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

Aug 3, 2026cond-mat.dis-nn

Tunneling the Loss Landscape: Bypassing Memorization with Monte Carlo Parameter Swapping

Grokking is a striking phenomenon in neural network training, where a model can undergo a prolonged period of pure memorization before abrupt generalization. While previous works have attempted to interpret it through classical machine learning mechanisms like weight norm, recent research draws an analogy from statistical physics, framing grokking as a form of computational glass relaxation. This theory defines the initial memorization as a result of fast cooling' where the training loss is reduced so quickly that a glass state is formed, followed by a slow relaxation' towards final generalization. Although providing a unifying framework for representative grokking theories, this perspective has remained largely at the theoretical on macroscopic level without direct empirical validation on training dynamics. Here we introduce a three-component framework to directly characterize the training dynamics via parameter mobility (PM), and two representative measurements from glassy dynamics: replica correlation (RC) and fractal dimension (FD). We demonstrate that standard optimization presents clear signatures of glass dynamics and inherently traps the grokking network in a kinetic arrested memorization state with a collapsed mobility, strong history dependence, and channel-like motions. This quantitative agreement motivates us to introduce State-Aware Monte Carlo Parameter Swapping (SAM-Swap), an optimization plug-in that can accelerate generalization, inspired by swap Monte Carlo algorithm widely used in glass dynamics. Comparing SAM-Swap, weight decay, and Gaussian gradient noise, we find that accelerated generalization is consistently associated with random exploration in the parameter space, similar to diffusion in physics.

Lai Shun Chan, Xiaotian Zhang, Yue Shang +2

Department of Physics, City University of Hong Kong, Hong Kong, China · Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA, USA · 3Innovation Campus Delaware, Air Liquide, Newark, DE, USA

Neural Network Generalization Markov Chain Monte Carlo Neural Network Memorization Neural Network Training Dynamics Grokking

Jul 29, 2026cond-mat.dis-nn

On the robustness of noisy solutions in non-convex neural networks

Optimization in non-convex neural network models is strongly influenced by the geometry of the solution space: sparse, isolated, point-like clusters are typically algorithmically inaccessible, whereas wide and flat regions can be found efficiently despite being relatively rare. At zero temperature this picture has been formalized in binary perceptrons through the overlap gap property (OGP), which limits algorithmic access to configurations with zero training error above a critical constraint density

α_{\rm OGP}

. Here we extend this description to finite temperature, where a positive training error is allowed and statistically penalized. We first show that the frozen one-step replica-symmetry-breaking solution, dominating the zero temperature equilibrium measure, survives at any finite temperature. We furthermore derive a general criterion, based on the smoothness of the single-pattern Gibbs weight near the decision boundary, that determines when a finite-temperature relaxation of the loss removes freezing. We then extend the OGP construction to finite temperature and show that dense, algorithmically accessible regions of finite-energy configurations persist beyond

α_{\rm OGP}

, up to a threshold

α_{\rm OGP}(ε)

that grows with the allowed training error

ε

. Finally, in the teacher-student setting, we show that these wide, finite-energy regions still retain good generalization. Using a finite energy message-passing algorithm, we demonstrate numerically that thermal noise enables effective generalization in the regime of constraint densities where both recovering the teacher and finding a zero temperature solution are computationally hard.

Enrico M. Malatesta, Alessandra Passalacqua, Riccardo Zecchina

Department of Computing Sciences, Bocconi University, Milano, Italy · Bocconi Institute for Data Science and Analytics (BIDSA)

Stochastic Optimization Neural Network Generalization Nonconvex Optimization

Jul 28, 2026stat.ML

More Data, Worse Decisions? Preference Reversals in Neural Networks under Gram Incompatibility

Neural networks increasingly combine data across populations, time periods, and operating conditions to improve generalization. This raises a reliability question: whether a model refitted on pooled data preserves an action ordering supported by both sources. Case-Based Decision Theory (CBDT) formalizes this requirement through its composition axiom, which requires source-supported preferences to survive their union. We study when this property holds for fixed-representation neural networks with ordinary least squares (OLS) output heads. First, we show that pooled refitting recomputes the inverse-Gram geometry used to weight source evidence, which can reverse shared preferences, and derive exact and approximate preservation conditions. Next, we introduce a scale-invariant Gram mismatch measure for prioritizing candidate pools and geometry-oriented regularization for shaping source geometry during training. Finally, we develop a three-stage audit that traces strict pairwise reversals through decision changes to task-defined utility loss. Experiments spanning a load-based bidding proxy and medical and financial decision proxies reveal stable and reversal-prone pooling regimes: the load audit identifies a measurable nonzero class of source-consensus-relative harmful decisions under the proxy utility, while cross-domain audits show that comparable mismatch can correspond to sharply different preservation rates. Geometry-oriented objectives occupy distinct descriptive accuracy-consistency-geometry-harm operating points. Together, the framework makes compositional reliability measurable and operational through screening, analytic certification, geometry-oriented training, and decision-consequence auditing.

