Neural Network Training Dynamics

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41 papers in the last four weeks, up 273% on the four weeks before. 0.4% of all new papers.

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Latest papers 349

Jun 11, 2026cs.LG

The Weight Norm Sets the Grokking Timescale: A Causal Delay Law

Grokking is the delayed onset of generalization in neural networks, arising long after they fit the training data. Whether the weight norm causes this delay is disputed: some studies report a critical norm at the transition, others observe grokking with no fixed norm at all. We settle this by intervening on the norm during training rather than only observing it. Under free training with weight decay, networks grok when the weight norm reaches a value Wc that varies little across seeds and learning rates (CV 1 to 2 percent) and grows with the modular base as a power law. When we instead clamp the norm to a fixed multiple rho of Wc and hold it there, the network still groks, but the delay follows T_grok proportional to exp(alpha rho). One exponent, alpha near 7.5, fits this delay across four moduli (R^2 = 0.996). Over the swept ranges the held norm moves the delay by about 19x and the learning rate by only about 2x, and holding the norm above Wc slows grokking rather than preventing it. A final LayerNorm removes the dependence by decoupling weight scale from the network function; without it the exponential law returns. This pinned-norm delay is the exponential counterpart to the logarithmic delay predicted for a freely contracting norm.

Truong Xuan Khanh, Doan Hoang Viet, Luu Duc Trung +1

H&K Research Studio / Clevix LLC, Hanoi, Vietnam · Bac A Bank, Hanoi · Banking Academy of Vietnam

Neural Network Generalization Neural Network Training Dynamics Grokking

Jun 11, 2026cs.LG

Different Layers, Different Manifolds: Module-Wise Weight-Space Geometry in Transformer Optimization

Weight-space geometry plays a central role in neural network optimization, yet manifold constraints are often applied uniformly across all weight matrices. In this work, we ask whether different transformer modules prefer different manifold geometries. We study Manifold Muon for GPT-2 pretraining and compare layer-wise assignments of Stiefel and DGram constraints across attention and MLP blocks. Our results show a clear asymmetry: constraining attention layers with Stiefel geometry while assigning DGram geometry to MLP layers gives the best performance among the tested configurations, whereas the inverted assignment and all-DGram configuration become unstable under the shared hyperparameter setting. We trace this failure to singular value growth in DGram-constrained attention weights, which can amplify attention logits and induce softmax saturation. These findings suggest that symmetry-aware and geometry-aware optimization for transformers should be module-specific rather than uniform.

Kirato Yoshihara

School of Engineering Science, The University of Osaka, Osaka, Japan.

Stiefel Manifold Optimization Neural Network Optimization Weight-Space Analysis Constrained Optimization Neural Network Training Dynamics

Jun 11, 2026cs.LG

Circuit Synchronization Precedes Generalization: A Causal Precursor to Grokking

Grokking is the delayed generalisation phenomenon where a transformer trained on modular arithmetic abruptly transitions from near-chance to near-perfect validation accuracy. It has been attributed to a Fourier-based algorithmic circuit, but its timing, causal structure, and controllability remain poorly understood. We introduce the Frequency Synchronization Degree (FSD), a normalised, permutation-tested metric for Fourier circuit synchronisation requiring no prior knowledge of the circuit. Across nine modular addition configurations (five primes, three seeds), FSD reaches its post-grokking level 500 to 3000 steps before grokking (mean lead 1722 steps, every configuration positive, sign-test p approx 0.004), and synchronises before a restricted-logit loss baseline in all nine cases, making it the earliest available predictor. We give direct causal evidence that the inter-phase gap is a regularisation phenomenon: forking training at the FSD-ceiling step and varying weight decay lambda produces monotonically earlier grokking, with delta-t proportional to 1/lambda. This law replicates across three primes (R-squared 0.89 to 0.99 on seed-averaged delta-t); per-run R-squared is unstable due to the chaotic transition, so we report error bars rather than single runs. Grokking occurs at a near-constant memorisation norm across lambda, grounding the constant in a threshold mechanism. This is not an artefact of applying a Fourier detector to a Fourier circuit: on the non-abelian group S5, a basis-faithful generalisation of FSD precedes grokking on all six seeds, while the original Fourier FSD does not. Using the FSD ceiling to schedule a weight-decay increase also accelerates grokking over a fixed schedule without destabilising training. An attention-only variant groks with a strong FSD precursor while an MLP-only model never groks.

Achyuthan Sivasankar

New York University

Weight Decay Frequency-Domain Feature Learning Neural Network Training Dynamics Grokking

Jun 11, 2026cs.LG

Weibull Weight-Scale Parameter Evolution under AdamW Training Dynamics

Building on a two-parameter Weibull framework for diagnosing transformer weight distributions, we study why the Weibull weight-scale parameter

λ

grows, overshoots, and then relaxes during AdamW training. We derive a leading-order three-force decomposition of the squared weight norm from the AdamW update: an alignment force measuring the correlation between weights and the adaptive update direction, an injection force from adaptive step magnitude, and a decay force from decoupled weight decay. On self-trained Pythia-70M models with ground-truth optimizer moments, alignment dominates the rise phase, contributing 88-94% of the absolute force budget across four random seeds and remaining robust to super-weight removal. Near saturation, alignment and decay approach balance, explaining the transition from weight-scale growth to relaxation. These force dynamics directly govern the squared-norm component underlying

λ(t)

; the remaining RMS-to-Weibull reconstruction offset is measurable and decomposes into bridge and integration components, totaling approximately 5-6% in densely sampled regions. To extend the analysis to real models where optimizer moments are unavailable, we introduce a spline displacement method that recovers the alignment force from sparse checkpoints with approximately 92-94% accuracy, about twice the naive two-point baseline. We further observe that the peak value of

λ(t)

varies with training-data coherence in our experiments, suggesting a data-dependent component of weight-scale growth that we leave to a controlled follow-up study. Code and data are available at https://github.com/tiexinding/NPM-Weibull-public.

