We find that this basic quantity can be applied to examine various features in neural generalization. The ratio between the magnitude potential with respect to a class and with respect to the entire data, computed at the logit layer, is informative of the representation of the point. In experiments, these ratios for individual training points are found to be correlated with the Feldman memorization scores. Magnitude potential ratios aggregated across points detect structural changes in the decision boundaries and provide a geometric indicator of grokking in modular arithmetic. Although the magnitude potential ratio and neural collapse are both closely associated with intra-class and inter-class geometric structure, the magnitude potential ratio remains informative even when neural collapse is explicitly suppressed.

Sahel Torkamani, Henry Gouk, Rik Sarkar

School of Informatics University of Edinburgh

Neural Network Generalization Neural Representation Geometry Neural Network Training Dynamics

Oct 1, 2026cs.LG

Why Does Train-Validation Separation Emerge? Update-Pressure Density Dynamics in Pretrained Backbones

Train-validation separation is the evolving difference between performance on observed training examples and a finite held-out validation set. We propose a dynamic structural account of how this gap develops during adaptation of pretrained models: continued fitting can shift update demand from broadly reusable support toward narrower support with weaker held-out transfer. A conditional local model links this shift to increasing heterogeneity in gradient allocation and train-validation separation. Fixed training probes make this structural evolution observable without validation examples entering the readouts; held-out performance is used separately to evaluate its relation to the gap. In a constructed hierarchy implemented with a residual multilayer perceptron (ResMLP), increasing the target share of example-private features from

p=.3

to

.5

to

.7

, while preserving the relative mixture

1{:}2{:}3{:}4

among the four shared feature levels, increases the final mean accuracy gap from

.185

to

.331

to

.527

across five runs per condition. Masked-input losses measured separately on training and validation examples expose the corresponding transfer asymmetry. The natural language processing (NLP) analysis uses 10-epoch runs of RoBERTa, DeBERTa, and Qwen on six datasets (90 runs): the training-probe-weighted within-class and overall dispersion readouts each have positive raw and smoothed level correlations with the accuracy gap in all 90 runs. Raw changes paired at approximately one-epoch intervals remain positively associated in 86/90 and 87/90 runs, respectively. A 40-epoch ResNet-18 study tests both readouts on three vision datasets. Together, controlled simulation, NLP, and vision support the dynamic structural account across settings, with real-model evidence testing its observable predictions under the specified monitors.

Yuchen Li, Mingyu Du, Zongqi Fan +2

School of Computer Science, The University of Sydney, Sydney, Australia · School of Computer Science and Engineering, UNSW Sydney, Sydney, Australia

Neural Network Generalization Transfer Learning Neural Network Training Dynamics

Sep 30, 2026cs.LG

Exact information accounting for SGD methods

As an alternative to the standard geometric analyses, we give an exact, information-theoretic analysis of stochastic gradient descent (SGD) and its variants. We show that a preconditioned SGD step is the posterior-mean update of a Gaussian Bayes model, and that its one-step regret splits into an intrinsic-time cost and a change in comparator information. The split extends to an identity for the objective itself. Convex convergence, strict-saddle-point escape, the link between flatness and generalization, the standard learning-rate schedules, adaptive optimizers, and the noisy, momentum, heavy-tailed, and gradient-free variants of SGD each correspond to a term or a special case of this identity. We measure its terms on synthetic and real training runs. On real networks it attributes the slack of classical convergence bounds to the terms their derivations drop and separates optimizers that reach the same training loss. That separation follows the number and consistency of their steps. Its relation to which of them generalizes better differs between networks. For gradient-free SGD the identity determines how a curvature preconditioner should enter the update. The sharpness-based generalization certificate it yields, with a data-independent isotropic prior, is vacuous at network scale unless the curvature spectrum is nearly flat across all parameters.

Akshay Balsubramani

Neural Network Generalization Stochastic Optimization Convergence Adaptive Gradient Methods Stochastic Gradient Descent

Sep 30, 2026stat.ML

Transferable Graph Metanetworks

A weight space network (or metanetwork) takes the weights of another neural network as input and predicts properties of it. Most prior work trains such models on input networks of one or a few fixed sizes and evaluates them in-distribution. The few attempts at out-of-distribution size generalization remain limited in scope and have achieved only modest success. Consequently, the potential efficiency gains of training on small networks and evaluating on much larger ones remain largely unrealized. We propose Transferable Graph Metanetworks, which extend the graph metanetwork paradigm with a set of modifications that make performance transferable across input networks of different widths. The modifications follow two principles: invariance to the ways in which networks of different widths represent the same function, and continuity, such that weights representing similar functions receive similar predictions. We further study whether size generalization is possible for input networks trained independently from random initialization. Empirically, our modifications significantly improve size generalization on every task we consider. Performance is strongest on input networks trained under the maximal-update parameterization (

μ

P), where it remains robust up to

42\times

the training width. Theoretically, we explain these observations with infinite-width limit theory: we prove size-generalization guarantees for our model on

μ

P-trained inputs, and explain why it can fail under other parameterizations.