Yanli Yan, Yuanzheng Li, Yong Zhao +2

Huazhong University of Science and Technology Wuhan, China

Neural Network Generalization Neural Representation Geometry Decision-Making under Uncertainty

Jul 27, 2026cs.LG

Mechanisms of Width Scaling in Normalized Residual Networks: The Effective Alignment Dimension

Existing theories of neural-network width characterize asymptotic limits, but provide limited guidance on whether an expansion direction identified from finite training data remains beneficial on unseen data. We study this problem for function-preserving residual expansion and introduce the effective alignment dimension, a measurable quantity describing the signal-noise geometry of activation gradients. By deriving the exact mean and variance of the inner product between independently estimated training and test gradients, we obtain a finite-sample upper bound on misalignment probability. The bound depends only on the effective alignment dimension and an effective sample size, requiring finite second moments and a nonzero population gradient, without covariance spectral assumptions or prescribed width-growth rates. We integrate this certificate into the train-test residual-expansion framework, yielding a high-probability condition for test-risk improvement. Experiments across width-controlled LLaMA-style Transformers, Pythia, and ResNet-20 show that wider models exhibit larger effective alignment dimensions and lower empirical misalignment. Direct residual interventions confirm that the alignment statistic predicts the sign and magnitude of held-out loss changes.

Jinhao Zhang, Zeyu Liu, Zicheng Yan +4

Beijing University of Posts and Telecommunications · Institute of Computing Technology, Chinese Academy of Sciences · University of Science and Technology of China +1

Neural Network Generalization Residual Learning Neural Scaling Laws Gradient Alignment

Jul 27, 2026cs.AI

Grokking on the Weight-Decay Clock: A Rate Hierarchy from Softly Broken Symmetries

Delayed generalization, or grokking, remains poorly understood despite extensive empirical study. We identify an exactly solvable late-time relaxation mechanism for grokking in linear models trained with full-batch heavy-ball optimization and weight decay, together with a locally quadratic extension to nonlinear neural networks. Our analysis reveals a distinguished population-active component of the empirical null space, which we call the grokking subspace. Along this subspace, the training predictions remain unchanged, leaving weight decay as the sole restoring force and giving rise to a slow dissipative relaxation governed by an exact discrete-time and continuous-time law. We show that only this subspace contributes to the slow asymptotic decay of the population risk and derive explicit iteration-scale predictions for the grokking time, recovering the familiar

(1-β)/(ηλ)

scaling in the weak-regularization regime. The theory further predicts distinct effects of optimizer choice, distinguishing coupled

L_2

regularization from decoupled weight decay, and yields causal predictions for interventions that modify the grokking component. We verify all theoretical identities without fitted parameters in a synthetic model where every subspace and relaxation rate is computable in closed form. We further observe genuine delayed generalization in modular addition, where the measured delay follows the predicted scaling and the late-time relaxation agrees closely with the theoretical clock.

Taeyoung Kim

Center for AI and Natural Sciences Korea Institute for Advanced Study Seoul 02455, Republic of Korea

Neural Network Generalization Weight Decay Grokking Gradient Descent Dynamics

Jul 26, 2026cs.LG

A Statistical Difference between Single-Layer Learning and Hierarchical Learning in Wide Neural Networks

Hierarchical neural networks are widely used in artificial intelligence, yet their mathematical properties remain incompletely understood. In the infinite-width limit, two different theoretical frameworks have been proposed. One reduces deep learning to kernel regression with a fixed kernel by assuming that the parameters remain close to their initialization, whereas the other allows the parameters to move away from their initialization, requiring the kernel itself to be optimized. In this paper, we study a three-layer neural network with a finite but large number of hidden units. We show that training the input-to-hidden weights yields a smaller generalization error than keeping them fixed. Furthermore, the latter setting exhibits singularities in the parameter space, whereas the former does not. These findings indicate that singularities play an essential role even in wide neural networks.

Sumio Watanabe

RIKEN Center for Advanced Intelligence Project 1-4-1 Nihonbashi, Chuo-ku, 103-0027, Tokyo, Japan

Neural Network Generalization Hierarchical Representation Learning Neural Network Training Dynamics

Jul 23, 2026quant-ph

Cautious optimism for deep parameterized quantum circuits

A central challenge in quantum machine learning is understanding the scaling behavior of parameterized quantum circuits (PQCs). In particular, it remains unclear how their performance on unseen data changes as the number of trainable parameters increases. Prior works have derived formal generalization guarantees for quantum models, but it is well-known that many such results do not fully characterize generalization behavior in practice. In this work, we show that gradient-based PQCs can exhibit improved performance on unseen data as model size increases, displaying the phenomenon of double descent. This contrasts with the traditional view that larger models lead to degraded generalization. We provide analytical results rigorously underpinning this behavior by leveraging add-one-in perturbation techniques and spectral properties of random matrices. We support these results with numerical experiments on re-uploading PQCs across several data sets and training set sizes, consistently observing the predicted double descent behavior. While other obstacles on the path toward practical quantum machine learning remain, our finding that deeper parameterized quantum circuits do not necessarily exhibit degraded performance provides reasons for cautious optimism.

Marie Kempkes, Elies Gil-Fuster, Carlos Bravo-Prieto +5

Leiden University, Niels Bohrweg 1, 2333 CA Leiden, Netherlands · Volkswagen Group Innovation, Berliner Ring 2, 38440 Wolfsburg, Germany · Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany +4

Double Descent Neural Network Generalization Quantum Machine Learning Variational Quantum Circuits

Jul 21, 2026cs.NI

NSMA: Neuro-Symbolic Manifold Alignment for Generalizable Adaptive Bitrate Streaming under Texture Shift