Tiexin Ding

Independent Researcher.

Weight-Space Analysis Adaptive Gradient Methods Neural Network Training Dynamics

Jun 10, 2026stat.ML

Phase Transitions in Attention: A Bayesian Theory of Copy Head Emergence

Attention is the key mechanism underlying in-context learning in transformers, and attention patterns have been observed empirically to emerge abruptly during training. We present a Bayesian theory of feature learning in attention; we then focus on how the copy subcircuit in the first layer of an induction head is learned by analyzing a single-layer softmax attention network trained on a copy task. We derive a closed-form posterior over the attention matrix and reduce it to a low-dimensional order parameter space. This reduction reveals a phase transition in the amount of training data, which we verify using both Bayesian sampling and standard training with Adam. We contrast our results with linear attention and find that softmax attention exhibits a \emph{first-order phase transition} while in linear attention an initial \emph{second-order phase transition} is followed by a smooth, continuous evolution toward the structured attention pattern (\emph{crossover}). Our work provides a first-principles theoretical account of the abrupt emergence of the copy subcircuit, reminiscent of the one observed in training large language models.

Itay Lavie, Kirsten Fischer, Andrey Lekov +3

Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel · John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA · Institute for Advanced Simulation (IAS-6), Computational and Systems Neuroscience, Jülich Research Center, Jülich 52428, Germany +3

Softmax Attention Phase Transitions Induction Heads Neural Network Training Dynamics

Jun 9, 2026cs.LG

Overcoming Rank Collapse in Feedback Alignment

Backpropagation (BP) is widely viewed as biologically implausible, in part because it requires feedback weights to be the transpose of forward weights for error propagation. Interestingly, when training a network with fixed random feedback weights to circumvent this issue, learning aligns the forward weights with the feedback weights, leading the backpropagated error signal to become an approximation of the standard gradient used by BP. This process, called Feedback Alignment (FA), occurs in MLPs and very shallow CNNs but does not scale well to deeper architectures. In this work, we first investigated differences between BP and FA models, trained on CIFAR10, specifically focusing on the effective rank of the signal. We found that the FA error has a considerably lower rank and hence is constrained to a lower-dimensional subspace compared to BP, limiting exploration of the parameter space. Motivated by this observation, we evaluated two mechanisms for increasing the effective dimensionality of FA: Muon, an optimiser that orthogonalises weight updates; and hidden activity normalisation, which promotes activation orthogonality. Across larger architectures and benchmarks, we find that these methods consistently improve over FA baselines, for example, on CIFAR100 with a Resnet-18, accuracy increases by 9 percentage points. Our results identify low-dimensional gradient dynamics as a key obstacle to scaling FA and suggest that inducing higher-dimensional update geometry is a promising route toward scaling alternatives to backpropagation.

Gauthier Boeshertz, Razvan Pascanu, Claudia Clopath

Department of Bioengineering Imperial College London · Mila

Backpropagation Neural Network Optimization Neural Network Training Dynamics

Jun 9, 2026cs.LG

Conservation Laws from Data Symmetry in Neural Networks

We explore whether intrinsic symmetries of the training data lead to conserved quantities during gradient-flow training of neural networks. Under the assumption that the loss function is analytic and non-polynomial, we prove that data symmetries generically do not induce any additional integrals of motion. For mean squared error (MSE) loss, on the other hand, there are situations in which data augmentation yields extra conserved quantities. We build a framework, utilizing \emph{tensorizable networks} to describe this phenomenon. Tensorizable networks are a family of architectures whose dependence on parameters and inputs can be separated using an intermediate representation. They include linear and polynomial networks, as well as Lightning Attention.

Jakob Galley, Vahid Shahverdi, Axel Flinth

Umeå University, Umeå, Sweden

Neural Network Optimization Neural Network Training Dynamics

Jun 9, 2026nlin.AO

Towards Critical Branching Mechanism in Recurrent Neural Networks

Criticality has been proposed as a key organizing principle in biological neural systems, yet its origin and relevance in artificial neural networks remain unclear. We analyze hidden-state dynamics in trained long short-term memory (LSTM) networks and show that small networks near their optimal training epochs (steps) exhibit scale-free avalanche statistics and branching parameters close to unity, indicative of near-critical dynamics, while larger models remain subcritical. To explain the coexistence of subcritical branching with robust

1/f^β

noise, we introduce a mixture branching process framework that links heterogeneous branching dynamics to long-range temporal correlations. These results identify critical-like behavior in LSTMs as an emergent, capacity-dependent dynamical regime.

