Neural Network Training Dynamics

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It examines how weights, gradients, activations, and predictions evolve during training, including why models learn some features before others, generalize, forget, or become unstable. Researchers analyze gradient-flow and stochastic-gradient dynamics using neural tangent kernels, mean-field limits, random matrix theory, and loss-landscape tools, alongside mechanistic experiments on tasks such as modular arithmetic and language-model pretraining. This work informs optimizer design, learning-rate and weight-decay schedules, architectures, and training diagnostics, with phenomena such as grokking, edge-of-stability behavior, and representation collapse receiving close attention. A central challenge is connecting tractable theories and early-warning signals to causal, predictive accounts of large finite networks, where optimizer, data, architecture, and noise interact.

Latest papers

520 papers77 papers in the last four weeks, up 93% on the four weeks before. 0.4% of all new papers.

Mar 2, 2026

Never Saddle for Reparameterized Steepest Descent as Mirror Flow

How does the choice of optimization algorithm shape a model's ability to learn features? To address this question for steepest descent methods --including sign descent, which is closely related to Adam --we introduce steepest mirror flows as a unifying theoretical framework. This framework reveals how optimization geometry governs learning dynamics, implicit bias, and sparsity and it provides two explanations for why Adam and AdamW often outperform SGD in fine-tuning. Focusing on diagonal linear networks and deep diagonal linear reparameterizations (a simplified proxy for attention), we show that steeper descent facilitates both saddle-point escape and feature learning. In contrast, gradient descent requires unrealistically large learning rates to escape saddles, an uncommon regime in fine-tuning. Empirically, we confirm that saddle-point escape is a central challenge in fine-tuning. Furthermore, we demonstrate that decoupled weight decay, as in AdamW, stabilizes feature learning by enforcing novel balance equations. Together, these results highlight two mechanisms how steepest descent can aid modern optimization.

Tom Jacobs, Chao Zhou, Rebekka Burkholz

CISPA Helmholtz Center for Information Security, Saarbrücken, Germany

Neural Network Training Dynamics Gradient Descent Dynamics Adaptive Gradient Methods Gradient Descent Mirror Descent Representation Learning

Mar 2, 2026

Quantitative Convergence of Wasserstein Gradient Flows of Kernel Mean Discrepancies

We study the quantitative convergence of Wasserstein gradient flows of Kernel Mean Discrepancy (KMD) (also known as Maximum Mean Discrepancy (MMD)) functionals. Our setting covers in particular the training dynamics of shallow neural networks in the infinite-width and continuous time limit, as well as interacting particle systems with pairwise Riesz kernel interaction in the mean-field and overdamped limit. Our main analysis concerns the model case of KMD functionals given by the squared Sobolev distance

\mathscr{E}^ν_{s}(μ)= \frac{1}{2}\lVert μ-ν\rVert_{\dot H^{-s}}^{2}

for any

s\geq 1

and

ν

a fixed probability measure on the

d

-dimensional torus. First, inspired by Yudovich theory for the

2d

-Euler equation, we establish existence and uniqueness in natural weak regularity classes. Next, we show that for

s=1

the flow converges globally at an exponential rate under minimal assumptions, while for

s>1

we prove local convergence at polynomial rates that depend explicitly on

s

and on the Sobolev regularity of

μ

and

ν

. These rates hold both at the energy level and in higher regularity classes and are tight for

ν

uniform. We then consider the gradient flow of the population loss for shallow neural networks with ReLU activation, which can be cast as a Wasserstein--Fisher--Rao gradient flow on the space of nonnegative measures on the sphere

\mathbb{S}^d

. Exploiting a correspondence with the Sobolev energy case with

s=(d+3)/2

, we derive an explicit polynomial local convergence rate for this dynamics. Except for the special case

s=1

, even non-quantitative convergence was previously open in all these settings. We also include numerical experiments in dimension

d=1

using both PDE and particle methods which illustrate our analysis.