Yuxin Ma, Adir Dayan, Yam Eitan +2

Department of Applied Mathematics and Statistics, Johns Hopkins University · Technion · NVIDIA

Neural Network Generalization Hypernetworks

Sep 30, 2026cs.LG

Forking: Sudden Overfitting Under Replay

This paper studies forking, a generalization failure discovered in NanoGPT autoresearch. Under data replay, models with an over-encoding n-gram memory branch show a sharp separation of training and validation loss at epoch boundaries, resembling the shape of forks. We study this phenomenon in a controlled vanilla NanoGPT setting and reproduce it in a DeepSeek-style model with Engram. Mechanistically, repeated updates sharpen the continuations observed in training while suppressing the probability of unseen continuations, whose loss grows with each pass. The n-gram module creates weakly interacting context-specific subspaces, amplifying this effect. Low-frequency contexts contribute most of the gap, whereas larger training budgets and heavily crowded tables suppress it. We also observe forking in short-budget, heavily repeated SFT and RL-like regimes. The contributions of this paper are twofold: (1) Forking reveals yet another curious phenomenon in deep learning, in addition to grokking and double descent. (2) Forking is an unexpected and unpleasant by-product of tricks proposed by autoresearch agents. While these agents produce an enormous number of results that seem useful, we should always be careful with their results.

Shanbin Yu, Shaoyang Guo, Haoran Zhao +2

MetaCircle (元环智能) · Tsinghua University · Peking University +1

Neural Network Generalization LLM Training Data Repetition in Language Model Training

Sep 30, 2026cs.LG

A Generalisation Signal Need Not Be a Model-Selection Signal

Model selection in computational biology often relies on validation data drawn from the training regime, even when deployment lies outside it. When validation no longer preserves which model is best, a natural alternative is to rank candidates using properties of the trained network itself. We test this idea using a novel, forward-only proxy motivated by the norm of the Hessian, alongside common Hessian measures, across molecular property, protein fitness, and drug-response tasks. Contrary to our hypothesis, geometry does not become more useful as validation Spearman correlation deteriorates: augmenting validation helps some shifts but significantly harms others. More surprisingly, the proxy still correlates with generalisation gap on most tasks even when Hessian trace and top-eigenvalue relationships are weak or reversed, yet this signal does not reliably identify the deployment-best model. A curvature bound need not preserve cross-model rankings, and low geometric scores can even favour collapsed predictors. Thus, a generalisation signal need not be a model-selection signal.

Aditya Nagarsekar, M P Ashish Bhat, Aadi Nesarkar +5

Department of CS&IS, BITS Pilani, K K Birla Goa Campus · LexisNexis Legal & Professional · Center for AI and Supercomputing, Mahindra University

Model Selection Neural Network Generalization Protein Fitness Prediction

Sep 28, 2026cs.LG

Reasoning with Neural Cellular Automata

Modern AI architectures used to solve visual reasoning tasks typically rely heavily on global connectivity and synchronization. As biological systems demonstrate, though, sophisticated computation can be performed in a more decentralized fashion. In this work, we test the reasoning capabilities of Neural Cellular Automata (NCAs), networks of recurrent cells that use strictly local connectivity and asynchronous updates. NCAs have been extensively studied in artificial life experiments, but it is unclear whether they can perform complex multi-step reasoning. We show that NCAs produce spatio-temporal dynamics capable of solving challenging visual reasoning tasks, including large mazes, Sudoku, and ARC-AGI-1. Furthermore, we provide evidence that NCAs generalize out-of-distribution when running with larger grids, longer rollouts, or parallel trials; and that the latter can be made more efficient via pruning of redundant trajectories. We find that these generalization capabilities depend on training with sample replay and stochastic perturbations, and that stochasticity remains beneficial at test time. Finally, we show that NCAs are robust reasoners capable of dynamically modulating compute to recover efficiently from damage, and that they can scale to solve reasoning in raw pixel space.