Feixiang Ren, Ling Feng

Department of Physics, National University of Singapore Singapore · Institute of High Performance Computing Agency for Science, Technology and Research (A*STAR) Singapore

Recurrent Neural Networks Long Short-Term Memory Networks Neural Network Training Dynamics

Jun 9, 2026cs.LG

Rank Collapse, Fixed Points, and the Renormalization Group Structure of MLP Residual Networks

The analogy between deep neural network forward passes and renormalization group (RG) flows has been repeatedly noted in the literature, but existing treatments remain qualitative: depth is described as a coarse-graining scale, attention is likened to a partition function, and representations are said to flow toward fixed points. No existing work has defined a measurable RG order parameter, tested it under controlled variation of the input distribution, or made quantitative predictions that are empirically verified. We study the simplest architecture for which the analogy is tractable: a pure MLP residual stack trained on masked token prediction over synthetic Markov chain sequences with known spectral properties. We report three findings. (i) The effective rank of the residual stream decreases monotonically with depth after training, consistent with progressive integration of irrelevant degrees of freedom. (ii) This rank collapse is selective: it occurs for chains with short correlation length approximately 1 but is absent for chains with long correlation length approximately 7, measured at the position level to control for mean-pooling artifacts. The network preserves exactly the degrees of freedom relevant to the prediction task, the content of the RG relevance criterion. (iii) Inter-layer kernel drift is concentrated at one or two specific transitions, with the remainder of the network near a fixed point, consistent with a discrete fixed-point plateau. Together these findings constitute the first quantitative, position-level evidence that MLP residual networks implement a selective coarse-graining procedure governed by the spectral structure of the input distribution.

Parviz Haggi-Mani, Irina Rish

Universit´e de Montr´eal · Mila - Quebec AI Institute

Multilayer Perceptrons Neural Network Training Dynamics

Jun 8, 2026cs.LG

Learning Entropy and Spatial Adaptation Dynamics of Multilayer Perceptrons for Structural Point Extraction

This paper extends the concept of Learning Entropy (LE) from temporal adaptive systems to spatial learning in multilayer perceptron networks (MLPs) applied to image data. Instead of evaluating image structure directly from gradients or covariance operators, as local neighborhood methods do, the proposed approach analyzes the learning process itself through Learning Entropy. An MLP is trained to predict the intensity of a center pixel from its surrounding spatial context, while LE is evaluated from the incremental adaptation of neural weights during learning across image-derived samples. The resulting Spatial Learning Entropy Maps (SLEM) identify unusual image points and regions that induce strong adaptation of the neural network and therefore have an important role in the learning process. The results indicate that spatial Learning Entropy provides a complementary perspective to conventional feature extraction and explainability methods by highlighting spatial locations that are particularly informative for network learning. Spatial Learning Entropy provides a complementary perspective to conventional feature extraction and explainability methods by identifying image points and regions according to their learning impact rather than their local structural properties. The proposed framework may open new directions for learning-driven image or scene analysis in computer vision, manufacturing, and robotics.

Jan Glaser, Ivo Bukovsky, Marcel Jirina

Czech Technical University in Prague, Prague, Czech Republic · University of South Bohemia in Ceske Budejovice, Ceske Budejovice, Czech Republic

Multilayer Perceptrons Neural Network Training Dynamics

Jun 8, 2026cs.LG

Preserving Plasticity in Continual Learning via Dynamical Isometry

Continual training of deep neural networks under non-stationarity often leads to a progressive loss of plasticity, eventually limiting further learning. We relate plasticity to the empirical Neural Tangent Kernel, and identify dynamical isometry (the condition that layer-wise Jacobian singular values remain close to one) as a key mechanism for preserving plasticity in continual learning. We revisit a class of networks that are almost-everywhere isometric while remaining universal Lipschitz function approximators, demonstrating that near-dynamical isometry is compatible with expressive nonlinear representations. For general architectures, we propose an efficient isometry-promoting regularization scheme and identify a novel mechanism by which it can reactivate dormant ReLU units. Building on this, we introduce AdamO, an Adam-style adaptive optimizer that decouples isometry regularization from gradient updates, analogous to AdamW. We further reinterpret prior plasticity-preserving approaches through the lens of dynamical isometry, showing that they target only a partial measure of isometry. Across supervised and reinforcement-learning continual-learning benchmarks designed to induce plasticity loss, our methods consistently match or outperform existing approaches.

Andries Rosseau, Robert Müller, Ann Nowé

AI Lab, Vrije Universiteit Brussel · Aganthos

Loss of Plasticity Continual Learning Adaptive Gradient Methods Neural Network Training Dynamics

Jun 8, 2026cs.LG

Learning Dynamics Reveal a Hierarchy of Weight-Induced Layerwise Gram Metrics

We study feed-forward ReLU networks with fixed readout and quadratic loss, and rewrite gradient descent as a collective dynamics of activation fields and conjugate fields on the training set. Working to first order in the learning rate inside a fixed activation chamber, we derive explicitly the one-, two- and three-hidden-layer cases, and then give the arbitrary-depth recursion. For one hidden layer the activation dynamics closes directly and the residual update is governed by the product of an input Gram matrix and a co-activation/backpropagation Gram matrix. For two hidden layers a conjugate field is required, but no nontrivial pullback Gram metric has yet appeared. For three hidden layers the first weight-induced pullback Gram metric enters the conjugate-field dynamics. At arbitrary depth, activation variations propagate forward through a recursive response operator

\cU_\ell^{αβ}

, while conjugate-field variations propagate backward through an effective transport operator

\cM_\ell^{αβ}

. Their contractions reconstruct a layerwise residual kernel

K_{αβ}^{(L)}=\sum_{\ell=1}^{L}Q_{αβ}^{(\ell-1)}S_{αβ}^{(\ell)}.

The resulting description exposes a duality between push-forward and pullback transport across every layer cut, and identifies the first Gram metrics as the lowest nontrivial terms in a broader hierarchy of activation-conditioned transport operators. We deliberately stop at the level of collective fields, conjugate fields, residual kernels and cut-wise transport metrics, leaving the later tensorial geometric formulation outside the scope of this paper.