Lénaïc Chizat, Maria Colombo, Roberto Colombo +1

Wasserstein Gradient Flows Maximum Mean Discrepancy Neural Network Training Dynamics

Mar 2, 2026

Uniform-in-time concentration in two-layer neural networks via transportation inequalities

We quantify, uniformly over time and with high probability, the discrepancy between the predictions of a two-layer neural network trained by stochastic gradient descent (SGD) and their mean-field limit, for quadratic loss and ridge regularization. As a key ingredient, we establish T p transportation inequalities (p

\in

{1, 2}) for the law of the SGD parameters, with explicit constants independent of the iteration index. We then prove uniform-in-time concentration of the empirical parameter measure around its mean-field limit in the Wasserstein distance W 1 , and we translate these bounds into prediction-error estimates against a fixed test function

Φ

. We also derive analogous concentration bounds in the sliced-Wasserstein distance SW 1 , leading to dimension-free rates.

Arnaud Guillin, Boris Nectoux, Paul Stos

LMBP · Université Clermont-Auvergne, CNRS UMR 6620, LMBP, Clermont-Ferrand, France

Neural Network Training Dynamics Stochastic Gradient Descent

Mar 1, 2026

A Basin-Selection Perspective on Grokking via Singular Learning Theory

Grokking, the abrupt transition from memorization to generalisation after extended training, suggests the presence of competing solution basins with distinct statistical properties. We study this phenomenon through the lens of Singular Learning Theory (SLT), a Bayesian framework that characterizes the geometry of the loss landscape. The key measure is the local learning coefficient (LLC) which quantifies the local degeneracy of the loss surface. SLT links lower-LLC basins to higher posterior mass concentration and lower expected generalisation error. Leveraging SLT, we develop a basin-selection perspective on grokking in quadratic networks: LLC ranks competing near-zero-loss basins by statistical preference, while the training-time transition between them is governed by optimisation dynamics. In this view, grokking corresponds to a transition from a higher-LLC (memorising) basin to a lower-LLC (generalising) basin that dominates the posterior. To support this, we derive analytic formulas for the LLC in shallow quadratic networks under both lazy and feature learning regimes. Empirically, we demonstrate that LLC trajectories estimated from training data track the onset of generalisation and provide an informative probe of the optimisation path.

Ben Cullen, Sergio Estan-Ruiz, Riya Danait +1

Department of Computer Science, University of Pisa, Italy · Department of Mathematics, Imperial College London, United Kingdom · Mathematical Institute, University of Oxford, United Kingdom +1

Grokking Neural Network Training Dynamics Singular Learning Theory

Mar 1, 2026

Scaling of learning time for high dimensional inputs

Representation learning from complex data typically involves models with a large number of parameters, which in turn require large amounts of data samples. In neural network models, model complexity grows with the number of inputs to each neuron, with a trade-off between model expressivity and learning time. A precise characterization of this trade-off would help explain the connectivity and learning times observed in artificial and biological networks. We present a theoretical analysis of how learning time depends on input dimensionality for a Hebbian learning model performing independent component analysis. Based on the geometry of high-dimensional spaces, we show that the learning dynamics reduce to a unidimensional problem, with learning times dependent only on initial conditions. For higher input dimensions, initial parameters have smaller learning gradients and larger learning times. We find that learning times have supralinear scaling, becoming quickly prohibitive for high input dimensions. These results reveal a fundamental limitation for learning in high dimensions and help elucidate how the optimal design of neural networks depends on data complexity. Our approach outlines a new framework for analyzing learning dynamics and model complexity in neural network models.

Carlos Stein Brito

NightCity Labs, Lisbon, Portugal

Neural Network Training Dynamics Scaling Laws Hebbian Learning Independent Component Analysis

Mar 1, 2026

When Does Margin Clamping Affect Training Variance? Dataset-Dependent Effects in Contrastive Forward-Forward Learning

Contrastive Forward-Forward (CFF) learning trains Vision Transformers layer by layer against supervised contrastive objectives. CFF training can be sensitive to random seed, but the sources of this instability are poorly understood. We focus on one implementation detail: the positive-pair margin in the contrastive loss is applied through saturating similarity clamping,

\min(s + m,\, 1)

. We prove that an alternative formulation, subtracting the margin after the log-probability, is gradient-neutral under the mean-over-positives reduction. On CIFAR-10 (