Mayalen Etcheverry, Pietro Miotti, Aidan Sirbu +7

Google Paradigms of Intelligence Team · School of Computer Science, McGill University · Mila - Quebec AI Institute +4

Neural Cellular Automata Neural Network Generalization Visual Reasoning Algorithmic Reasoning

Sep 28, 2026cs.LG

First Learn, Then Memorize: The Spectral Bias of Diffusion Models

Diffusion models trained on a finite dataset first learn to generate novel, high-quality samples and only much later collapse onto their training set. We identify the mechanism behind this separation of timescales and the object that probes it. The training dynamics of the score function are governed---exactly, and at any width---by the Gram matrix of the Neural Tangent Kernel (NTK) evaluated on the noisy training data, so the timescales of generalization and of memorization must be encoded in its spectrum. We show that they are, and that the structure responsible has no analogue in standard kernel settings. The use of multiple noise realizations per sample (

m

noised copies at a fixed noise level) in the score-matching loss is what restructures the Gram matrix spectrum into two distinct parts. The first, of large eigenvalues, carries the global features of the target distribution and is present already for

m=1

. The second, which the repeated noising creates, consists of the smallest eigenvalues and is supported on eigenvectors aligned with the sample-specific noise directions; it sets a memorization timescale parametrically larger in the training set size

n

. We establish this picture on two fronts. Analytically, we solve the spectrum in the lazy high-dimensional limit for both linear (

n \asymp d

) and polynomial (

n \asymp d^k

) sample complexities, and prove through a bias--variance decomposition that the first bulk minimizes the approximation error while the second drives the error associated with memorization. Empirically, we show the same two-bulk structure in Convolutional NTKs on CelebA and in finite-width U-Nets trained well beyond the lazy regime, and we make the link causal: truncating the Gram matrix at rank

r

tunes the generalization--memorization transition, and an

L_2

penalty targeting the second bulk suppresses memorization in feature-learning U-Nets.

Raphaël Urfin, Tony Bonnaire, Giulio Biroli +1

Laboratoire de Physique de l’École normale supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université Paris Cité, F-75005 Paris, France · Université Paris-Saclay, CNRS, Institut d’Astrophysique Spatiale, 91405 Orsay, France · Department of Computing Sciences, Bocconi University, Milano, Italy

Neural Network Generalization Memorization in Generative Models Neural Network Training Dynamics Neural Tangent Kernel Diffusion Models

Sep 28, 2026cs.LG

Universality and Generalization of Causal Transformers Across Context Lengths

Long contexts are central to modern transformer systems, but most expressivity results choose a different network for each fixed sequence length. We study whether one masked transformer can approximate causal token-to-token maps uniformly over sequences of arbitrary length sampling a fixed normalized horizon. To relate sampling resolutions, we model tokens by

α

-Hölder sequences or, more generally, a common modulus of continuity. Our notion of continuity across resolutions characterizes the causal families admitting uniform approximation on these compact input classes by a single transformer with length-independent parameters. The result extends to the infinite-length mean-field limit, where tokens form continuous curves and masked attention becomes a causal time integral. For bounded regression with target maps satisfying a

β

-smooth stability condition defined using regular test functions, quantitative approximation yields a generalization bound: exact empirical risk minimization over suitably sized bounded-weight transformers gives root mean-square prediction error

O((\log\log N/\log N)^{β/(d+2)})

from

N

iid labeled sequences. The bound holds at fixed confidence on the same sampling distribution, with

d

the token dimension and no maximum-length factor. Finally, experiments on physical time series support the Hölder-regular token model at observed scales, with dataset-dependent fitted exponents, whereas text input embeddings provide a contrasting case. Native and dense sampling, shuffled controls, and refinement checks delimit this empirical regularity regime.

Takashi Furuya, Maarten V. de Hoop, Gabriel Peyré

Doshisha University, RIKEN AIP · Rice University · CNRS, ENS, PSL Université

Neural Network Generalization Transformer Expressivity Length Generalization Long-Context Modeling Time Series Transformers

Sep 27, 2026stat.ML

Neural Scaling Laws of Transformer Operator Network

Transformers have emerged as powerful architectures for learning solution operators of physical systems. Empirically the prediction error has been observed to decrease when the data size and model size increase, suggesting neural scaling behavior. Yet a theoretical understanding of such scaling laws for transformer-based operator learning remains limited. In this work, we develop a theoretical framework for characterizing the approximation and generalization errors of transformer-based operator learning. Our analysis builds on a local-to-global approximation principle that is naturally aligned with the softmax attention mechanism and yields discretization-invariant output functions. On approximation theory, we derive a universal approximation error of transformer-based operator learning for Hölder-regular operators. On generalization theory, we establish a power scaling law between the prediction error and the training data size. The rate of convergence represented by the scaling exponent explicitly reflects the dimensions of the input and output domains, the regularity of the underlying functions and operators, and crucially, the intrinsic dimension of the input function class. By exploiting this intrinsic low-dimensional structure, our analysis yields a power-law generalization rate for operator learning, in contrast to the logarithmic-type power-law rates appearing in existing analyses of operator learning with feedforward neural networks. Numerical experiments validate the predicted power-law scaling and confirm that the convergence rate varies systematically with the intrinsic dimension of the input function class.