Claudio Nordio

Backpropagation ReLU Neural Networks Neural Network Training Dynamics Gradient Descent Dynamics

Jun 8, 2026cs.LG

Integrating Out, Twice:The Open-System Case That Neural-Network Ensemble Theory Is Missing

Averaging a neural network over its random parameters and marginalizing a Gaussian sector are the same operation, the Schur complement of the eliminated block, and when that block is closed it returns a covariance and its inverse. That is all a network ensemble produces, the closed case. The open case is missing, and nuclear reaction theory has it worked out. Projecting a scattering problem onto a chosen set of channels, with the rest carrying probability irreversibly to a continuum, leaves a non-Hermitian effective generator that conserves and itemizes exactly what it loses: the nuclear optical model and its generalized optical theorem. I set the two cases side by side using only the moments of a distribution, the algebra of Gaussians, and block inversion, no field theory, and give the closed-case dictionary in full: the neural tangent kernel is the Fisher sensitivity kernel, the infinite-width Gaussian limit is the Gaussian-process emulator, and the lazy-to-feature transition is the validity boundary of a reduced-basis emulator. I then test the open export on a truncated attention map, a token-level transfer operator, and a sparse expert router, and report a mostly negative result. The conserved flux ledger ports wherever openness is genuinely present, but its distinctive content is absent, an artifact of the chosen partition, or pinned near a floor by the training objective, and the operationally useful uncertainty turns out to be epistemic, living in the closed half of the correspondence, not the open one. The negative has a structural reason this note makes precise: the open case needs an eliminated sector with a continuous spectrum and wave-like, not relaxational, dynamics, which mainstream learning's finite or dissipative objects do not supply. This is a note, not a result; its main finding is that negative one, and its value is the map that locates it.

Jin Lei

Department of Physics, Tongji University, Shanghai 200092, China · Southern Center for Nuclear-Science Theory (SCNT), Institute of Modern Physics, Chinese Academy of Sciences, Huizhou 516000, Guangdong Province, China

Ensemble Learning Neural Network Training Dynamics

Jun 8, 2026cs.LG

Hybridizing Equilibrium Propagation with Ising Machines for Efficient Energy-Based Learning

The rapid evolution of artificial intelligence has led to substantial advances in deep neural networks. Nonetheless, conventional GPU-based training remains highly energy-demanding, motivating the exploration of physical dynamics and compatible energy-based learning schemes, such as equilibrium propagation (EP). EP-based training, however, frequently suffers from convergence to local minima due to phase-space contraction. Here we introduce an Ising-dynamics-inspired equilibrium-propagation framework in which dissipative Hopfield relaxation is replaced by an extended phase-space dynamics with conjugate variables. The resulting training paradigm keeps the local two-phase learning rule of EP while changing the physical route by which neural states reach equilibrium. We show that this dynamics lowers effective energy barriers, accelerates convergence, improves noise robustness, and trains deep convolutional Hopfield networks on MNIST, FashionMNIST, and CIFAR-10 with performance comparable to backpropagation.

Chen-Rui Fan, Bo Lu, Xing-Yu Wu +2

School of Artificial Intelligence, Beijing Normal University, Beijing 100875, China · Laboratory for Advanced Computing and Intelligence Engineering, Information Engineering University, Zhengzhou 450001, China · School of Physical Science and Technology, Beijing University of Posts and Telecommunications, Beijing 100876, China

Equilibrium Propagation Hopfield Networks Energy-Based Models Neural Network Training Dynamics

Jun 8, 2026cs.LG

Beyond Neural Collapse: Task-Intrinsic Geometry Governs Neural Representations in Modular Arithmetic

While neural collapse (NC) predicts that a

K

-class-balanced classifier should organize terminal representations as a

(K-1)

-dimensional simplex equiangular tight frame (ETF), modular addition consistently enters a different regime: networks compress to a two-dimensional cyclic geometry in which both classifier weights and token embeddings lie on circles. We refine the explanation of this phenomenon in three directions. First, we formalize a layerwise non-uniform training mechanism: downstream classifier weights are driven by dense cross-entropy gradients into a rank-2 equiangular configuration before upstream embeddings fully reorganize, and once this classifier plane forms, backpropagated feature gradients constrain embedding motion to the same plane while weight decay suppresses orthogonal components. Second, after this subspace locking, the induced in-plane dynamics admit an entropy-regularized transport interpretation on

S^1

; combined with modular-addition labels, this reduces embedding formation to phase alignment, whose minimizers are single-frequency characters of

\mathbb{Z}/P\mathbb{Z}

and hence equal-angle points on a circle. Third, we quantify why this solution prevails over NC: a simplex ETF gains only an

O(1)

advantage in cross-entropy, whereas the cyclic rank-2 solution enjoys a

Θ(K)

advantage under Schatten or weight-decay surrogates, yielding a critical threshold

λ_{\mathrm{crit}} = Θ(1/K)

. Our results explain both why classifier weights move first and why embeddings subsequently align with them, showing that grokking on modular arithmetic is governed not by maximal separation alone but by a task-structured trade-off between separation, symmetry, and complexity.

