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.

Jul 13Week of Sep 28

Latest papers 204

May 5, 2026cs.LG

Information Plane Analysis of Binary Neural Networks

Information plane (IP) analysis has been suggested to study the training dynamics of deep neural networks through mutual information (MI) between inputs, representations, and targets. However, its statistical validity is often compromised by the difficulty of estimating MI from samples of high-dimensional, deterministic representations. In this work, we perform IP analyses on binary neural networks (BNNs) where activations are discrete and MI is finite. We characterise the finite-sample behaviour of the plug-in entropy estimator and identify regimes for sample size

N

and representation dimensionality

D

under which MI estimates are reliable. Outside these regimes, we show that empirical MI estimates saturate to

\log_2 N

, rendering IP trajectories uninformative. Restricting attention to the reliable regime, we train 375 BNNs to investigate the existence of late-stage compression phases and the relationship between compressed representations and generalisation performance. Our results show that while late-stage compression is frequently observed, compressed latent representations do not consistently correlate with improved generalization performance. Instead, the relationship between compression and generalisation is highly dependent on task, architecture, and regularisation.

Maximilian Nothnagel, Bernhard C. Geiger

Graz University of Technology, Graz, Austria · Know Center Research GmbH, Graz, Austria · Signal Processing and Speech Communication Laboratory, Graz University of Technology, Graz, Austria +1

Neural Network Generalization Binary Neural Networks Mutual Information Estimation Neural Network Training Dynamics Mutual Information

May 2, 2026cs.LG

A Theory of Generalization in Deep Learning

We present a non-asymptotic theory of generalization in deep learning where the empirical neural tangent kernel partitions the output space. In directions corresponding to signal, error dissipates rapidly; in the vast orthogonal dimensions corresponding to noise, the kernel's near-zero eigenvalues trap residual error in a test-invisible reservoir. Within the signal channel, minibatch SGD ensures that coherent population signal accumulates via fast linear drift, while idiosyncratic memorization is suppressed into a slow, diffusive random walk. We prove generalization survives even when the kernel evolves

\mathcal{O}(1)

in operator norm, the full feature-learning regime. This theory naturally explains disparate phenomena in deep learning theory, such as benign overfitting, double descent, implicit bias, and grokking. Lastly, we derive an exact population-risk objective from a single training run with no validation data, for any architecture, loss, or optimizer, and prove that it measures precisely the noise in the signal channel. This objective reduces in practice to an SNR preconditioner on top of Adam, adding one state vector at no extra cost; it accelerates grokking by

5 \times

, suppresses memorization in PINNs and implicit neural representations, and improves DPO fine-tuning under noisy preferences while staying

3 \times

closer to the reference policy.

Elon Litman, Gabe Guo

Stanford University

Neural Network Generalization Benign Overfitting Neural Network Training Dynamics Neural Tangent Kernel

May 1, 2026cs.CV

Benchmarking ResNet Backbones in RT-DETR: Impact of Depth and Regularization under environmental conditions

Visual perception plays a central role in competitive robotics, where environmental variations can directly affect real-time detection performance. The related literature on transformer-based detectors lack information regarding the impact of backbone scale and environmental settings on model performance. This work presents a comparative evaluation of RT-DETR for detecting round objects under environmental and hyperparameter variations relevant to competitive robotics. Four ResNet backbones (ResNet18, ResNet34, ResNet50, and ResNet101) were compared using dropout rates, analyzing their effect on confidence and accuracy. All models were trained under the same configuration and evaluated under changes in lighting and background contrast. Environmental conditions primarily impact prediction confidence, while inference latency remains largely unaffected and classification accuracy stays consistently high, approaching or above 1.00 in most cases. Two distinct behaviors were observed. Under illumination variation, ResNet50 achieves the best trade-off, combining near-perfect accuracy, confidence values up to approximately 0.869 and latency around 0.058-0.059 ms. Under background variation, ResNet34 provides the most balanced performance, reaching near-perfect accuracy and higher confidence values up to approximately 0.887. These results indicate that the optimal architecture depends on the type of environmental variation, with intermediate-depth models offering the best balance between performance and efficiency.

Pamela Barboza, Víctor Castelli, Belén Pereira +3

1, Robotics and Artificial Intelligence Laboratory, Technological University of Uruguay, Rivera, Uruguay

Neural Network Generalization Detection Transformer Object Detection Real-Time Object Detection

Apr 30, 2026stat.ML

Adaptive Norm-Based Regularization for Neural Networks

In this paper, we study norm-based regularization methods for neural networks. We compare existing penalization approaches and introduce two regularization strategies that extend classical ridge- and lasso-type penalties to neural network models. The first strategy modifies weight decay by incorporating the covariance structure of the input features into a ridge-type

\ell_2

penalty, allowing regularization to account for feature dependence. The second combines an

\ell_1

sparsity penalty with covariance-aware

\ell_2

regularization, producing neural network weights that are both sparse and structurally informed. Monte Carlo simulations are used to evaluate these methods under different data-generating settings, followed by two real-data applications on building cooling-load prediction and leukemia cell-type classification from high-dimensional gene expression data. Across simulated and real-data examples, the proposed regularizers improve predictive performance on unseen data and provide more effective complexity control than standard norm-based penalties, particularly when features are correlated or high-dimensional.