For decades, ABR has kept two kinds of intelligence apart. Neural policies learn rich behaviors yet forget them the moment the environment changes; rules never learn, and never forget. Every prior attempt to combine them has kept this separation, letting rules supervise, constrain, or override the network from outside. We dissolve the boundary itself. But no union can be trusted before it can be tested, and ABR has never known how to measure what its policies learn or forget. The field's yardstick is bandwidth statistics, and we show it misleads. Identical statistics can hide entirely different outcomes, while wildly different statistics can hide similar ones. We replace the yardstick before building the bridge, with Texture-Aware Generalization Evaluation, a protocol that judges a policy by its whole training journey across traces whose temporal nature is laid bare. What truly breaks a policy is invisible. No statistic reveals it, no feature extracts it, yet rules walk through it untouched, for they reason from physics and owe the data nothing. So we build the bridge. Neuro-Symbolic Manifold Alignment (NSMA) embeds rule decisions as anchors inside the latent space of the neural policy, so that it keeps learning where learning pays, and can no longer forget what rules have always known. Generalization cannot be argued, only survived. We raise NSMA on 3G traces alone and release it, without fine-tuning, into eight unseen datasets spanning 4G, 5G, and WiFi, and onto a real-world player. It outperforms every state-of-the-art baseline. And when we open its latent space to ask why, probing and visualization return the same answer the design promised. https://tinyzqh.github.io/NSMA/

Zhiqiang He, Zhi Liu

The University of Electro-Communications Tokyo, Japan

Neural Network Generalization Domain Generalization Neurosymbolic Learning

Jul 13, 2026cs.LG

Requential Coding: Pushing the Limits of Model Compression with Self-Generated Training Data

Compression is fundamental to intelligence. A model that can represent its training data as a short code has discovered regularities that enable generalization. Large neural networks may learn functions far simpler than their parameter counts suggest, but it is challenging to construct codes that realize this simplicity. Parameter-based methods such as quantization produce code lengths that scale with model size, insensitive to how much information the parameters store. Prequential coding bypasses this issue by compressing the training trajectory, but codes the exact data sequence regardless of how much the model learns, yielding large codes when the data has high entropy. We introduce requential coding, where a teacher model selects training samples drawn from the student's own distribution. The student's code records only these selections, which cost bits only where teacher and student disagree. The resulting code length is independent of parameter count and data entropy, and often orders of magnitude shorter than the prequential counterpart, with an advantage that grows with scale. This compression sheds light on phenomena inaccessible to prior compressors. Holding loss fixed, larger models and ensembles compress to much smaller sizes despite more parameters. Plugged into a PAC-Bayes bound, the requential code yields state-of-the-art generalization guarantees for billion-parameter LLMs, outperforming bounds built on aggressive post-training quantization even granted zero error. The bound tightens with scale in the compute-optimal regime, as models become increasingly compressible relative to dataset size. The same code predicts that models gradually overfit when trained for multiple epochs. It also isolates the learnable information in a dataset from its unpredictable, random content, revealing that lower-entropy text holds far more learnable structure than higher-entropy image data.

Shikai Qiu, Marc Finzi, Yujia Zheng +2

New York University · Carnegie Mellon University

Model Compression Neural Network Generalization

Jul 13, 2026cs.LG

How to Tame Grokking: Representation Geometry as a Control Signal

Grokking is a phenomenon in which neural networks initially memorize training data and only later exhibit strong generalization after prolonged optimization. Despite extensive recent study, the factors influencing the emergence and timing of grokking remain incompletely understood. We investigate the relationship between representation geometry and delayed generalization. We find that dimensionality collapse consistently precedes the onset of grokking in all evaluated settings. Motivated by these observations, we introduce Geometric Dimensionality Regularization (GeomDR), a simple spectral regularizer that modifies the effective dimensionality of hidden representations during training. Across modular addition, modular division, and permutation composition tasks, GeomDR consistently alters grokking dynamics and can substantially accelerate the onset of generalization depending on the intervention schedule and target dimensionality. In several settings, grokking is accelerated by up to 52 times relative to standard AdamW training. Similar qualitative effects are observed in both multilayer perceptrons and transformers. Together, these results suggest that representation geometry can serve as an effective control signal for grokking and provide evidence that geometric interventions offer a practical approach for studying and influencing delayed generalization in neural networks.

Maksim A Kazanskii

Independent Researcher

Neural Network Generalization Spectral Regularization Representation Collapse Neural Representation Geometry Neural Network Training Dynamics Grokking

Jul 13, 2026cs.LG

Random Label Prediction Heads for Studying Memorization in Deep Neural Networks

We introduce a straightforward yet effective method to empirically study memorization in deep neural networks for classification tasks. Our approach augments each training sample with auxiliary random labels, which are then predicted by a random label prediction head (RLP-head). RLP-heads can be attached at arbitrary depths of a network, predicting random labels from the corresponding intermediate representation and thereby enabling analysis of how memorization capacity evolves across layers. By interpreting the RLP-head performance as an empirical estimate of Rademacher complexity, we obtain a direct measure of both sample-level memorization and model capacity. We leverage this random label accuracy metric to analyze generalization and overfitting in different models and datasets. Building on this approach, we further propose a novel regularization technique based on the output of the RLP-head, which demonstrably reduces memorization. Interestingly, our experiments reveal that reducing memorization can either improve or impair generalization, depending on the dataset and training setup. These findings challenge the traditional assumption that overfitting is equivalent to memorization and suggest new hypotheses to reconcile these seemingly contradictory results. The source code is available at https://github.com/MarlonBecker/RandomLabelHeads

Marlon Becker, Jonas Konrad, Luis Garcia Rodriguez +1

University of Münster, Germany

Neural Network Generalization Neural Network Memorization Statistical Learning Theory