Hu Tan, Kuo Gai, Shihua Zhang

Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China; and School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China · Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS), Shanghai, China · Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China; School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China; and Key Laboratory of Systems Health Science of Zhejiang Province, School of Life Science, Hangzhou Institute for Advanced Study, University of Chinese Academy of Sciences, Chinese Academy of Sciences, Hangzhou 310024, China

Neural Representation Geometry Classification Neural Collapse Neural Network Training Dynamics Modular Arithmetic Grokking

Jun 7, 2026cs.LG

Between Amnesia and Chaos: A Memory Stability Expressivity Trilemma for Trainable Dissipative Oscillator Networks

Physical reservoir computing harnesses nonlinear mechanical dynamics but, by convention, freezes the substrate and trains only a linear readout, presuming the substrate is not usefully trainable. We revisit that premise for networks of nonlinear oscillators whose mass, damping, and stiffness are learned end-to-end through a symplectic integrator. Our central result is a trilemma: memory horizon, gradient stability, and dynamical expressivity cannot be simultaneously maximized, because all three are governed by the damping. The backward gradient decays at a rate set by the damping, capping how far back credit can propagate, while forward sensitivities grow exponentially in the largest Lyapunov exponent, so usable gradients require damping above a stability floor. Since the Lyapunov exponent falls as damping rises while the memory ceiling falls as the horizon grows, stable training is confined to a band that contracts with horizon and closes at a critical point. We test every step on a twenty-oscillator network. A damping sweep finds the largest Lyapunov exponent monotone and crossing zero at a well-defined stability floor, confirming the theorem's key assumption. A compute-matched comparison of learned versus frozen substrate on delayed recall across nine horizons shows the learned substrate dominating at short horizons and the advantage closing and reversing near a horizon of eleven steps, the predicted signature of band closure; trained models settle near the stability floor, seeking the edge of chaos unprompted. The analytic ceiling overestimates the empirical crossover roughly fivefold, a gap between detectable and learnable gradient that we report rather than tune away. The contribution is a confirmed account of when training a physical substrate beats freezing it.

Caleb Munigety

Independent Researcher

Dynamical Systems Lyapunov Stability Reservoir Computing Neural Network Training Dynamics

Jun 6, 2026cs.LG

Fourier fractal dimension to predict the generalization of deep neural networks

Predicting the generalization performance of deep neural networks without relying on hold-out validation data is a fundamental challenge in machine learning. While Stochastic Gradient Descent (SGD) drives the optimization of these highly parameterized models, its heavy-tailed, non-Gaussian dynamics induce complex, scale-invariant trajectories in the parameter space. In this paper, we propose a novel generalization measure based on the Fourier fractal dimension of the network's weight variations. By analyzing the characteristic function of the Lévy-driven stochastic differential equations in the frequency domain, we extract a metric that robustly captures the geometric complexity of the learning process. Furthermore, we introduce a customized Fourier-based optimizer designed to actively regularize this fractal dimension during training. Extensive empirical evaluations on the CIFAR-10, SVHN, and MNIST datasets demonstrate that our proposed Fourier generalization measure exhibits a strong correlation with the actual generalization gap. Our method achieves state-of-the-art Kendall rank correlation coefficients, outperforming a wide array of existing norm-based, margin-based, and PAC-Bayesian measures. Ultimately, this work highlights the potential of frequency-domain fractal analysis as both a powerful predictor for model generalizability and a principled foundation for developing more stable optimization algorithms.

Joao B. Florindo, Davi Wanderley Misturini

Institute of Mathematics, Statistics and Scientific Computing - University of Campinas Rua Sérgio Buarque de Holanda, 651, Cidade Universitária “Zeferino Vaz” - Distr. Barão Geraldo, CEP 13083-859, Campinas, SP, Brasil

Neural Network Generalization Neural Network Training Dynamics Stochastic Gradient Descent

Jun 5, 2026cs.LG

Breaking the Bubble: Asynchronous Pipeline Parallel Training with Bounded Weight Inconsistency

Pipeline parallelism is essential for training large neural networks, but existing schedules trade off throughput, memory, and optimization consistency. Synchronous pipelines preserve forward/backward weight consistency but suffer from bubbles; asynchronous pipelines remove bubbles but introduce weight-version mismatch, typically requiring weight stashing, prediction, or correction mechanisms. We introduce PACI (Pipeline Asynchronous training with Controlled Inconsistency), a bubble-free asynchronous pipeline method that bounds forward/backward version drift without weight stashing, prediction, additional parameter copies, or global synchronization. The key idea is to use local gradient accumulation as a version-control mechanism: by slowing parameter-version evolution relative to pipeline delay, PACI limits the number of optimizer updates crossed by any micro-batch while preserving steady-state utilization. In GPT-style language-model pretraining, PACI matches the stability and final perplexity of synchronous 1F1B-flush, retains the same peak memory footprint, achieves fully utilized pipeline throughput, and improves training time-to-accuracy by up to

1.69\times

over the fastest flush baseline. These results show that forward/backward inconsistency need not be eliminated: when explicitly bounded, it can be safely traded for substantial efficiency gains.

Itay Elam, Eliron Rahimi, Avi Mendelson +1

Department of Computer Science Technion - Israel Institute of Technology · School of Electrical and Computer Engineering Ben-Gurion University of the Negev

Neural Network Training Dynamics Pipeline Parallelism

Jun 5, 2026cs.LG

Second-Order Path Kernel Interpolation Formulas in Machine Learning

Understanding how training data shape neural network predictions is a central problem in modern learning theory. In 2020, Pedro Domingos proposed an interpolation formula valid for every model learned by deterministic gradient descent. It expresses the model's prediction as an integral, along the optimization path, of a data-dependent kernel that aligns the model's gradients at the test and training data. Such a first-order characterization remains valid for models trained with batch-based stochastic optimization. In this paper, we develop second-order forms of these interpolation formulas. We show that the leading path-kernel interpolation is supplemented by a curvature-weighted interpolation term. For stochastic gradient descent, an additional sampling-induced component appears, coupling the curvature of the prediction with the covariance of mini-batch gradient noise. We also extend the representation to stochastic gradient descent with momentum, where the interpolation structure is preserved but with the weights modified by a memory-related factor. Moreover, we establish a concentration estimate for the terminal prediction, identifying the fluctuation scale around the expected second-order representation. Together, these results provide a refinement of the path-kernel interpretation of neural network prediction.