Muhammad Qasim, Farrukh Javed

Department of Statistics, Lund University, Lund, Sweden

Neural Network Generalization

Apr 30, 2026math.ST

Decoupled Descent: Exact Test Error Tracking Via Approximate Message Passing

In modern parametric model training, full-batch gradient descent (and its variants) suffers due to progressively stronger biasing towards the exact realization of training data; this drives the systematic ``generalization gap'', where the train error becomes an unreliable proxy for test error. Existing approaches either argue this gap is benign through complex analysis or sacrifice data to a validation set. In contrast, we introduce decoupled descent (DD), a novel theory-based training algorithm that satisfies a train-test identity -- enforcing the train error to asymptotically track the test error for stylized Gaussian mixture models. Within this specific regime, leveraging approximate message passing theory, DD iteratively cancels the biases due to data reuse, rigorously demonstrating the feasibility of zero-cost validation and

100\%

data utilization. Moreover, DD is governed by a low-dimensional state evolution recursion, rendering the dynamics of the algorithm transparent and tractable. We validate DD on XOR classification, yielding superior performance compared to GD; additionally, we implement noisy MNIST and non-linear probing of CIFAR-10, demonstrating that even when our stylized assumptions are relaxed, DD narrows the generalization gap compared to GD.

Max Lovig

Statistics and Data Science, Yale University

Neural Network Generalization Gradient Descent

Apr 29, 2026cs.LG

Multiple Additive Neural Networks for Structured and Unstructured Data

This paper extends and explains the Multiple Additive Neural Networks (MANN) methodology, an enhancement to the traditional Gradient Boosting framework, utilizing nearly shallow neural networks instead of decision trees as base learners. This innovative approach leverages neural network architectures, notably Convolutional Neural Networks (CNNs) and Capsule Neural Networks, to extend its application to both structured data and unstructured data such as images and audio. For structured data the advantages of capsule neural networks as feature extractors are used and combined with MANN as a classifier. MANN's unique architecture promotes continuous learning and integrates advanced heuristics to combat overfitting, ensuring robustness and reducing sensitivity to hyperparameter settings like learning rate and iterations. Our empirical studies reveal that MANN surpasses traditional methods such as Extreme Gradient Boosting (XGB) in accuracy across well-known datasets. This research demonstrates MANN's superior precision and generalizability, making it a versatile tool for diverse data types and complex learning environments.

Janis Mohr, Jörg Frochte

Interdisciplinary Institute for Applied Artificial Intelligence and Data Science Ruhr Bochum University of Applied Sciences 42579 Heiligenhaus, Germany

Neural Network Generalization

Apr 28, 2026cs.LG

Investigation into In-Context Learning Capabilities of Transformers

Transformers have demonstrated a strong ability for in-context learning (ICL), enabling models to solve previously unseen tasks using only example input output pairs provided at inference time. While prior theoretical work has established conditions under which transformers can perform linear classification in-context, the empirical scaling behavior governing when this mechanism succeeds remains insufficiently characterized. In this paper, we conduct a systematic empirical study of in-context learning for Gaussian-mixture binary classification tasks. Building on the theoretical framework of Frei and Vardi (2024), we analyze how in-context test accuracy depends on three fundamental factors: the input dimension, the number of in-context examples, and the number of pre-training tasks. Using a controlled synthetic setup and a linear in-context classifier formulation, we isolate the geometric conditions under which models successfully infer task structure from context alone. We additionally investigate the emergence of benign overfitting, where models memorize noisy in-context labels while still achieving strong generalization performance on clean test data. Through extensive sweeps across dimensionality, sequence length, task diversity, and signal-to-noise regimes, we identify the parameter regions in which this phenomenon arises and characterize how it depends on data geometry and training exposure. Our results provide a comprehensive empirical map of scaling behavior in in-context classification, highlighting the critical role of dimensionality, signal strength, and contextual information in determining when in-context learning succeeds and when it fails.