Jul 11, 2026q-bio.NC

Emergent Generalization by Representation Learning in Artificial Neural Networks

Dimensionality reduction has proven powerful for identifying neural manifolds, which are low-dimensional structures underlying high-dimensional neural activity. These low-dimensional representations have improved the interpretability of population-level coding. Yet whether such low-dimensional representations are biologically relevant and confer functional advantages in learning systems, or merely reflect neuron-level activity, remains contested in neuroscience. We show that an explicit information bottleneck forcing a recurrent neural network to learn a low-dimensional representation is necessary for rotational and out-of-distribution generalisation in a time-series prediction task. Using information-theoretic measures of causal emergence, we characterise the dynamics of this representation across the memorisation-to-generalisation transition, finding a non-monotonic trajectory which shows an initial decrease, a minimum, and a subsequent rise to a maximum, even as prediction loss falls monotonically. This trajectory scales with task complexity, and the magnitude of emergent structure reliably predicts generalisation performance. Analysis of CA1 hippocampal activity in mice learning an alternating maze task reveals analogous non-monotonic emergence dynamics that track behavioural performance. Together, these findings indicate that the ability of neural networks to learn compact, distributed and emergent representations confers a functional advantage for generalisation, supporting a causal role for learned representations in cognition.

Hardik Rajpal, Dan Goodman

1I-X Centre for AI in Science, Imperial College London, W12 0BZ, UK · Department of Electrical and Electronic Engineering, Imperial College London, SW7 2AZ, UK

Neural Network Generalization Representation Learning Information Bottleneck Recurrent Neural Networks Neural Encoding

Jul 10, 2026cs.LG

Similarity search generalisation in contrastive learning with InfoNCE loss

Similarity search is a primary application of embedding models trained by contrastive learning. For one of the most popular contrastive learning loss functions, InfoNCE, we show that the population risk with

k

negative samples is

O(1/k)

close to an expected cross-entropy which quantifies deviation between i) a softmax similarity search over unseen data using the learned embedding function, and ii) an idealised softmax search over the same data but using similarity implicitly represented in the positive sample generator. This complements existing interpretations of InfoNCE in the

k\to\infty

limit which are phrased in terms of mutual information, and alignment versus uniformity in embeddings. To quantify generalisation performance, we introduce a new continuity bound for the InfoNCE loss, obtained via Gâteaux differentiation. The bound preserves the structure of averaging over negative samples present in the loss function and features an ``inverse temperature'' parameter which can be tuned to account for the algorithmic temperature. For embedding functions which are Lipschitz in a parameter, this yields a simple demonstration that the averaging effect of

k

negative samples in the InfoNCE loss carries over to stabilisation of the generalisation error as

k

grows.

Nick Whiteley

School of Mathematics, University of Bristol, U.K.

Neural Network Generalization Contrastive Learning Statistical Learning Theory

Jul 8, 2026cs.LG

A law of robustness for two-layer neural networks with arbitrary weights

Bubeck, Li and Nagaraj conjectured that, for generic data, any two-layer neural network with

m

neurons that fits

n

noisy labels must have Lipschitz constant at least of order

\sqrt{n/m}

, with no restriction on the size of the weights. Bubeck and Sellke proved a universal version of this law for Lipschitz-parameterized classes, but under a polynomial bound on the parameters; at depth three that boundedness hypothesis is genuinely necessary. The two-layer unbounded-weight case requires a different argument. We prove the conjectured law, up to one logarithmic factor, for every continuous piecewise-linear activation, in particular for ReLU networks. For data drawn uniformly from

\mathbb{S}^{d-1}

,

d\ge3

, or from

N(0,I_d/d)

, labels in

[-1,1]

with noise level

σ^2>0

, and any width-

m

two-layer network with arbitrary real weights, biases and affine skip connection, fitting the data

\varepsilon

below the noise floor forces

\mathrm{Lip}(f)\ge c\,\varepsilon\sqrt{n/(\bar m\log(C\bar m nd/\varepsilon))}

,

\bar m=(K-1)m+1

, with high probability. A realized-kink-count version holds on the same event: every realized two-layer piecewise-linear function with

k(f)\le n

distinct kink hyperplanes obeys the bound with

\bar m

replaced by

k(f)+1

, irrespective of how many redundant hidden units parameterize it. The proof replaces parameter-space covering, impossible for unbounded weights, by a function-space covering. The central deterministic ingredient is a rigidity lemma: on

B_2

, and on

\mathbb{S}^{d-1}

for

d\ge3

, the coefficient of each canonical kink is controlled by the Lipschitz constant of the realized function, because kinks on distinct hyperplanes cannot cancel at generic points. Rigidity genuinely fails at

d=2

, and an explicit two-layer ReLU interpolant with

O(1)

Lipschitz constant at width

2n

matches the law at the overparameterized endpoint.

Yitzchak Shmalo

Neural Network Generalization Multilayer Perceptrons Neural Network Memorization Neural Network Robustness Adversarial Robustness Lipschitz Neural Networks

Jul 8, 2026cs.LG

Architecture Generalization with MetaNCA

Self-organization is an emergent property of life, driven by the collective behavior of individual components acting on local information. Biological neurons, through local interactions transmitted through synapses, are able to learn efficiently and can adapt their connections over an organism's lifespan. Motivated by these desirable properties of adaptability and local interaction, neural cellular automata (NCA) models have been successful at learning morphogenesis solely through local update rules, demonstrating stability over many updates and robustness to perturbations. In this work, we introduce Meta Neural Cellular Automata (MetaNCA), a framework that learns local rules which self-organize the weights of artificial neural networks. A learned rule network iteratively updates the weights of a task network using only local interactions on the computation graph. We propose a novel Weight Transformer architecture for the local rule network, which uses linear attention to aggregate signals from neighboring weights and hidden states. Once trained, the rule network generates task networks of diverse architectures without backpropagation. We show that MetaNCA generates weights for feedforward MLPs, CNNs, and ResNets on MNIST and CIFAR-100, scaling to networks of 2 million parameters. We further show that MetaNCA generalizes to architectures not seen during meta-training, and that architectural diversity in the training phase strengthens this generalization.