Jin Guo, Roy Y. He, Jean-Michel Morel

Department of Mathematics, City University of Hong Kong, Hong Kong · Division of Industrial Data Science, Lingnan University, Hong Kong

Neural Network Training Dynamics

Jun 5, 2026cs.LG

Beyond Linear and Overcomplete Regimes: A Mean-Field Analysis of Bottleneck Autoencoders

Autoencoders (AEs) learn low-dimensional representations by mapping data into a latent space while minimizing reconstruction error. Despite their empirical success, theoretical understanding remains limited and largely restricted to linear models or settings without a bottleneck. In this work, we study nonlinear AEs with a fixed finite-dimensional bottleneck in the mean-field (MF) regime. We derive explicit MF learning dynamics for both encoder and decoder, providing a tractable characterization of training in the nonlinear setting. We show that, over finite time horizons, the empirical risk of finite-width networks trained with stochastic gradient descent closely tracks the MF risk trajectory with high probability. At optimality, we further establish that the finite-width risk converges to the MF optimum, demonstrating that finite networks are sufficiently expressive to approximate the infinite-width solution.

Santanu Das, Ramyak Bilas, Pascal Esser +1

STCS department Tata Institute Of Fundamental Research Mumbai, India-400005 · Department of Mathematics Indiana University Bloomington Indiana, USA-47405 · Department of Mathematics Ludwig-Maximilians-Universität München Germany, München-80799 +1

Autoencoders Mean-Field Theory Neural Network Approximation Theory Neural Network Training Dynamics

Jun 4, 2026cs.NE

Quantifying Uncertainty In Wide Two-Layer Neural Networks: On The Law Of The Limiting Fluctuation Process

Uncertainty quantification in neural networks prediction is a main issue for usual applications. Our approach seeks at reducing computation costs by directly evaluating uncertainty using PDE's information on the asymptotic variance, rather than the deep ensemble method which may be seen as a Monte Carlo estimation of the prediction, requiring the training of multiple networks. We thus study the law of the limiting process describing the random fluctuations around the mean-field limit of wide two-layer neural networks trained by stochastic gradient descent in a weak-noise regime. Building on a recent trajectorial central limit theorem, in which this limit is characterized as the weak solution of a linear stochastic evolution equation, we identify its law explicitly. More precisely, we show that it is a centered Gaussian process in the dual of a weighted Sobolev space, and we derive a closed covariance representation for the finite-dimensional distributions obtained by testing it against smooth functions. This covariance is expressed through the solution of a backward transport equation with a nonlocal source term, whose coefficients are driven by the mean-field trajectory. As a consequence, by testing against the activation function at a fixed input, we obtain an expression for the limiting variance of the corresponding network-output fluctuations. We illustrate this result numerically on a one-dimensional regression example.

Arnaud Descours, Arnaud Guillin, Geoffrey Lacour +3

LSAF · LMBP · MaIAGE +1

Uncertainty Quantification Neural Network Training Dynamics

Jun 4, 2026cs.LG

Deciphering Two Training Clocks in Grokking via Deep Linear Network Theory with Conditional ReLU Reduction

Grokking suggests that fitting the training data and learning a simple underlying rule may occur on different time scales. We formalize this phenomenon by separating the fast decay of the classification loss from the slower simplification of the learned representation, and we call the resulting pair of stopping times two training clocks. For deep linear networks, we show that a post-margin gap-growth or one-step tail-contraction condition reduces the cross-entropy loss to level epsilon on a logarithmic time scale. In contrast, when layerwise weight decay is present, the induced regularization on the end-to-end map can be expressed as a Schatten-type penalty; under a sharp late-time Kurdyka-Lojasiewicz tail, this structural energy closes on a polynomial time scale. The two clocks, therefore, separate fitting from representation simplification. We then explain how the same mechanism can appear in ReLU MLPs. In regions where the activation patterns on the training set remain fixed, the network reduces to a linear model in the active coordinates. In a two-layer ReLU embedding model, chain-rule estimates further show that the classifier head can receive larger effective gradients than the embedding block under controlled downstream norms. This supports a two-stage mechanism in which the classifier fits first, while the representation continues to simplify later. We use modular addition as the main experimental setting. The deep linear theory provides the rigorous core of the analysis. But the ReLU results are formulated as conditional reductions that account for empirical behavior without claiming a global proof for nonlinear training dynamics.

Hu Tan, Kuo Gai, Shihua Zhang

1State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China · School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China · 3Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS), Shanghai, China +1

ReLU Neural Networks Deep Linear Networks Neural Network Training Dynamics Grokking

Jun 3, 2026cs.LG

A prism hierarchy of learning regimes in large linear autoencoders

Theoretical studies of machine learning models commonly consider different limiting regimes in which the learning dynamics of gradient descent becomes theoretically tractable. It is, however, desirable to have a systematically obtained picture of qualitatively different extreme learning regimes for a particular type of models. In this paper we propose such a picture for large weight-tied linear autoencoders characterized by input and latent dimensions, initialization magnitude, and training set size. This model is nonlinear in the weights and its gradient flow does not have a general theoretical solution. We show that at the level of the formal loss-expansion hierarchy, its extreme regimes are naturally associated with faces of a triangular prism. In particular, there are five basic extreme regimes associated with the 2-faces of the prism: (1) large-data, (2) small-data, (3) mean-field, (4) narrow-latent, and (5) free. For regimes (1,2,3,4), we derive and rigorously characterize the limiting train and population dynamics under gradient flow, rigorously prove convergence to these limits, and get good agreement with experimental results. The resulting limits reveal spectral critical slowing, initialization-driven trapping away from the PCA optimum, and regimes in which population loss converges exponentially while training loss relaxes only algebraically.