Rushil Chandrupatla, Leo Bangayan, Sebastian Leng

Neural Network Generalization Transformer Benign Overfitting In-Context Learning Scaling Laws

Apr 27, 2026cs.LG

Unveiling the Backdoor Mechanism Hidden Behind Catastrophic Overfitting in Fast Adversarial Training

Fast Adversarial Training (FAT) has attracted significant attention due to its efficiency in enhancing neural network robustness against adversarial attacks. However, FAT is prone to catastrophic overfitting (CO), wherein models overfit to the specific attack used during training and fail to generalize to others. While existing methods introduce diverse hypotheses and propose various strategies to mitigate CO, a systematic and intuitive explanation of CO remains absent. In this work, we innovatively interpret CO through the lens of backdoor. Through validations on pathway division, diverse feature predictions, and universal class distinguishable triggers in CO, we conceptualize CO as a weak trigger variant of unlearnable tasks, unifying CO, backdoor attacks, and unlearnable tasks under a common theoretical framework. Guided by this, we leverage several backdoor inspired strategies to mitigate CO: (i) Recalibrate CO affected model parameters using vanilla fine tuning, linear probing, or reinitialization-based techniques; (ii) Introduce a weight outlier suppression constraint to regulate abnormal deviations in model weights. Extensive experiments support our interpretation of CO and show the efficacy of the proposed mitigation strategies.

Mengnan Zhao, Lihe Zhang, Tianhang Zheng +2

School of Computer Science and Technology, Anhui University, Hefei 230601, China · School of Information and Communication Engineering, Dalian University of Technology (DUT), Dalian 116024, China · School of Computer Science and Technology, Zhejiang University, Hangzhou 310058, China +1

Adversarial Training Neural Network Generalization Adversarial Robustness

Apr 27, 2026cs.LG

Mitigating Error Amplification in Fast Adversarial Training

Fast Adversarial Training (FAT) has proven effective in enhancing model robustness by encouraging networks to learn perturbation-invariant representations. However, FAT often suffers from catastrophic overfitting (CO), where the model overfits to the training attack and fails to generalize to unseen ones. Moreover, robustness oriented optimization typically leads to notable performance degradation on clean inputs, and such degradation becomes increasingly severe as the perturbation budget grows. In this work, we conduct a comprehensive analysis of how guidance strength affects model performance by modulating perturbation and supervision levels across distinct confidence groups. The findings reveal that low confidence samples are the primary contributors to CO and the robustness accuracy trade off. Building on this insight, we propose a Distribution-aware Dynamic Guidance (DDG) strategy that dynamically adjusts both the perturbation budget and supervision signal. Specifically, DDG scales the perturbation magnitude according to the sample confidence at the ground truth class, thereby guiding samples toward consistent decision boundaries while mitigating the influence of learning spurious correlations. Simultaneously, it dynamically adjusts the supervision signal based on the prediction state of each sample, preventing overemphasis on incorrect signals. To alleviate potential gradient instability arising from dynamic guidance, we further design a weighted regularization constraint. Extensive experiments on standard benchmarks demonstrate that DDG effectively alleviates both CO and the robustness accuracy trade off.

Mengnan Zhao, Lihe Zhang, Bo Wang +3

Anhui University, Anhui, China · Dalian University of Technology, Liaoning, China · Zhejiang University, Zhejiang, China +1

Adversarial Training Neural Network Generalization Adversarial Robustness

Apr 27, 2026cs.LG

Self-Abstraction Learning for Effective and Stable Training of Deep Neural Networks

Training large-scale deep neural networks effectively and stably is essential for applying deep learning across various fields. However, conventional methods, which rely on training a single large network, often encounter challenges such as gradient vanishing, overfitting and unstable learning. To overcome these limitations, we introduce Self-Abstraction Learning (SAL), a hierarchical framework. In SAL, networks are arranged by structural complexity, where the simplest topmost network is trained first and its hidden and output layers serve as guidance for the successively more complex networks below. This top-down sequential guidance effectively mitigates optimization issues, enabling stable training of deep architectures. Various experiments across MLP, CNN, and RNN architectures demonstrate that SAL consistently outperforms conventional methods, ensuring robust generalization even in data-scarce and complex network regimes.

Wonyong Cho, Taemin Kim, Jungmin Kim +2

Department of Mathematics, University of Seoul, Seoul 02504, South Korea · Department of Applied Artificial Intelligence, Hansung University, Seoul 02876, South Korea

Neural Network Generalization Neural Network Optimization Neural Network Training Dynamics

Apr 23, 2026cs.NE

L-System Genetic Encoding for Scalable Neural Network Evolution: A Comparison with Direct Matrix Encoding