Meet Barot, Daniel Berenberg, Sina Khajehabdollahi

Mythos Scientific, New York, USA · Independent Scholar

Neural Cellular Automata Neural Network Generalization Meta-Learning Local Learning

Jul 7, 2026stat.ML

A Function-Space Dichotomy for Compositional Learning: Exponential Sub-Optimality of the Neural Tangent Kernel

A persistent empirical observation is that trained neural networks outperform their neural tangent kernel (NTK) limit on tasks with compositional structure, yet a quantitative account of

\textbf{when}

and

\textbf{by how much}

has been lacking. Working on the unit circle, we give such an account through a dichotomy between two complexity measures of the target: its

\textbf{Fourier complexity}

, which controls NTK kernel regression, and its

\textbf{architectural complexity}

, which controls learning over depth-

L

, width-

w

ReLU networks with the variation norm of the weights bounded by

R

. We first characterize the minimax rate of the architecture class

\mathcal{C}_{L,w,R}

, pinning it down up to a single factor of

L

: between

Ω(Lw^2R^2/n)

and

\tilde{O}(L^2w^2R^2/n)

. We then show the NTK estimator sits

\textbf{exponentially}

above this floor whenever the two complexities decouple: for the depth-

L

iterated sawtooth, NTK regression needs

Ω(4^L)

samples while the minimax floor is polynomial in

L

. Numerical experiments confirm the theoretical claims: on bandlimited smooth targets, the NTK is competitive or better, while on the hypercube sparse-parity model, a standard two-layer network beats the NTK by four to six orders of magnitude in test error. The gap is thus a function-space property, a mismatch between the kernel's smoothness bias and the target's compositional structure, rather than a generic kernel-versus-network phenomenon.

Arkaprabha Ganguli, Emil Constantinescu

Mathematics & Computer Science Division, Argonne National Laboratory, Lemont, IL, USA

Neural Network Generalization Kernel Regression ReLU Neural Networks Neural Network Approximation Theory Neural Tangent Kernel

Jul 7, 2026cs.LG

Leveraging Extragradient for Effective Sharpness-Aware Minimization in Deep Learning

Generalization remains a pivotal challenge in deep learning, where traditional optimizers like Stochastic Gradient Descent (SGD) often converge to sharp minima, leading to overfitting and reduced performance on unseen data. Building on Sharpness-Aware Minimization (SAM), for seeking flat minima associated with improved generalization, we propose the Extragradient-Inspired Sharpness-Aware Minimization (EISAM), a novel optimizer that enhances generalization via the extragradient technique. EISAM uses a two-step update process: a prediction step investigating the geometry of the loss landscape and a perturbation step that refines updates with a base optimizer. This approach achieves better generalization performance than SAM. Crucially, EISAM reduces sensitivity to the perturbation radius, enhancing robustness, and simplifying the tuning across diverse settings. Extensive experiments on benchmark datasets demonstrate that EISAM consistently outperforms SGD, Adaptive Moment Estimation (Adam), and SAM in test accuracy and training efficiency across various architectures. Theoretical analysis further confirms that EISAM tightens the generalization bound by steering parameters toward flatter minima with reduced curvature. Accompanied by a thorough hyperparameter analysis, EISAM offers practical tuning guidance, establishing it as a robust, scalable, and broadly applicable optimization solution that advances both the theory and practice in deep learning.

Yao Fu, Chunxia Zhang, Junmin Liu +3

School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, China · SGIT AI Lab, State Grid Corporation of China, Xi’an, 710054, China · School of Software, Xi’an Jiaotong University, Xi’an, 710049, China +2

Neural Network Generalization Sharpness-Aware Minimization Flat Minima

Jul 7, 2026cs.LG

Drift Happens: An Empirical Study of Neural Architecture Robustness to Temporal Distribution Shift

Real-world data distributions evolve over time, inducing temporal distribution shift that can substantially degrade the reliability of deployed machine learning systems. However, the extent to which architectural choices and their associated inductive biases affect temporal robustness remains insufficiently understood. We present a systematic empirical comparison of temporal robustness across three heterogeneous, time-indexed domains encompassing image classification, multi-label text classification, and text regression tasks. Using a unified evaluation framework based on temporal drift matrices, we train models on cumulative historical data and evaluate their performance on both earlier and later time periods, thereby quantifying cross-temporal generalization. Our study spans model families ranging from simple multilayer perceptrons and convolutional networks to recurrent networks and pretrained Transformer-based encoders. Collectively, the results show that architectural inductive biases systematically shape temporal robustness: models whose inductive biases lead them to exploit localized, highly discriminative features attain the highest in-distribution accuracy, yet those features are often the ones that change most over time, so these models degrade fastest, while pretrained encoders that draw on coarser, more stable representations drift more gradually. These observations offer practical guidance for selecting architectures for real-world systems subject to temporal drift.