Haoran Yan, Zhongjie Shi, Yuanzhe Xi +2

School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA · Department of Mathematics, Emory University, Atlanta, GA 30322, USA · School of Computational Science and Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA

Neural Network Generalization Neural Network Approximation Theory Operator Learning Neural Scaling Laws

Sep 24, 2026cs.LG

An Analytical Theory of Auxiliary Learning

Auxiliary learning is an optimization paradigm in which a neural network's performance on a target task is improved by jointly training it on additional tasks. However, the mechanisms behind this improvement remain poorly understood. We study this problem using a teacher-student framework and derive a closed system of differential equations describing the dynamics of online stochastic gradient descent in the large-input limit. For linear networks, we obtain a closed-form expression for the generalization error to leading order in the learning rate, quantifying how task correlations and label noise determine the benefit of auxiliary learning. For non-linear activation functions, we develop a fluctuation-dissipation analytical theory that establishes a general relation linking the main and auxiliary errors to the corresponding single-task error. Numerical experiments support the theoretical predictions and show how auxiliary tasks improve generalization by balancing the forcing dynamics towards the optimal solution with gradient noise.

Federico Milanesio, Alessandro Ingrosso, Matteo Osella

Department of Physics and INFN University of Turin Via Giuria 1, 10125 Turin, Italy · Donders Centre for Neuroscience Radboud University Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands

Statistical Physics of Learning Neural Network Generalization Neural Network Training Dynamics

Sep 22, 2026cs.LG

A Spectral Theory of Grokking: Weight Decay induces Feature Learning

In grokking an early fit to the training data separates from a much later improvement in generalization. During this delay, training can move from a fixed neural tangent kernel (NTK) regime to one in which task-relevant kernel eigendirections continue to evolve. We provide a quantitative theory for how this transition from lazy to rich learning can produce delayed generalization. For homogeneous networks trained with squared loss and

L_2

weight decay, we show that a finite residual remains after memorization, with larger residual fractions in target components associated with smaller NTK eigenvalues. These residuals feed back into the dynamics of the NTK itself, and projecting the resulting dynamics onto task-relevant spectral directions yields a reduced system in which residual-driven kernel growth competes with weight decay. This system predicts that the grokking timescale is controlled by the product of learning rate and weight decay, that feature learning slows logarithmically near a critical decay above which task-aligned NTK structure can no longer support generalization, and that stronger decay can prevent fitting altogether. We test these predictions in modular addition. In a homogeneous MLP, task-aligned Fourier structure continues to emerge in the NTK after training accuracy has saturated, and an 84

\times

90-grid of trained networks across varying learning rate and weight decay recovers the predicted phase geometry and inverse-product scaling of the generalization time with learning rate and weight decay. A one-block Transformer shows similar macroscopic phase structure in a 42

\times

45-grid, as well as the same transition-time scaling despite violating exact homogeneity. Together, these results provide a mechanistic derivation connecting post-fit feature learning to both the onset of generalization and its phase structure in the learning rate and weight decay plane.

Lenz Pracher, Pascal de Jong, Oskar Lieshaus +2

Arnold-Sommerfeld-Center for Theoretical Physics, Ludwig-Maximilians-Universität München, Munich, Germany · Department of Applied Physics, Standford University, Standford, CA, USA · Department of Mathematics, University of Cambridge, Cambridge, UK

Neural Network Generalization Weight Decay Neural Tangent Kernel Grokking

Sep 22, 2026stat.ML

Generalized Deep Regression for Repeated Measurements

In this paper, we study the estimation of a marginal regression function from independent units with repeated binary, count, or continuous responses using ReLU deep neural networks. In the model, we assume that the dependence is generated by an unobserved random mean function within each unit. We then fit a neural network with a convex generalized regression loss. We show an oracle inequality by separating conditional measurement variation from between-unit variation. In addition, we prove that with

n

units and

m

measurements per unit, ReLU networks can attain an integrated mean squared error of order

n^{-1}+(nm)^{-2β/(2β+d)}

, up to logarithmic factors, over

β

-Hölder classes. We also derive a weighted oracle inequality for unequal cluster sizes and a rate for compositionally smooth functions. For pointwise ensemble inference, we give a projection central limit theorem and prove infinitesimal jackknife consistency under an explicit asymptotic linearity condition. Simulations and real data examples are provided to support our theoretical findings and practical implications.