An artificial world of barriers and plains scattered with food is used to test the feasibility of using genetic algorithms to optimize hebbian neural networks to perform on problems without apriori knowledge of the problem domain. A formal L-System based genetic alphabet for neural networks, titled Lsys, and a neural network genetic modeling tool titled Wp1hgn are introduced. Lsys and Matrix neural network topology genetic encoding methods are compared across 24 experimental runs. Lsys encoding achieved a mean maximum food count of 3802 +- 197 at generation 1000 across 8 runs with varied parameters, compared to 1388 +- 610 for Matrix encoding, a 2.74x performance advantage with an 8.5-fold improvement in consistency as measured by coefficient of variation (5.2% vs 44.0%). All 8 Lsys populations successfully learned to navigate the environment, while 4 of 8 Matrix populations failed to achieve competitive performance at any point during 1000 generations. When transferred to a novel maze environment, Lsys populations demonstrated immediate robust generalization, achieving a mean maximum food count of 2455 +- 176 compared to 422 +- 212 for Matrix populations, a 5.82x advantage that exceeded the training world performance gap. A MatrixLSG control condition, in which initial populations were generated using Lsys genotypes and then evolved using Matrix operators, demonstrated that the performance advantage of Lsys encoding derives primarily from the genetic algorithm operating on the compressed symbolic Lsys alphabet throughout evolution rather than from initial population structure. Lsys encoding is shown to provide faster convergence, higher peak performance, dramatically greater reliability, and superior generalization to novel environments compared to Matrix encoding across all experimental conditions tested.

Alexander Stuy, Nodin Weddington

Florida State University

Neural Network Generalization Evolutionary Optimization Genetic Algorithms Neuroevolution

Apr 22, 2026cs.LG

Explicit Dropout: Deterministic Regularization for Transformer Architectures

Dropout is a widely used regularization technique in deep learning, but its effects are typically realized through stochastic masking rather than explicit optimization objectives. We propose a deterministic formulation that expresses dropout as an additive regularizer directly incorporated into the training loss. The framework derives explicit regularization terms for Transformer architectures, covering attention query, key, value, and feed-forward components with independently controllable strengths. This formulation removes reliance on stochastic perturbations while providing clearer and fine-grained control over regularization strength. Experiments across image classification, temporal action detection, and audio classification show that explicit dropout matches or outperforms conventional implicit methods, with consistent gains when applied to attention and feed-forward network layers. Ablation studies demonstrate stable performance and controllable regularization through regularization coefficients and dropout rates. Overall, explicit dropout offers a practical and interpretable alternative to stochastic regularization while maintaining architectural flexibility across diverse tasks.

Vidhi Agrawal, Illia Oleksiienko, Alexandros Iosifidis

Faculty of Information Technology and Communications,Tampere University, Tampere, 33720, Finland · Department of Electrical and Computer Engineering, Aarhus University, Aarhus, 8000, Denmark

Neural Network Generalization

Apr 22, 2026cs.LG

ILDR: Geometric Early Detection of Grokking

Grokking describes a delayed generalization phenomenon in which a neural network achieves perfect training accuracy long before validation accuracy improves, followed by an abrupt transition to strong generalization. Existing detection signals are indirect: weight norm reflects parameter-space regularization and consistently lags the transition, while GrokFast's slow gradient EMA, used without gradient amplification, is unstable across seeds with standard deviation exceeding mean lead time. We propose the Inter/Intra-class Distance Ratio (ILDR), a geometric metric computed on second-to-last layer representations as the ratio of inter-class centroid separation to intra-class scatter. ILDR provides an early detection signal: it rises and crosses a threshold at 2.5 times its baseline before the grokking transition appears in validation accuracy, indicating early geometric reorganization in representation space. Grounded in Fisher's linear discriminant criterion, ILDR requires no eigendecomposition and runs in O(|C|^2 + N). It is evaluated exclusively on held-out data, making it robust to memorization effects. Across modular arithmetic and permutation group composition (S5), ILDR leads the grokking transition by 9 to 73 percent of the training budget, with lead time increasing with task algebraic complexity. Over eight random seeds, ILDR leads by 950 +/- 250 steps with a coefficient of variation of 26 percent, and post-grokking variance drops by 1696 times, consistent with a sharp phase transition in representation space. Using ILDR as an early stopping trigger reduces training by 18.6 percent on average. Optimizer interventions triggered at the ILDR threshold demonstrate bidirectional control over the transition, suggesting ILDR tracks representational conditions underlying generalization rather than a downstream correlate.

Shreel Golwala

Virginia Tech

Neural Network Generalization Early Stopping Neural Representation Geometry Grokking

Apr 21, 2026cs.LG

Generalization and Membership Inference Attack a Practical Perspective

With the emergence of new evaluation metrics and attack methodologies for Membership Inference Attacks (MIA), it becomes essential to reevaluate previously accepted assumptions. In this paper, we revisit the longstanding debate regarding the correlation between MIA success rates and model generalization using an empirical approach. We focused on employing augmentation techniques and early stopping to enhance model generalization and examined their impact on MIA success rates. We found that utilizing advanced generalization techniques can significantly decrease attack performance, potentially by up to 100 times. Moreover, combining these methods not only improves model generalization but also reduces attack effectiveness by introducing randomness during training. Additionally, our study confirmed the direct impact of generalization on MIA performance through an analysis of over 1K models in a controlled environment.