Robin Holzinger, Riccardo Colletti

Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, USA

Distribution Shift Robustness Neural Network Generalization Neural Network Robustness Temporal Distribution Shift

Jul 7, 2026stat.ML

Width-Robust Learnability in Mean-Field Bayesian Neural Networks

Infinite-width limits are a standard way to reason about neural networks, but it is not automatic that the limiting learner has the same complexity-theoretic inductive bias as large finite networks. We study this question for Bayesian neural networks at the mean-field, or critical feature-learning, scaling. The central quantity is the \emph{reduced entropy}

s_\infty(y,\varepsilon)=\limsup_N -\frac{1}{N}\log π_N^0(L\le \varepsilon),

the intensive prior cost of representing a target function

y

to population mean-squared error

\varepsilon

. Our main result is a width-robust learnability theorem. At fixed depth, a family of Boolean-cube targets is learnable from polynomially many samples at infinite width if and only if it is learnable at polynomial width, if and only if its reduced entropy is polynomially bounded. Equivalently, up to polynomial slack in accuracy, the Bayesian mean-field learner generalizes exactly on the targets that can be represented by polynomial-size networks. The forward direction is proved by a form of subsampling: from the infinitely many hidden neurons in the mean-field solution, one can select polynomially many representatives and still preserve the learned function on every input simultaneously. At the critical scaling this subsampling has both an active'' component, which keeps the data-dependent low-dimensional statistics, and a lazy'' component, which resamples the entropy-dominated directions from the prior. Thus the infinite-width mean-field limit gives a clean analytic description of learning without introducing spurious width-dependent generalization power.

Dmitry Vaintrob, Kaarel Hänni

Principles of Intelligence

Bayesian Neural Networks Neural Network Generalization Sample Complexity Statistical Learning Theory

Jul 5, 2026cond-mat.dis-nn

Broken Ergodicity and the Violation of the Fluctuation-Dissipation Theorem Lead to Generalization Beyond Overfitting in Machine Learning

The remarkable ability of modern neural networks to generalize improves with increasing network capacity, even when the number of model parameters or effective degrees of freedom exceeds the number of training data points. This phenomenon is all the more surprising given that generalization error diverges when the number of model parameters approaches a critical value from below. Here we use dynamical mean field theory to show that this so-called "double descent" behavior is the outcome of a phase transition in the stochastic field theory describing the training process. We calculate the critical exponents and scaling function of the double descent phase transition, and show that it is marked by a breakdown of the fluctuation-dissipation theorem associated with broken ergodicity. The corresponding response function has the same functional form as the simple London model of the superconducting transition, with the rigidity of the wave function corresponding to the neural network's ability to generalize accurately.

Chan Li, Nigel Goldenfeld

Department of Physics, University of California San Diego, 9500 Gilman Drive, La Jolla, California 92093, USA. · Halıcıoğlu Data Science Institute, University of California San Diego, 9500 Gilman Drive, La Jolla, California 92093, USA

Double Descent Statistical Physics of Learning Neural Network Generalization Phase Transitions Neural Network Training Dynamics

Jul 3, 2026cs.LG

Statistically Meaningful Geometry (SMG) Beyond the Euclidean Paradigm, with Application to Generative AI

Conventional uniform convergence bounds and empirical risk minimization break down in massive over-parameterized models, such as large language transformers and biological sequence networks. With near-infinite unconstrained internal degrees of freedom, their optimization landscapes develop flat vertical gauge valleys, rendering classical generalization metrics vacuous and inducing severe pathologies, specifically generative hallucination and catastrophic forgetting. We introduce the Statistically Meaningful Geometry (SMG) framework, an information-geometric paradigm lifting deterministic parametric models into infinite-dimensional non-parametric Orlicz statistical manifolds. Modeling the total state space as a differential fiber bundle (

\mathcal{M}, \mathcal{B}, π, \mathcal{V}, \mathcal{H}, ω

), we establish a Two-Fold Inference Paradigm. We formalize an Ehresmann connection 1-form

ω

as a dynamic geometric filter that strips away vertical gauge noise (Structural Internal Directions, or SID) and isolates learning trajectories along the strictly non-degenerate horizontal distribution (Statistical Variational Directions, or SVD

χ

). We prove that under connection-filtered pre-training, out-of-distribution predictive variance is strictly upper-bounded by the finite diameter of the identifiable quotient base manifold

\mathcal{B}

, establishing a hard geometric containment of generative hallucinations. By projecting downstream updates onto the orthogonal complement of the historical horizontal carriage, we formalize the SMG Sequential Adaptation Flow, proving the total non-asymptotic elimination of catastrophic forgetting. SMG replaces empirical fine-tuning heuristics with coordinate-free topological constraints, bridging advanced differential geometry with structural reliability in AI.

Bing Cheng, Yi-Shuai Niu, Howell Tong +1

Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China · AMSS Center for Forecasting Science, Chinese Academy of Sciences, Beijing, China · Beijing Institute of Mathematical Sciences and Applications (BIMSA), Beijing, China +4

Neural Network Generalization Information Geometry Continual Learning for LLMs Hallucination Mitigation

Jul 2, 2026stat.ML

Benign Overfitting Does Not Occur in Diffusion Models

Benign overfitting and double descent have come to shape our understanding of generalization in deep learning, establishing that overfitting is not only compatible with good generalization but can actively benefit it. Diffusion models share much of the machinery of standard deep learning, so it is natural to assume that they also exhibit these properties. In this work, we show that this assumption is largely incorrect. We first establish fundamental impossibility results showing that, unless the sample size grows exponentially with the data dimension, overfitting and good generalization cannot occur simultaneously. Consequently, the population loss follows a classical U-shaped curve in model complexity rather than exhibiting double descent. Analyzing a simplified setting, we identify a key difference between regression and score matching: regression benefits from an alignment between the target and the empirical covariance; score matching admits no such alignment, leaving overfitting irreparably harmful. We further identify implicit regularization stemming from time-smoothness of the score and early stopping during training as mechanisms that prevent such overfitting and verify our findings with high-dimensional image generation experiments. Our results reveal that generalization in diffusion models is governed by mechanisms distinct from those of traditional regression, motivating the development of new theory.