Kexuan Li

Bristol Myers Squibb

Neural Network Generalization ReLU Neural Networks Nonparametric Regression

Sep 16, 2026cs.LG

Enhanced Agriculture-informed Neural Network by Domain Knowledge

Accurate prediction of nitrous oxide (N2O) emissions from agriculture is important for assessing environmental impacts and supporting sustainable farming. However, prediction remains difficult because N2O emissions result from complex interactions among soil properties, climate, biochemical processes, and management practices, while high-quality observations are limited. Deep learning models can capture nonlinear relationships but often lack physical interpretability and may generalize poorly across environmental conditions. We propose the Knowledge-enhanced Agriculture-informed Neural Network (KAINN), a hybrid neural-mechanistic framework that extends the Agriculture-informed Neural Network by incorporating domain knowledge about fertilizer diffusion, soil respiration, and water-filled porosity. We evaluate KAINN using CNN, LSTM, and Transformer architectures across multiple growing seasons and input-feature configurations. The results show that KAINN generally provides lower root mean square error and mean absolute error and higher R-squared values than purely data-driven models and the original AINN. Analysis of the learned interfaces also shows smoother and more physically consistent parameter trajectories with reduced uncertainty. These findings demonstrate that incorporating environmental knowledge into neural networks can improve the reliability, interpretability, and generalization of agricultural N2O-emission predictions.

Ci Lin, Futong Li, Rose Chong-Wu +2

University of Ottawa, Canada

Neural Network Generalization

Sep 14, 2026cs.CV

Assessing nnU-Net Generalization across Brain Tumor Populations in BraTS-GoAT 2026

BraTS-GoAT evaluates tumor segmentation across heterogeneous populations. We trained a conventional 3D nnU-Net on 1,351 labeled cases using five-fold cross-validation and 1,000 epochs per fold. The final predictor averaged all folds and applied test-time mirroring. On pooled official validation, global DSC values were 0.7805, 0.8288, and 0.8854 for enhancing tumor (ET), tumor core (TC), and whole tumor (WT). Under matched fold-0 inference, mean regional Dice decreased from 0.9058 on source out-of-fold (OOF) cases to 0.8310 on pooled validation (difference--0.0747). Mirroring gave small single-fold gains but no clear ensemble benefit; a residual-encoder alternative reached 0.8282 mean Dice. In labeled OOF predictions, failure cases had substantially smaller reference ET volumes; after adjustment for ET and WT volume, lower Dice remained associated with more disconnected ET components and a smaller fraction of ET contained in the largest component.

Tristan Kirscher, Vivian Metzger, Philippe Meyer +1

ICube, Institut Strauss · ICube Laboratory, CNRS UMR 7357, University of Strasbourg, Strasbourg, France · Institut Strauss +2

Neural Network Generalization U-Net 3D Medical Image Segmentation Medical Image Benchmarks Brain Tumor Segmentation

Sep 14, 2026cs.LG

Structured Features Overfit Where Random Features Grok

Xu, Vardi and Safran (ICML 2026) prove that over-parameterized ridge regression over an unstructured random Gaussian feature map groks, with the delay between memorization and generalization growing as

1/λ

in the weight decay. We show that on a structured feature map the same delay does not appear. For a band-limited Fourier feature map over

\mathbb{Z}_p^2

carrying a single-character target that lies inside the expressible class, enlarging the band at fixed positive weight decay drives peak held-out accuracy monotonically from

1.00

to

0.07

, with no memorize-then-generalize regime anywhere along the sweep. The degradation is not an interpolation effect. It sets in at capacity ratio

q/n = 0.638

, far below the interpolation threshold, on separate grounds from the exact null space that appears above it. What does have a sharp boundary is the active support. Holding the nominal dimension fixed and masking the band back to

1089

active modes restores held-out accuracy of

1.000

with zero variance across seeds, while the full

4225

-mode band collapses to

0.185

. The number of active modes acts through the teacher-weighted spectrum of the empirical Gram matrix and not through the capacity ratio, which makes this a statement about feature geometry and not a restatement of double descent.

Chon-Fai Kam, Miloud Bessafi, Frederic Cadet

University Paris City & University of Reunion, Paris, France; Dipartimento di Fisica e Chimica · Emilio Segr`e, Universit`a degli Studi di Palermo, Italy · EnergyLab, University of Reunion, Saint-Denis, France +2

Neural Network Generalization Ridge Regression Grokking

Sep 14, 2026cond-mat.dis-nn

Hierarchical Prototype Emergence in Modern Hopfield Models

Hierarchical correlations are a universal feature of any realistic model of data, and the question of how associative memory models may learn these correlations and generalize beyond them to construct new sensible images is an important step towards understanding more complex modern architectures such as diffusion models. We consider a hierarchical model for memories which are sampled and stored in a dense Hopfield network with polynomial activation. We analytically derive conditions for each level of this hierarchy to be locally stable - that is they are local energy minima. We use prototype reconstruction as a minimal model of generalization and we find that it takes only a quasi-polynomial amount of information to generalize beyond particular memories and even particular groups in the hierarchy. We observe a qualitatively analogous phase diagram in the number of memories, sharpness of the activation function (polynomial degree) for data from Fashion-MNIST.