Fateme Rahmani, Mahdi Jafari Siavoshani, Mohammad Hossein Rohban

Department of Computer Engineering Sharif University of Technology Tehran, Iran

Neural Network Generalization Membership Inference Attacks

Apr 21, 2026cs.LG

Generalization at the Edge of Stability

Training modern neural networks often relies on large learning rates, operating at the edge of stability, where the optimization dynamics exhibit oscillatory and chaotic behavior. Empirically, this regime often yields improved generalization performance, yet the underlying mechanism remains poorly understood. In this work, we represent stochastic optimizers as random dynamical systems, which often converge to a fractal attractor set (rather than a point) with a smaller intrinsic dimension. Building on this connection and inspired by Lyapunov dimension theory, we introduce a novel notion of dimension, coined the `sharpness dimension', and prove a generalization bound based on this dimension. Our results show that generalization in the chaotic regime depends on the complete Hessian spectrum and the structure of its partial determinants, highlighting a complexity that cannot be captured by the trace or spectral norm considered in prior work. Experiments across various MLPs and transformers validate our theory while also providing new insights into the recently observed phenomenon of grokking.

Mario Tuci, Caner Korkmaz, Umut Şimşekli +1

INRIA, CNRS, Département d’Informatique de l’Ecole Normale Supérieure / PSL, France · Department of Computing, Imperial College London, United Kingdom

Intrinsic Dimensionality Neural Network Generalization Edge of Stability Neural Network Training Dynamics Generalization Bounds

Apr 21, 2026cs.LG

Benign Overfitting in Adversarial Training for Vision Transformers

Despite the remarkable success of Vision Transformers (ViTs) across a wide range of vision tasks, recent studies have revealed that they remain vulnerable to adversarial examples, much like Convolutional Neural Networks (CNNs). A common empirical defense strategy is adversarial training, yet the theoretical underpinnings of its robustness in ViTs remain largely unexplored. In this work, we present the first theoretical analysis of adversarial training under simplified ViT architectures. We show that, when trained under a signal-to-noise ratio that satisfies a certain condition and within a moderate perturbation budget, adversarial training enables ViTs to achieve nearly zero robust training loss and robust generalization error under certain regimes. Remarkably, this leads to strong generalization even in the presence of overfitting, a phenomenon known as \emph{benign overfitting}, previously only observed in CNNs (with adversarial training). Experiments on both synthetic and real-world datasets further validate our theoretical findings.

Jiaming Zhang, Meng Ding, Shaopeng Fu +2

Division of CEMSE, King Abdullah University of Science and Technology · School of Statistics, Renmin University of China · Department of Computer Science and Engineering, State University of New York at Buffalo +1

Adversarial Training Neural Network Generalization Vision Transformer Neural Network Robustness Adversarial Robustness Benign Overfitting

Apr 19, 2026stat.ML

PAC-Bayes Bounds for Gibbs Posteriors via Singular Learning Theory

We derive explicit non-asymptotic PAC-Bayes generalization bounds for Gibbs posteriors, that is, data-dependent distributions over model parameters obtained by exponentially tilting a prior with the empirical risk. Unlike classical worst-case complexity bounds based on uniform laws of large numbers, which require explicit control of the model space in terms of metric entropy (integrals), our analysis yields posterior-averaged risk bounds that can be applied to overparameterized models and adapt to the data structure and the intrinsic model complexity. The bound involves a marginal-type integral over the parameter space, which we analyze using tools from singular learning theory to obtain explicit and practically meaningful characterizations of the posterior risk. Applications to low-rank matrix completion and ReLU neural network regression and classification show that the resulting bounds are analytically tractable and substantially tighter than classical complexity-based bounds. Our results highlight the potential of PAC-Bayes analysis for precise finite-sample generalization guarantees in modern overparameterized and singular models.

Chenyang Wang, Yun Yang

Department of Statistics, University of Illinois Urbana-Champaign · Department of Mathematics, University of Maryland, College Park

Neural Network Generalization PAC-Bayesian Generalization Bounds Singular Learning Theory Statistical Learning Theory

Apr 17, 2026cs.LG

When Do Early-Exit Networks Generalize? A PAC-Bayesian Theory of Adaptive Depth

Early-exit neural networks enable adaptive computation by allowing confident predictions to exit at intermediate layers, achieving 2-8

\times

inference speedup. Despite widespread deployment, their generalization properties lack theoretical understanding -- a gap explicitly identified in recent surveys. This paper establishes a unified PAC-Bayesian framework for adaptive-depth networks. (1) Novel Entropy-Based Bounds: We prove the first generalization bounds depending on exit-depth entropy

H(D)

and expected depth

\mathbb{E}[D]

rather than maximum depth

K

, with sample complexity

\mathcal{O}((\mathbb{E}[D] \cdot d + H(D))/ε^2)

. (2) Explicit Constructive Constants: Our analysis yields the leading coefficient

\sqrt{2\ln 2} \approx 1.177

with complete derivation. (3) Provable Early-Exit Advantages: We establish sufficient conditions under which adaptive-depth networks strictly outperform fixed-depth counterparts. (4) Extension to Approximate Label Independence: We relax the label-independence assumption to

ε

-approximate policies, broadening applicability to learned routing. (5) Comprehensive Validation: Experiments across 6 architectures on 7 benchmarks demonstrate tightness ratios of 1.52-3.87

\times

(all

p < 0.001

) versus

>

100

\times

for classical bounds. Bound-guided threshold selection matches validation-tuned performance within 0.1-0.3%.