Tyler Farghly, Benjamin Dupuis, Alain Durmus +1

1INRIA - École Normale Supérieure - PSL Research University - CNRS, France · 2 École Polytechnique - CMAP - IP Paris, France

Double Descent Neural Network Generalization Benign Overfitting Diffusion Models

Jul 2, 2026stat.ML

Born Discrete, Made Smooth: Variational Formulation of Shallow Neural Networks

Although neural networks are remarkably effective, their underlying optimization principles remain theoretically elusive, often characterized by non-convex landscapes and stochastic heuristics. In this work, we propose a paradigm shift by replacing the discrete training problem of shallow neural networks with a well-posed continuum variational surrogate. We identify a family of

λ

-convex functionals over parameter densities in weighted Sobolev spaces and prove that these variational problems are globally well-posed, stable, and exhibit unexpected almost

C^3

regularity. Unlike existing Wasserstein-based or Mean-Field approaches, which often face limited regularity and discretization challenges, our formulation provides direct access to elliptic regularity and convex analysis. This allows us to prove that the optimal parameter density can be obtained by solving a single linear system, bypassing iterative optimization entirely. We establish explicit generalization error controls at a rate of

1/α

relative to the regularization parameter, and prove that finite-width networks of size

N

achieve the continuum optimum at an

O(1/N)

rate. This perspective bridges the gap between the Neural Tangent Kernel (NTK) and feature-learning regimes, providing a principled framework for understanding over-parameterization through the lens of variational calculus.

Matej Benko, Pierre Bousquet, Iwona Chlebicka +1

Institute of Mathematics, Faculty of Mechanical Engineering, Brno University of Technology · Technická 2896/2, 616 69, Brno, Czech Republic · Université de Toulouse, INSA Toulouse, CNRS, IMT, F-31062 Toulouse Cedex 9, France +2

Shallow Neural Networks Neural Network Generalization Neural Network Optimization

Jul 1, 2026cs.LG

From Approximation to Emergence: A Theory of Deep Learning

Deep learning has outgrown any single mathematical explanation. From Approximation to Emergence develops a unified, proof-oriented account of modern deep learning theory, tracing a path from the classical foundations of approximation, optimization, and generalization to the contemporary mechanisms of overparameterization, robustness, generative modeling, transformers, in-context learning, scaling laws, interpretability, alignment, and emergence. Rather than presenting isolated results, the book organizes a broad literature into a coherent research narrative: each theory is examined through the object it controls, the assumptions that make it valid, and the phenomena it leaves unexplained. Written for researchers, graduate students, and mathematically trained practitioners, this monograph offers a rigorous map of deep learning theory as it stands today: powerful, incomplete, and increasingly centered on the question of how learned mechanisms arise from scale, data, architecture, and training.

Zhilin Zhao

Sun Yat-sen University

Neural Network Generalization Deep Learning Optimization In-Context Learning Neural Network Approximation Theory Neural Scaling Laws

Jun 30, 2026cs.LG

Radial Suppression Accelerates Algorithmic Generalization: A Geometric Analysis of Delayed Generalization

Why do neural networks memorize algorithmic training data long before they generalize? We present a geometric case study demonstrating that, on tasks where generalization requires discovering structured low-dimensional circuits, the memorization-generalization delay is driven by radial inflation of hidden representations under cross-entropy optimization. We formalize a radial-angular decomposition of activation-space dynamics and derive three testable propositions: (i) that penalizing radial inflation induces anisotropic, data-dependent weight regularization; (ii) that it suppresses radial gradient energy below the isotropic random baseline, forcing predominantly angular updates; and (iii) that it biases convergence toward flatter minima. To empirically validate these propositions, we study a single-hyperparameter norm penalty that softly constrains activations to a sqrt(d)-radius hypersphere. On modular arithmetic, this penalty accelerates grokking up to 6x across MLPs and Transformers, and halves training steps for a 10M-parameter nanoGPT on 3-digit addition.

Srijan Tiwari, Aditya Chauhan, Manjot Singh

Indian Institute of Technology Roorkee

Neural Network Generalization Neural Representation Geometry Neural Network Memorization Neural Network Training Dynamics Feedforward Neural Networks Grokking

Jun 30, 2026cs.LG

Revisiting the Volume Hypothesis

Modern deep neural networks often contain far more parameters than needed to fit their training data, yet they achieve impressive generalization. A common explanation for this success is the implicit bias of stochastic gradient descent (SGD). An alternative volume hypothesis posits that, within low training-loss regions, loss-landscape basins leading to strong generalization occupy much larger regions of weight space than basins that generalize poorly, and therefore SGD is simply more likely to land in the former. Recent experimental explorations of this idea present seemingly contradictory results. While in one set of experiments randomly sampling the network weights until achieving zero training error yielded poor generalization, molecular dynamics density estimates supported the volume hypothesis. We observe that these experiments were performed at different dataset size regimes, and explore an intermediate regime using the Replica Exchange Wang-Landau algorithm to estimate the joint density of states over training and test accuracies in binary networks. Across several architectures and datasets, we show that the generalization advantage of gradient learning over random sampling training generally diminishes as the training data size grows, suggesting a resolution of the paradox.

Ari Pakman, Lior Kreimer, Yakir Berchenko

Department of Industrial Engineering and Management, Ben-Gurion University of the Negev, Beer Sheva, Israel.