2 \times 2

factorial,

n{=}7

seeds per cell), clamping produces

5.90\times

higher pooled test-accuracy variance (

p{=}0.003

) with no difference in mean accuracy. Analyses of clamp activation rates, layerwise gradient norms, and a reduced-margin probe point to saturation-driven gradient truncation at early layers. The effect does not transfer cleanly to other datasets: on CIFAR-100, SVHN, and Fashion-MNIST, clamping produces equal or lower variance. Two factors account for the discrepancy. First, positive-pair density per batch controls how often saturation occurs. Second, task difficulty compresses seed-to-seed spread when accuracy is high. An SVHN difficulty sweep confirms the interaction on a single dataset, with the variance ratio moving from

0.25\times

at high accuracy to

16.73\times

under aggressive augmentation. In moderate-accuracy regimes with many same-class pairs per batch, switching to the gradient-neutral subtraction reference removes this variance inflation at no cost to mean accuracy. Measuring the layer-0 clamp activation rate serves as a simple check for whether the problem applies.

Joshua Steier

Independent Researcher

Neural Network Training Dynamics Supervised Contrastive Learning Forward-Forward Learning

Feb 28, 2026

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 27, 2026

On Emergences of Non-Classical Statistical Characteristics in Classical Neural Networks

Inspired by measurement incompatibility and Bell-family inequalities in quantum mechanics, we propose the Non-Classical Network (NCnet), a simple classical neural architecture that stably exhibits non-classical statistical behaviors under typical and interpretable experimental setups. We find non-classicality, measured by the

S

statistic of CHSH inequality, arises from gradient competitions of hidden-layer neurons shared by multi-tasks. Remarkably, even without physical links supporting explicit communication, one task head can implicitly sense the training task of other task heads via local loss oscillations, leading to non-local correlations in their training outcomes. Specifically, in the low-resource regime, the value of

S

increases gradually with increasing resources and approaches toward its classical upper-bound 2, which implies that underfitting is alleviated with resources increase. As the model nears the critical scale required for adequate performance,

S

may temporarily exceed 2. As resources continue to grow,

S

then asymptotically decays down to and fluctuates around 2. Empirically, when model capacity is insufficient,

S

is positively correlated with generalization performance, and the regime where

S

first approaches

2

often corresponding to good generalization. Overall, our results suggest that non-classical statistics can provide a novel perspective for understanding internal interactions and training dynamics of deep networks.

Hanyu Zhao, Yang Wu, Yuexian Hou

Tianjin University, China

Neural Network Training Dynamics Neural Network Generalization

Feb 27, 2026

Optimizer-Induced Low-Dimensional Drift and Transverse Dynamics in Transformer Training

We analyze cumulative parameter trajectories of transformer training under AdamW and identify a dominant low-dimensional drift direction ("backbone") that captures 60--80% of long-horizon displacement from initialization. This direction is highly stable over rolling training windows yet reorients gradually across phases, particularly following objective reweighting. Per-batch gradients exhibit near-noise-floor alignment with the backbone, whereas optimizer-integrated updates align strongly with it, indicating that the structure emerges from accumulated optimizer dynamics rather than instantaneous gradient geometry. Replacing AdamW with SGD-family optimizers eliminates this structure, and reducing

β_2

smoothly degrades backbone dominance and reheating recoverability. Reheating experiments show that transverse probe modes can be transiently re-excited without substantially altering accumulated backbone drift. These results provide a trajectory-level characterization of optimizer-induced geometric structure in transformer training and shift attention from instantaneous gradient properties to cumulative update dynamics.

Yongzhong Xu

Transformer Training Dynamics Neural Network Training Dynamics Transformer Adaptive Gradient Methods Intrinsic Dimensionality

Feb 25, 2026

Don't stop me now: How Validation Criteria Affect Checkpoint Selection and Early Stopping

Checkpoint selection is a standard component of neural network training, yet the validation criterion used to select a checkpoint is often chosen heuristically. Moreover, the same criterion may be used either only to rank checkpoints after completion of a predefined training run or also to determine when training should stop, thereby affecting both the selected checkpoint and the set of checkpoints available for selection. In this work, we systematically investigate the role of validation criteria under these two settings. We separately vary the training loss, the validation criterion, and the target evaluation metric, and compare post-hoc checkpoint selection, in which training proceeds for all predefined epochs, with patience-based early stopping, in which the validation criterion also controls training termination. We consider three Cross-Entropy, C-Loss, and PolyLoss as training losses, and accuracy, macro-F1, and Matthews correlation coefficient as target metrics. Selection quality is assessed through the relative gap between the test performance of the validation-selected checkpoint and the best-observed test performance for the same target metric over the complete predefined training run.