Aditya Cowsik, Adithya Sriram

Department of Physics, Stanford University, Stanford, CA 94305, USA

Neural Network Generalization Hopfield Networks Hierarchical Memory Associative Memory

Sep 14, 2026cs.LG

Theoretical Guarantees for One-Shot Magnitude Pruning and Compute-Adaptive Early Exit

We study compute reduction in neural networks through a unified partial versus full computation view, captured by one-shot magnitude pruning in the static regime and early exit in the adaptive regime. In an asymptotic single-neuron model, we prove a concentration theorem for one-shot magnitude pruning with explicit rates. We also introduce the conditional perceptron for early exit and show that its excess generalization error decays as a power of the compute gap, with an exponent that grows to infinity as the alignment between partial and full computations tends to one. We then extend the analysis to deep networks, characterizing how pruning-induced distortions accumulate with depth and deriving a corresponding compute-accuracy tradeoff for frozen-backbone early exit under a neural network Gaussian process model. Numerical simulations corroborate the predicted scaling laws.

Erdem Koyuncu

Department of Electrical and Computer Engineering, University of Illinois Chicago, IL, USA.

Neural Network Generalization Early-Exit Neural Networks Neural Network Pruning

Sep 8, 2026cs.LG

Explaining f-Divergence-Based Regularization via Local Curvature and Sharpness-Aware Minimization

Divergence-based regularization and Sharpness-Aware Minimization (SAM) are two prominent approaches for improving generalization in deep learning, both motivated by robustness to perturbations. However, their relationship has remained largely unexplored. Building on classical second-order expansions of

f

-divergences, we show that the two methods are locally consistent under parameter-space perturbations: both induce curvature-sensitive penalties, with divergence regularization yielding a Fisher-weighted quadratic form and SAM penalizing sharpness through the dominant Hessian eigenvalue. For negative log-likelihood objectives with exponential-family output distributions, this correspondence becomes especially transparent, since the Fisher and Gauss-Newton matrices coincide. We further show that the same local geometric perspective extends to input-space perturbations, where divergence-based regularization is defined through transformations of the input. In this setting, the regularizer induces a pullback quadratic form on the input space, providing a more general perturbation framework than standard SAM while preserving the same local sensitivity interpretation. To validate the analysis empirically, we use the asymmetric

α

-skew Jensen-Shannon divergence (JSD) family as a controlled testbed. Its local curvature coefficient scales as

α(1-α)

and is maximized at the symmetric point

α=\tfrac12

, which recovers the standard JSD. Loss-landscape visualizations in the input-perturbation regime show that stronger induced curvature penalization is associated with flatter local minima. Experiments on four benchmark datasets further demonstrate that both accuracy and negative log-likelihood are consistently best near this regime of maximal curvature penalization.

Nour Jamoussi, Marios Kountouris

Communication Systems Department EURECOM France · Department of Computer Science and Artificial Intelligence, University of Granada, Spain

Neural Network Generalization Sharpness-Aware Minimization

Sep 8, 2026cs.LG

Nearly Tight Rademacher Bounds for Sparsely Activated Neural Networks

An input may activate few hidden units even when different inputs collectively use an entire network. We study the statistical complexity of this input-dependent sparsity in the one-hidden-layer ReLU model of Awasthi et al. (COLT 2024). For width

s

, at most

k

active units per input, and effective weight and bias bounds

W,B

, every size-

m

sample in the class's fixed radius-

R

input domain satisfies

\mathcal{R}(S)\le CWR\min\{k,\sqrt{sk/m}\log^{3/2}(2m)\}+kB/\sqrt m

. A support-preserving cover and a single normalized chaining argument remove the previous explicit dimension factor, up to logarithms. Lower bounds on appropriate i.i.d. marginals match up to those logarithms, showing how changing active units across inputs retains a width dependence. The input domain matters: zero-bias networks sparse on the entire ball have at most

2k

nonzero units and complexity

O(kWR/\sqrt m)

, whereas bias bounds comparable to

WR

restore the worst-case rate on that same domain in only logarithmic dimension. A spherical-cap construction proves the latter claim without assuming sparsity merely on the sampling support. For a specified normalized bounded loss and biases comparable to

WR

, we also obtain agnostic minimax excess-risk bounds of order

\min\{1,\sqrt{s/(km)}\}

up to logarithms.