Neural Network Generalization Stochastic Gradient Descent

Jun 29, 2026cs.LG

Informational Frustration in Neural Manifolds: Shannon Bottlenecks and the Limits of Learnability

Why overparameterised deep networks generalise so remarkably well remains one of the most stubborn open questions in machine learning theory. Classical frameworks like VC dimension and Rademacher complexity predict catastrophic overfitting in modern models, leaving a massive theoretical gap between theory and reality. In this paper, we bridge this divide by introducing a unified framework that links information theory, topology, and statistical mechanics to map the hard limits of deep learning. Central to our approach is the Entropic Learnability Horizon (ELH): a fundamental law stating that a network can only truly learn a target function if the Shannon entropy of the data manifold outpaces the topological entropy of the function's decision boundary, balanced by the von Neumann entropy of the network's weight space. We establish the Shannon-Topological Bottleneck Theorem, proving that when a target boundary's geometric complexity exceeds this informational horizon, the system undergoes a sudden entropic phase transition. It falls into a state of Informational Frustration - a glassy, rigid memorization phase where generalization becomes thermodynamically impossible. Using this lens, we show that the enigmatic phenomenon of "grokking" is actually an Entropic Release, where weights abruptly reorganise to unlock the bottleneck. Finally, we translate this theory into practice with Entropic Gradient Descent (EGD), an optimization algorithm that dynamically manages weight entropy to keep learning on track. Ultimately, this work repositions entropy not just as a tool for tracking uncertainty but as the fundamental physical currency that dictates whether a machine can learn.

Srinivasa Rao P., Vangmayi P Reddy

Curlvee Technolabs, India, Former Scientist, C-DAC · Former Scientist, C-DAC · Indian Institute of Technology Madras, India

Statistical Physics of Learning Neural Network Generalization Information Bottleneck Statistical Learning Theory Entropy Regularization

Jun 29, 2026stat.ML

A Stochastic--Geometric Theory of Scaling Laws in Grokking

Delayed generalization (\ie~grokking) refers to the phenomenon in which a neural network fits its training data early in training but only begins to generalize after a prolonged delay, often through an abrupt transition. Despite extensive empirical study, its underlying mechanism remains poorly understood. In this work, we first theoretically characterize a shell--core topological configuration of the reachable solution space induced by Adam's optimization dynamics with weight-shrinkage regularization, supported by empirical evidence. This optimization-induced topological configuration gives rise to grokking. In model's parameter space, random initialization solutions concentrate on a thin outer spherical shell, enclosing another spherical shell of memorization solutions, which in turn contains a core corresponding to the generalization solutions. Leveraging stopping-time theory, we then analyze the geometry of this topological configuration and the solution transition time at which optimization trajectories escape the memorization manifold and first reach the boundary of the generalization manifold. Our theoretical analysis derives grokking scaling laws for the learning rate, batch size, and

\ell_2

regularization coefficient, which are further validated through experiments and shown to recover results from prior literature.

Róisín Luo, Christian Gagné, Jonas Ngnawé +2

Research Ireland – Centre for Research Training in AI · University of Galway · Universit´e Laval +2

Neural Network Generalization Neural Network Optimization Neural Network Memorization Neural Scaling Laws Grokking

Jun 28, 2026stat.ML

Generalization Analysis of Transformers in Distribution Regression

In recent years, models based on the Transformer architecture have seen widespread applications and have become one of the core tools in the field of deep learning. Numerous successful techniques, such as parameter-efficient fine-tuning and efficient scaling, have been proposed surrounding their applications to further enhance performance. However, the success of these strategies has always lacked the support of rigorous mathematical theory. To study the underlying mechanisms behind Transformers and related techniques, we first propose a Transformer learning framework motivated by distribution regression, with distributions being inputs, connect a two-stage sampling process with natural language processing, and present a mathematical formulation of the attention mechanism called attention operator. We demonstrate that by the attention operator, Transformers can compress distributions into function representations without loss of information. Moreover, with the advantages of our novel attention operator, Transformers exhibit a stronger capability to learn functionals with more complex structures than convolutional neural networks and fully connected networks. Finally, we obtain a generalization bound within the distribution regression framework. Through the aforementioned theoretical results, we further discuss some successful techniques emerging with large language models (LLMs), such as prompt tuning, parameter-efficient fine-tuning, and efficient scaling. We also provide theoretical insights behind these techniques within our novel analysis framework.

Peilin Liu, Ding-Xuan Zhou

School of Mathematics and Statistics, University of Sydney, Sydney, NSW 2006, Australia

Neural Network Generalization Transformer Attention Statistical Learning Theory

Jun 27, 2026cs.LG

How Far Can Sharpness and Complexity Jointly Explain Generalization?

Sharpness and complexity are two central factors in the generalization analysis of deep neural networks. Existing quantitative evaluations of generalization measures have largely focused on individual scalar measures, leaving the joint explanatory power of sharpness and complexity largely unexplored. This work studies how far sharpness and complexity can jointly explain generalization. We use linear regression and introduce a Pareto-based analysis to quantitatively evaluate the joint explanatory power of these two factors. Beyond the existing parameter-level definitions, we further propose realizations of sharpness and complexity that are closer to function space and less dependent on raw parameter representations. We find that function-oriented definitions of these two quantities expand the explanatory scope of the two-factor view beyond what is achieved by existing parameter-level metrics. Overall, our results support the sharpness-complexity perspective as an informative lens for understanding generalization across diverse settings. At the same time, the remaining failures indicate that whether this two-factor view can serve as a complete theory of generalization remains open.

Ziyu Cheng, Xitong Zhang, Longxiu Huang +1

Michigan State University

Neural Network Generalization Statistical Learning Theory