Dongxin Guo, Jikun Wu, Siu Ming Yiu

The University of Hong Kong, Hong Kong, China · Brain Investing Limited, Hong Kong, China

Neural Network Generalization PAC-Bayesian Generalization Bounds Adaptive Inference Dynamic Neural Networks Early-Exit Neural Networks

Apr 16, 2026cs.LG

Stability and Generalization in Looped Transformers

Looped transformers promise test-time compute scaling by spending more iterations on harder problems, but it remains unclear which architectural choices let them extrapolate to harder problems at test time rather than memorize training-specific solutions. We introduce a fixed-point based framework for analyzing looped architectures along three axes of stability -- reachability, input-dependence, and geometry -- and use it to characterize when fixed-point iteration yields meaningful predictions. Theoretically, we prove that looped networks without recall have countable fixed points and cannot achieve strong input-dependence at any spectral regime, while recall combined with outer normalization reliably produces a regime in which fixed points are simultaneously reachable, locally smooth in the input, and supported by stable backpropagation. Empirically, we train single-layer looped transformers on chess, sudoku, and prefix-sums and find that downstream performance tracks the framework's predictions across tasks and architectural configurations. We additionally introduce internal recall, a novel recall placement variant, and show that it becomes competitive with -- and on sudoku, substantially better than -- standard recall placement once outer normalization is applied.

Asher Labovich

Neural Network Generalization Dynamical Systems Recurrent Transformers

Apr 16, 2026cs.CV

Chaotic CNN for Limited Data Image Classification

Convolutional neural networks (CNNs) often exhibit poor generalisation in limited training data scenarios due to overfitting and insufficient feature diversity. In this work, a simple and effective chaos-based feature transformation is proposed to enhance CNN performance without increasing model complexity. The method applies nonlinear transformations using logistic, skew tent, and sine maps to normalised feature vectors before the classification layer, thereby reshaping the feature space and improving class separability. The approach is evaluated on greyscale datasets (MNIST and Fashion-MNIST) and an RGB dataset (CIFAR-10) using CNN architectures of varying depth under limited data conditions. The results show consistent improvement over the standalone (SA) CNN across all datasets. Notably, a maximum performance gain of 5.43% is achieved on MNIST using the skew tent map with a 3-layer CNN at 40 samples per class. A higher gain of 9.11% is observed on Fashion-MNIST using the sine map with a 3-layer CNN at 50 samples per class. Additionally, a strong gain of 7.47% is obtained on CIFAR-10 using the skew tent map at 200 samples per class. The consistent improvements across different chaotic maps indicate that the performance gain is driven by the shared nonlinear and dynamical properties of chaotic systems. The proposed method is computationally efficient, requires no additional trainable parameters, and can be easily integrated into existing CNN architectures, making it a practical solution for data-scarce image classification tasks.

Anusree M, Akhila Henry, Pramod P Nair

1*Department of Mathematics, Amrita Vishwa Vidyapeetham, Amritapuri, India.

Neural Network Generalization Convolutional Neural Networks Few-Shot Image Classification Data-Efficient Learning

Apr 16, 2026cs.LG

CLion: Efficient Cautious Lion Optimizer with Enhanced Generalization

Lion optimizer is a popular learning-based optimization algorithm in machine learning, which shows impressive performance in training many deep learning models. Although convergence property of the Lion optimizer has been studied, its generalization analysis is still missing. To fill this gap, we study generalization property of the Lion via algorithmic stability based on the mathematical induction. Specifically, we prove that the Lion has a generalization error of

O(\frac{1}{Nτ^T})

, where

N

is training sample size, and

τ>0

denotes the smallest absolute value of non-zero element in gradient estimator, and

T

is the total iteration number. In addition, we obtain an interesting byproduct that the SignSGD algorithm has the same generalization error as the Lion. To enhance generalization of the Lion, we design a novel efficient Cautious Lion (i.e., CLion) optimizer by cautiously using sign function. Moreover, we prove that our CLion has a lower generalization error of

O(\frac{1}{N})

than

O(\frac{1}{Nτ^T})

of the Lion, since the parameter

τ

generally is very small. Meanwhile, we study convergence property of our CLion optimizer, and prove that our CLion has a fast convergence rate of

O(\frac{\sqrt{d}}{T^{1/4}})

under

\ell_1

-norm of gradient for nonconvex stochastic optimization, where

d

denotes the model dimension. Extensive numerical experiments demonstrate effectiveness of our CLion optimizer.