Eugene Golikov, Yaroslav Gusev, Dmitry Yarotsky

Applied AI Institute Moscow, Russia · Steklov Mathematical Institute of Russsian Academy of Sciences Moscow, Russia

Autoencoders Neural Network Training Dynamics Statistical Learning Theory Gradient Descent Dynamics

Jun 3, 2026cs.AI

Position: Don't Just "Fix it in Post": A Science of AI Must Study Training Dynamics

What would it mean to have a scientific understanding of AI? Models are not static objects: they are snapshots of time-evolving processes shaped by data, objectives, architectures, and optimization dynamics. Yet much of AI research treats models as fixed artifacts, analyzing behaviors after training rather than asking why they emerge. This position paper argues that a science of AI must move beyond post-hoc fixes and study the training dynamics that produce model behavior. Such a science should support progressively stronger forms of understanding: predicting outcomes from early training signals, intervening when trajectories go wrong, and ultimately designing training procedures that more reliably produce desired properties. Scaling laws have made prediction routine for loss; the challenge is extending this success to capabilities, biases, robustness, and safety-relevant behaviors. We articulate requirements for such theories grounded in the history and philosophy of science, examine progress in mechanistic interpretability, fairness, memorization, and simplicity bias, and identify concrete open problems.

Stella Biderman, Mohammad Aflah Khan, Niloofar Mireshghallah +3

1EleutherAI · 2Max Planck Institute for Software Systems · 3Carnegie Mellon University +3

Neural Network Training Dynamics

Jun 3, 2026cs.LG

When Both Layers Learn: Training Dynamics of Representing Linear Models via ReLU Networks

In this paper, we study the gradient descent dynamics for jointly training both layers of a one-hidden-layer ReLU network to fit a linear target function. Concretely, we consider a realizable setting where inputs are drawn i.i.d. from a Gaussian distribution and labels follow a planted linear model. This stylized framework captures salient features of end-to-end training in inverse problems and certain auto-encoder models. Despite its apparent simplicity, the dynamics remain poorly understood, in part because the loss landscape contains multiple non-strict saddle points, making it unclear why gradient descent from random initialization reliably escapes bad stationary regions. We provide a detailed characterization of the optimization landscape and prove that gradient descent from a moderately small random initialization-simultaneously training both layers-converges to a global minimizer at a linear rate with order-wise optimal sample complexity. Our analysis tracks the trajectory through three phases: an alignment phase in which hidden weights progressively align with the planted direction while the output weights maintain the correct sign pattern; a growth phase in which the norms of both layers increase while preserving alignment; and a local refinement phase in which the aligned neurons rapidly converge to the planted direction, yielding fast local convergence. To rigorously show that GD avoids non-strict saddles, we develop trajectory-level control arguments for the end-to-end dynamics. In addition, we establish novel uniform concentration results that hold along the entire trajectory, and are essential for obtaining order-wise optimal sample complexity. We corroborate our theory with extensive experiments across a range of configurations.

Berk Tinaz, Changzhi Xie, Mahdi Soltanolkotabi

Department of Electrical and Computer Engineering University of Southern California Los Angeles, CA, USA · Department of Computer Science University of Southern California Los Angeles, CA, USA

ReLU Neural Networks Neural Network Optimization Neural Network Training Dynamics Gradient Descent Dynamics

Jun 2, 2026cs.LG

Spectral Scaling Laws of Muon

Orthonormalized update rules have rapidly become a leading choice of optimizer for training large language models, with recent open-source state-of-the-art models adopting Muon. To keep these updates tractable, Muon performs the orthonormalization with the Newton--Schulz (NS) iteration. Since NS is only approximate, directions with small singular values fail to be orthonormalized. In Muon, NS is applied to the momentum matrix at every step, yet little is known about how the singular value spectrum of these momentum matrices behaves during training, or how that behavior changes with model size. We present the first systematic study of this question. Tracking singular value quantiles of the momentum buffer across layers in models ranging from 77M to 2.8B parameters, we observe a consistent picture: after a short burn-in, the quantiles stabilize at a value determined by the layer type and model size. These stabilization values follow remarkably clean power laws in model size, with layer-dependent exponents. Layers up to mid-late depth scale very mildly with model size

M

(around

M^{-0.25}

), so the standard 5-step NS configuration used at academic scale will continue to orthonormalize them at much larger scales. Some of the late layers, however, scale much more aggressively (up to

M^{-0.96}

) and will fall into the NS failure regime at frontier scale unless one uses more NS iterations or better-tuned coefficients. NS iterations are computationally expensive at scale; our laws give practitioners a principled, layer-aware recipe for choosing the minimum NS configuration that still orthonormalizes the directions that matter -- avoiding unnecessary computation without sacrificing update quality.