Andrea Apicella, Francesco Isgrò, Andrea Pollastro +1

Department of Information Engineering, Electrical Engineering, and Applied Mathematics (DIEM), University of Salerno, Via Giovanni Paolo II, 132, Fisciano (Salerno), 84084, Italy. · Department of Electrical Engineering and Information Technology, University of Naples Federico II, Via Claudio 21, Naples, 80125, Italy.

Early Stopping Neural Network Training Dynamics

Feb 25, 2026

The Error of Deep Operator Networks Is the Sum of Its Parts: Branch-Trunk and Mode Error Decompositions

Operator learning has the potential to strongly impact scientific computing by learning solution operators for differential equations, potentially accelerating multi-query tasks such as design optimization and uncertainty quantification by orders of magnitude. Despite proven universal approximation properties, deep operator networks (DeepONets) often exhibit limited accuracy and generalization in practice, which hinders their adoption. Understanding these limitations is therefore crucial for further advancing the approach. This work analyzes performance limitations of the classical DeepONet architecture. It is shown that the approximation error is dominated by the branch network when the internal dimension is sufficiently large, and that the learned trunk basis can often be replaced by classical basis functions without a significant impact on performance. To investigate this further, a modified DeepONet is constructed in which the trunk network is replaced by the left singular vectors of the training solution matrix. This modification yields several key insights. First, for examples involving the KdV and Burgers equations, a spectral bias in the branch network is observed, with coefficients of dominant, low-frequency modes learned more effectively. Second, through the interplay of the singular-value weighting and the optimizer's neglect of small modes, the branch error is dominated by modes with large and intermediate singular values. Third, using a shared branch network for all mode coefficients, as in the standard architecture, improves generalization of small modes compared to a stacked architecture in which coefficients are computed separately. Finally, detrimental coupling between modes in parameter space is identified, which weakens with increasing network width.

Alexander Heinlein, Johannes Taraz

Delft Institute of Applied Mathematics, Delft University of Technology, Delft, 2628CD, The Netherlands

Deep Operator Networks Neural Operators Neural Network Generalization Neural Network Training Dynamics

Feb 25, 2026

When Learning Hurts: Fixed-Pole RNN for Real-Time Online Training

Recurrent neural networks (RNNs) can be interpreted as discrete-time state-space models, where the state evolution corresponds to an infinite-impulse-response (IIR) filtering operation governed by both feedforward weights and recurrent poles. While, in principle, all parameters including pole locations can be optimized via backpropagation through time (BPTT), such joint learning incurs substantial computational overhead and is often impractical for applications with limited training data. Echo state networks (ESNs) mitigate this limitation by fixing the recurrent dynamics and training only a linear readout, enabling efficient and stable online adaptation. In this work, we analytically and empirically examine why learning recurrent poles does not provide tangible benefits in data-constrained, real-time learning scenarios. Our analysis shows that pole learning renders the weight optimization problem highly non-convex, requiring significantly more training samples and iterations for gradient-based methods to converge to meaningful solutions. Empirically, we observe that for complex-valued data, gradient descent frequently exhibits prolonged plateaus, and advanced optimizers offer limited improvement. In contrast, fixed-pole architectures induce stable and well-conditioned state representations even with limited training data. Numerical results demonstrate that fixed-pole networks achieve superior performance with lower training complexity, making them more suitable for online real-time tasks.

Alexander Morgan, Ummay Sumaya Khan, Lingjia Liu +1

dept. of ECE, Virginia Tech, Alexandria, VA, USA

Recurrent Neural Networks Neural Network Training Dynamics Neural Network Optimization Online Learning Linear RNNs

Feb 24, 2026

Motivation is Something You Need

This work introduces a novel training paradigm that draws from affective neuroscience. Inspired by the interplay of emotions and cognition in the human brain and more specifically the SEEKING motivational state, we design a dual-model framework where a smaller base model is trained continuously, while a larger motivated model is activated intermittently during predefined "motivation conditions". The framework mimics the emotional state of high curiosity and anticipation of reward in which broader brain regions are recruited to enhance cognitive performance. Exploiting scalable architectures where larger models extend smaller ones, our method enables shared weight updates and selective expansion of network capacity during noteworthy training steps. Empirical evaluation on the image classification task demonstrates that, not only does the alternating training scheme efficiently and effectively enhance the base model compared to a traditional scheme, in some cases, the motivational model also surpasses its standalone counterpart despite seeing less data per epoch. This opens the possibility of simultaneously training two models tailored to different deployment constraints with competitive or superior performance while keeping training cost lower than when training the larger model.