Feihu Huang, Guanyi Zhang, Songcan Chen

College of Computer Science and Technology, Nanjing University of Aeronautics and Astronautics, Nanjing, China · MIIT Key Laboratory of Pattern Analysis and Machine Intelligence, Nanjing, China

Stochastic Optimization Sign-Based Optimization Neural Network Generalization Deep Learning Optimization Nonconvex Stochastic Optimization

Apr 13, 2026cs.LG

VISTA: Validation-Informed Trajectory Adaptation via Self-Distillation

Deep learning models may converge to suboptimal solutions despite strong validation accuracy, masking an optimization failure we term Trajectory Deviation. This is because as training proceeds, models can abandon high generalization states for specific data sub-populations, thus discarding previously learned latent features without triggering classical overfitting signals. To address this problem we introduce VISTA, an online self-distillation framework that enforces consistency along the optimization trajectory. Using a validation-informed Marginal Coverage score, VISTA identifies expert anchors, which are earlier model states that retain specialized competence over distinct data regions. A coverage-weighted ensemble of these anchors is integrated online during training, regularizing the loss landscape and preserving mastered knowledge. When evaluated across multiple benchmarks, VISTA demonstrates improved robustness and generalization over standard training and prior self-distillation methods, while a lightweight implementation reduces storage overhead by 90% without performance loss.

Eli Corn, Daphna Weinshall

Department of Computer Science, Hebrew University of Jerusalem, Jerusalem, Israel.

Neural Network Generalization Deep Learning Optimization Self-Distillation Neural Network Training Dynamics

Mar 26, 2026stat.ML

The Rules-and-Facts Model for Simultaneous Generalization and Memorization in Neural Networks

A key capability of modern neural networks is their capacity to simultaneously learn underlying rules and memorize specific facts or exceptions. Yet, theoretical understanding of this dual capability remains limited. We introduce the Rules-and-Facts (RAF) model, a minimal solvable setting that enables precise characterization of this phenomenon by bridging two classical lines of work in the statistical physics of learning: the teacher-student framework for generalization and Gardner-style capacity analysis for memorization. In the RAF model, a fraction

1 - \varepsilon

of training labels is generated by a structured teacher rule, while a fraction

\varepsilon

consists of unstructured facts with random labels. We characterize when the learner can simultaneously recover the underlying rule - allowing generalization to new data - and memorize the unstructured examples. Our results quantify how overparameterization enables the simultaneous realization of these two objectives: sufficient excess capacity supports memorization, while regularization and the choice of kernel or nonlinearity control the allocation of capacity between rule learning and memorization. Experiments on a CIFAR10-based RAF task show that the qualitative trade-off persists for fixed kernels, whereas representations learned by deeper neural networks substantially mitigate it, preserving factual recall while improving generalization. The RAF model provides a theoretical foundation for understanding how modern neural networks can infer structure while storing rare or non-compressible information.

Gabriele Farné, Fabrizio Boncoraglio, Lenka Zdeborová

Statistical Physics of Computation Laboratory, École Polytechnique Fédérale de Lausanne (EPFL) CH-1015 Lausanne

Neural Network Generalization Neural Network Memorization

Mar 19, 2026cs.LG

Rigorous Error Certification for Neural PDE Solvers: From Empirical Residuals to Solution Guarantees

Uncertainty quantification for partial differential equations is traditionally grounded in discretization theory, where solution error is controlled via mesh/grid refinement. Physics-informed neural networks fundamentally depart from this paradigm: they approximate solutions by minimizing residual losses at collocation points, introducing new sources of error arising from optimization, sampling, representation, and overfitting. As a result, the generalization error in the solution space remains an open problem. Our main theoretical contribution establishes generalization bounds that connect residual control to solution-space error. We prove that when neural approximations lie in a compact subset of the solution space, vanishing residual error guarantees convergence to the true solution. We derive deterministic and probabilistic convergence results and provide certified generalization bounds translating residual, boundary, and initial errors into explicit solution error guarantees.