Gagik Magakyan, Pablo Parrilo, Asuman Ozdaglar

MIT

Newton-Schulz Iteration Momentum Methods Muon Optimizer Second-Order Optimization Neural Network Training Dynamics LLM Training

Jun 2, 2026cs.LG

Neural Networks Provably Learn Spectral Representations for Group Composition

Understanding how structured internal structure emerges during neural network training is central to the study of deep learning. We investigate this phenomenon through the group composition task, where a two-layer neural network is trained to predict

g_1 \star g_2

for elements of a finite group

G

. By lifting the projected gradient flow to the Fourier domain, we demonstrate that the training dynamics are governed by a Riemannian gradient ascent on a representation-theoretic energy functional. We prove that, under random initialization, this flow drives each neuron to converge almost surely toward a single irreducible representation, while the cross-layer Fourier coefficients achieve a rotational rank-one alignment. This framework provides a representation-theoretic account of feature learning and characterizes a novel low-rank compression phenomenon for matrix-valued group representations. Moreover, for Abelian groups, we provide a complete population-level description: random initialization promotes uniform diversification across nontrivial representations and induces Haar-uniform phases, jointly approximating the indicator via a majority-vote mechanism. We further prove that both phase alignment and representation competition emerge with exponential convergence rates.

Jianliang He, Leda Wang, Fengzhuo Zhang +2

Department of Statistics and Data Science, Yale University

Representation Learning Neural Network Training Dynamics

Jun 1, 2026cs.LG

Beyond $\ell_2$ -norm and $\ell_\infty$ -norm: A Curvature-Inspired $\ell_p$ -Norm Scheme for Deep Neural Networks

The existing optimizers for deep neural networks (DNNs) typically rely on either the

\ell_2

norm or the

\ell_\infty

norm, resulting in optimizers that do not adapt well to substantial changes in curvature across parameter dimensions. Generally, the training process of DNNs often exhibits strong curvature anisotropy in the early period, whereas in the later period, the training process of DNNs tends to move toward flatter regions with weaker anisotropy. Particularly, optimizers based on the

\ell_2

-norm are usually dominated by high-curvature directions, restricting updates of optimizers along with lower curvature direction and thus leading to a slower convergence rate. While optimizers based on the

\ell_\infty

-norm are prone to oscillations in flatter regions, due to the coordinate-wise updates of the same magnitude. To address these two extreme cases generated by

\ell_2

and

\ell_\infty

norms, we propose a novel

\ell_p

-norm scheme with a dynamical value of

p

and incorporate it into stochastic gradient descent (SGD) and SGD with momentum (SGDM), leading to two novel optimizers with better generalization performance:

{\ell_p}

-SGD (LPSGD) and

{\ell_p}

-SGDM (LPSGDM). Particularly, the resulting optimizers suppress the dominance of high-curvature directions in the early period by utilizing a large

p

(

p>2

), followed by a gradual decrease of

p

toward 2 to enable more stable and refined updates, where the latter process is motivated by the cosine annealing strategy. We establish theoretical guarantees of the resulting algorithms and analyze that both LPSGD and LPSGDM achieve an

O(T^{-1/2})

convergence rate for the nonconvex setting. Extensive experiments are conducted on benchmark datasets, including CIFAR-10, CIFAR-100, and ImageNet-1K, with multiple DNNs such as VGG-11, ResNet-18, and ResNet-50.

Jianhao Xu, Zhuang Yang

School of Computer Science and Technology, Soochow University, Suzhou, 215006, China

Neural Network Optimization Nonconvex Stochastic Optimization Neural Network Training Dynamics Stochastic Gradient Descent

May 30, 2026cs.LG

Looped Transformers with Layer Normalization Provably Learn the Power Method

Transformers have achieved remarkable success across a wide range of applications, and a growing body of work suggests that part of their strength comes from their ability to learn and execute algorithmic procedures. However, our understanding of how transformers learn such algorithms remains limited, especially in the presence of layer normalization (LN). In this work, we study principal component prediction as a concrete testbed for understanding the training dynamics of transformers with LN. We prove that a looped linear transformer with LN, trained by gradient descent, converges to a solution that implements the power method, with each self-attention layer performing one power iteration. Notably, the model is trained only for principal component prediction, rather than being explicitly supervised to implement the power method. Our finding thus reveals an "algorithmic implicit bias" of looped transformers with LN: principal-component prediction can in principle be achieved by many mechanisms, yet gradient descent selects one that realizes the power method. We further provide a concrete comparison between transformers with and without LN: even with layerwise guidance from power iterations, a transformer without LN cannot exactly learn the power method, whereas the corresponding transformer with LN can, leading to a provable performance gap in principal component prediction. Our results provide, to our knowledge, the first theoretical analysis of the training dynamics of looped and single-layer transformers with LN, and shed light on the role of LN in transformer models.

Lyumin Wu, Chenyang Zhang, Yuan Cao

School of Computing & Data Science, The University of Hong Kong

Recurrent Transformers Layer Normalization Neural Network Training Dynamics

May 30, 2026cs.LG

On the Difficulty of Learning a Meta-network for Training Data Selection

Synthetic data are increasingly used to train neural networks, yet distributional mismatch with real data limits their effectiveness when used indiscriminately. A common strategy is to learn data weights via bi-level optimization, which we refer to as Meta-learning for Training-data Selection (MTS). Interestingly, in practice, MTS often performs below expectation. We identify two obstacles in properly training MTS: a poor gradient signal-to-noise ratio (GSNR), which causes optimization difficulties, and lack of informative features that correlates with data quality. We present a mathematical analysis of MTS, which reveals the dynamics of normalized data weights and the relation between disparate data quality and poor GSNR. The analysis suggests a a simple yet effective solution: increasing the batch size. Further, we propose a set of informative features that capture the positions of training data in their distributions and training dynamics. Experiments across four benchmarks show consistent improvements, achieving average gains of 5.49% over training without selection and 2.89% over the strongest baseline.

Zilin Du, Junqi Zhao, Boyang Albert Li

College of Computing and Data Science, Nanyang Technological University, Singapore.

Meta-Learning Training Data Selection Bilevel Optimization Neural Network Training Dynamics Synthetic Data Selection