Xiaoyu Li, Zhizhou Sha, Jiaojiao Jiang +2

University of New South Wales · University of Texas at Austin · University of Sydney

Neural Network Generalization ReLU Neural Networks Activation Sparsity Statistical Learning Theory

Sep 7, 2026cs.LG

A Theoretical Analysis of Generalization Dynamics in Neural Networks under Gradient Descent with Weight Decay

Understanding generalization remains a central challenge in machine learning because it requires jointly considering data, architecture, and training dynamics. In this paper, we develop a theoretical framework that characterizes how these factors jointly shape generalization performance throughout training. More precisely, we study a broad class of neural networks trained under the

\ell^2

loss by gradient descent (GD) with weight decay, and prove the convergence of GD to a neighbourhood of the global minimizers of the empirical loss. By partitioning the space based on the input data, we then decompose the population error into data error, optimization error, and prediction variation error, and bound them separately. In particular, for the prediction variation error, which measures the oscillations of the learned function, we propose (local) approximate homogeneity and derive explicit cellwise and layerwise bounds for its evolution along the training trajectory. These bounds yield two important implications: a necessary condition of improved generalization explains differences in layerwise generalization behavior; a sufficient condition describes delayed generalization and provides a theoretical characterization of grokking.

Yuqing Wang, Ioannis G. Kevrekidis, Mikhail Belkin

Johns Hopkins University · University of California San Diego

Neural Network Generalization Neural Network Training Dynamics Gradient Descent Dynamics Generalization Bounds

Aug 31, 2026cs.LG

Measuring Memory and Generalization as Separable Geometric Channels: The Topo^2 Framework

Deep networks trained on noisy labels simultaneously generalize on clean data and memorize flipped labels. These are usually conflated as pressures on one capacity. We present Topo^2, a measurement framework that makes them causally separable, measurable, and law-governed. Persistent-homology H1 structure of the representation space separates into a within-class manifold channel (a function of the training stopping point) and a cross-class channel (a monotone readout of memorized flipped samples). An intervention, the FM0 prescription (zero loss on flipped samples from epoch 0), reaches each setting's generalization ceiling while memorizing essentially nothing. Within the framework we establish a law set with graded evidence: (L2) FM0 separation prescription (9/9); (L1) the within-channel as a training-position function (mid-rise 6/6; convergence-back CIFAR 3/3, SVHN 2/3); (L3) a ring-construction identity (definitional, not a law); and TLS (memory-generalization topological layering): memory is causally additive, anchored (silencing clean collapses the representation), invertible (stripping memory restores near-ceiling generalization), and quantitatively billable (the memorization cost law, effective slope coefficient C ~ 0.38 at the reference capacity: CIFAR-10 0.3801 / SVHN 0.3806 / CIFAR-100 0.384 / VGG 0.3715, capacity-dependent in general and traced to clean-sample feature displacement). We also publish the framework's boundaries: a falsification ledger of nine dead ends, and an instrument-vindication section that excludes six families of global statistics as explanations of the within-channel. The framework turns "memorization" from an ill-defined capacity into a measurable, separable, invertible topological layer.

Zhanbo Zhang, Ming Liu, Qing Wang

Anhui iFlytek Yinglian Technology Co., Ltd.

Neural Network Generalization Neural Network Memorization Noisy-Label Learning Topological Representation Learning

Aug 26, 2026cs.LG

Mapping the Emergence of Regularization-Driven Dynamics in Grokking

For overparameterized neural networks, many solutions can fit the training data equally well while behaving very differently on unseen samples. Grokking separates training fit from visible generalization, providing a window for studying how this selection develops during training. We sweep short, fixed-duration weight decay (WD) perturbations across the pre-generalization plateau and measure how they shift later generalization time. Across three grokking tasks, these shifts are unordered early in the plateau but later form a stable dose ordering before visible generalization, with stronger WD increases leading to earlier generalization and stronger WD decreases leading to later generalization. Test-loss barriers between perturbed and baseline generalization checkpoints collapse toward zero while the ordered timing effects persist. A similar response reorganization is observed under

\ell_1

regularization in the grokking setting of Junior et al. (2025). Drawing on Waddington's developmental landscape as an analogy, we call this combination of increasingly constrained solution selection and persistent dose-ordered timing shifts the canalization of grokking solution selection. Together, our response maps and loss-barrier measurements reveal a dynamical reorganization before visible generalization that is consistent with the theoretical picture of regularization-driven motion along a stable slow manifold (Boursier et al., 2025).