Amartya Mukherjee, Maxwell Fitzsimmons, David C. Del Rey Fernández +1

Department of Applied Mathematics, University of Waterloo, Waterloo, ON, Canada

Neural Network Generalization Neural PDE Solvers PDE Solving Physics-Informed ML Generalization Bounds

Mar 5, 2026cs.LG

Asymptotic Behavior of Multi--Task Learning: Implicit Regularization and Double Descent Effects

Multi--task learning seeks to improve the generalization error by leveraging the common information shared by multiple related tasks. One challenge in multi--task learning is identifying formulations capable of uncovering the common information shared between different but related tasks. This paper provides a precise asymptotic analysis of a popular multi--task formulation associated with misspecified perceptron learning models. The main contribution of this paper is to precisely determine the reasons behind the benefits gained from combining multiple related tasks. Specifically, we show that combining multiple tasks is asymptotically equivalent to a traditional formulation with additional regularization terms that help improve the generalization performance. Another contribution is to empirically study the impact of combining tasks on the generalization error. In particular, we empirically show that the combination of multiple tasks postpones the double descent phenomenon and can mitigate it asymptotically.

Ayed M. Alrashdi, Oussama Dhifallah, Houssem Sifaou

Department of Electrical Engineering, College of Engineering, University of Ha’il, P.O. Box 2440, Ha’il, 81441, Saudi Arabia · John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA · Department of Electrical and Electronic Engineering, King’s College London – Strand, London WC2R 2LS – London, UK

Double Descent Neural Network Generalization Multi-Task Learning Multi-Task Optimization Statistical Learning Theory Implicit Regularization

Feb 28, 2026cs.LG

To Use or not to Use Muon: How Simplicity Bias in Optimizers Matters

While Adam has long been the ubiquitous default optimizer for deep neural networks, Muon has recently seen rapid adoption due to its superior training speed. Although much of the literature focuses on validating the benefits of Muon, our work investigates the potential downsides of the mechanism driving this speedup. On the theoretical front, we analyze the learning dynamics of simplified Muon on deep linear networks and linear attention. Our analysis reveals that Muon gains speed by avoiding saddle points, but does so at the expense of the simplicity bias characteristic of Gradient Descent (GD), where the complexity of the functional solution learned grows sequentially. Experiments demonstrate the consequences of losing the simplicity bias, showing that Muon struggles to uncover common underlying structure across tasks and may be prone to fitting spurious features. More broadly, this paper serves as a reminder that faster optimization is rarely a free lunch; improvements in optimization can come at the cost of changes in the inductive biases that shape generalization.

Sara Dragutinović, Yedi Zhang, Rajesh Ranganath

Courant Institute School of Mathematics, Computing, and Data Science, New York University · Gatsby Computational Neuroscience Unit, University College London

Neural Network Generalization Muon Optimizer Deep Linear Networks Implicit Bias Neural Network Training Dynamics

Feb 23, 2026cs.LG

A Theory of How Pretraining Shapes Inductive Bias in Fine-Tuning

Pretraining and fine-tuning are central stages in modern machine learning systems. In practice, feature learning plays an important role across both stages: deep neural networks learn a broad range of useful features during pretraining and further refine those features during fine-tuning. However, an end-to-end theoretical understanding of how choices of initialization impact the ability to reuse and refine features during fine-tuning has remained elusive. Here we develop an analytical theory of the pretraining fine-tuning pipeline in diagonal linear networks, deriving exact expressions for the generalization error as a function of initialization parameters and task statistics. We find that different initialization choices place the network into four distinct fine-tuning regimes that are distinguished by their ability to support feature learning and reuse and therefore by the task statistics for which they are beneficial. In particular, a smaller initialization scale in earlier layers enables the network to both reuse and refine its features, leading to superior generalization on fine-tuning tasks that rely on a subset of pretraining features. We demonstrate empirically that the same initialization parameters impact generalization in ResNets trained on CIFAR-100 and SVHN as well as Transformers trained on modular arithmetic tasks. Overall, our results demonstrate an alytically how data and network initialization interact to shape fine-tuning generalization, highlighting an important role for the relative scale of initialization across different layers in enabling continued feature learning during fine-tuning.

Nicolas Anguita, Francesco Locatello, Andrew M. Saxe +4

Department of Engineering, University of Cambridge · Institute of Science and Technology, Austria (ISTA) · Gatsby Computational Neuroscience Unit and Sainsbury Wellcome Centre, UCL +1

Neural Network Generalization Representation Learning Fine-Tuning Neural Network Initialization Statistical Learning Theory

Feb 1, 2026cs.LG

Theoretical Analysis of Measure Consistency Regularization for Partially Observed Data

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

Yinsong Wang, Shahin Shahrampour

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

Neural Network Generalization Learning with Missing Data Statistical Learning Theory Consistency Regularization

Jan 27, 2026cs.LG

To Grok Grokking: Provable Grokking in Ridge Regression

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

Mingyue Xu, Gal Vardi, Itay Safran

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

Neural Network Generalization Ridge Regression Grokking Gradient Descent Dynamics Generalization Bounds

Nov 21, 2025physics.flu-dyn

Addressing A Posteriori Performance Degradation in Neural Network Subgrid Stress Models

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

Andy Wu, Sanjiva K. Lele

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

Neural Network Generalization Data Augmentation Computational Fluid Dynamics