Yiming Lin, Yuxuan Wang

School of Artificial Intelligence, University of Chinese Academy of Sciences

Neural Network Generalization Neural Network Training Dynamics Grokking

Aug 12, 2026cs.LG

TESLA: Taylor Expansion of Sinusoidal Learnable Activations

The parity problem--deciding whether the number of ones in a binary vector is odd or even--remains challenging for standard neural networks due to linear inseparability and the need for global interactions. We propose TESLA, an activation defined as a learnable combination of sine and cosine terms, enabling explicit control over polynomial degree and selective amplification of high-order components. Theoretically, we show that constraining TESLA's coefficients yields Lipschitz/Rademacher complexity bounds and shapes the training dynamics to emphasize higher-frequency structure. Empirically, on parity with input length n = 32, TESLA attains strong generalization with 100K training samples (approximately 0.002% of the 2^32 input space) and remains robust under heavy corruption, retaining high accuracy with up to 30% label noise. We also compare against periodic and frequency-based baselines (SIREN, SNAKE, and Fourier feature embeddings) on parity and Forrelation. Beyond synthetic structure, TESLA delivers comparable performance on ImageNet-100, indicating that activation-level degree control transfers to more general vision workloads. Code: https://github.com/KAU-QuantumAILab/TESLA

Daehwa Ko, Jaehyeon Kim, Seunghyun Ham +1

Korea Aerospace University, Goyang, Republic of Korea

Neural Network Generalization Neural Network Activation Functions Neural Network Training Dynamics

Aug 9, 2026cs.LG

No Unique Minimizer, No Problem: On the Consistency of Robust Neural Classifiers

Neural network classifiers trained by cross-entropy minimization are highly sensitive to label noise and adversarial contamination. While robust alternatives offer bounded influence and resistance to corruption, their statistical foundations in the deep learning setting are insufficient due to a fundamental difficulty: neural parameterizations are non-identifiable, so the population loss minimizer is an equivalence class of parameters, not a unique point. We develop a consistency theory for robust neural classifiers based on the S-divergence family that requires no identifiability assumption. Casting training as stochastic optimization over a non-identifiable parameter space, we prove that empirical S-divergence minimizers converge to the population-optimal equivalence class under mild regularity conditions, and verify these conditions for three architecture choices. We further establish that limit points of the robust training algorithm are stationary points of the empirical objective. Experiments on vision and language benchmark datasets confirm that S-divergence training maintains clean-data accuracy while exhibiting performance competitive with existing robust methods.

Subhabrata Majumdar, Anand Deo, Partha Pratim Saha +1

Indian Institute of Management Bangalore · Independent Researcher · Indian Statistical Institute, Kolkata

Neural Network Generalization Neural Network Robustness Robust Loss Functions Learning with Noisy Labels Statistical Learning Theory Robust Estimation

Aug 7, 2026cs.LG

Mathematical Principles and Experimental Discoveries of the Emergence of Symbolic Patterns in Artificial Neural Networks

Artificial Neural networks (ANNs) are often treated as black-box models, making explainability a central challenge in deep learning. Many engineering methods have been proposed to approximately explain the ANN from various perspectives, such as feature attribution and visualization. However, it remains a long-standing open question whether the complex inference logic of an ANN can be explained exhaustively and concisely as sparse symbolic patterns. This raises a deeper inquiry: does the emergence of symbolic patterns reflect a natural law rather than chance? Here, we show that across a broad class of ANNs trained on diverse tasks, their inference logic can indeed be reformulated as sparse symbolic interactions. We further prove that two common mathematical criteria, which are implicitly required across tasks, lead to the emergence of such sparse symbolic interactions. Empirical evidence confirms that the two criteria hold for the majority of input samples in diverse models. Furthermore, the faithfulness of these interactions is also demonstrated by their strong sample-to-sample and model-to-model transferability, as well as their ability to explain the overall generalization power of ANNs. Our theoretical analysis and extensive experiments provide a solid foundation for symbolic explanations of ANNs, and offer novel insights into the ANN's generalization power. Our findings also highlight the potential of communicative learning, a paradigm in which the inference logic of an ANN can be directly inspected and tuned at the level of symbolic patterns, thus complementing traditional end-to-end learning paradigm. Finally, the observed emergence of symbolic patterns in ANNs suggests that similar symbolic representations may also emerge in other types of black-box systems under certain conditions, because our proof does not depend on any specific ANN architecture.

Quanshi Zhang, Qihan Ren, Siyu Lou

School of Computer Science, Shanghai Jiao Tong University, Shanghai, China

Neural Network Generalization Neural Network Interpretability