Mehdi Acheli, Walid Gaaloul

SAMOVAR, Telecom SudParis Institut Polytechnique de Paris, Palaiseau, France

Neural Network Training Dynamics Conditional Computation

Feb 23, 2026

Path-conditioned training: a principled way to rescale ReLU neural networks

Despite recent algorithmic advances, we still lack principled ways to leverage the well-documented rescaling symmetries in ReLU neural network parameters. While two properly rescaled weights implement the same function, the training dynamics can be dramatically different. To offer a fresh perspective on exploiting this phenomenon, we build on the recent path-lifting framework, which provides a compact factorization of ReLU networks. We introduce a geometrically motivated criterion to rescale neural network parameters which minimization leads to a conditioning strategy that aligns a kernel in the path-lifting space with a chosen reference. We derive an efficient algorithm to perform this alignment. In the context of random network initialization, we analyze how the architecture and the initialization scale jointly impact the output of the proposed method. Numerical experiments illustrate its potential to speed up training.

Arthur Lebeurrier, Titouan Vayer, Rémi Gribonval

ENS de Lyon, CNRS, Inria, Université Claude Bernard Lyon 1, LIP, UMR 5668, 69342, Lyon cedex 07, France · Inria, Rennes, France · Inria, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP, UMR 5668, 69342, Lyon cedex 07, France

ReLU Neural Networks Neural Network Training Dynamics Neural Network Initialization

Feb 23, 2026

The Confusion is Real: GRAPHIC -- A Network Science Approach to Confusion Matrices in Deep Learning

Explainable artificial intelligence has emerged as a promising field of research to address reliability concerns in artificial intelligence. Despite significant progress in explainable artificial intelligence, few methods provide a systematic way to visualize and understand how classes are confused and how their relationships evolve as training progresses. In this work, we present GRAPHIC, an architecture-agnostic approach that analyzes neural networks on a class level. It leverages confusion matrices derived from intermediate layers using linear classifiers. We interpret these as adjacency matrices of directed graphs, allowing tools from network science to visualize and quantify learning dynamics across training epochs and intermediate layers. GRAPHIC provides insights into linear class separability, dataset issues, and architectural behavior, revealing, for example, similarities between flatfish and man and labeling ambiguities validated in a human study. In summary, by uncovering real confusions, GRAPHIC offers new perspectives on how neural networks learn. The code is available at https://github.com/Johanna-S-Froehlich/GRAPHIC.

Johanna S. Fröhlich, Bastian Heinlein, Jan U. Claar +3

Friedrich-Alexander-Universität Erlangen-Nürnberg Erlangen, Germany · Technical University of Darmstadt Darmstadt, Germany

Neural Network Interpretability Explainable Artificial Intelligence Neural Network Training Dynamics Classification

Feb 23, 2026

Leap+Verify: Regime-Adaptive Speculative Weight Prediction for Accelerating Neural Network Training

We introduce Leap+Verify, a framework that applies speculative execution -- predicting future model weights and validating predictions before acceptance -- to accelerate neural network training. Inspired by speculative decoding in language model inference and by the Automatically Scalable Computation (ASC) architecture for program execution, Leap+Verify decomposes training into three dynamically detected regimes (chaotic, transition, stable) using activation-space cosine similarity as a real-time Lyapunov proxy signal. Within each regime, analytic weight predictors (momentum, linear, quadratic extrapolation) attempt to forecast model parameters K training steps ahead; predictions are accepted only when validated against a held-out loss criterion. We evaluate Leap+Verify on GPT-2 124M and Qwen 2.5-1.5B trained on WikiText-103 across five random seeds, sweeping prediction depth K in {5, 10, 25, 50, 75, 100}. Momentum-based prediction (Adam moment extrapolation) fails catastrophically at both scales, with predicted losses exceeding actuals by 100-10,000x -- a universal norm explosion in optimizer-state extrapolation. Finite-difference predictors (linear, quadratic) succeed where momentum fails: at 124M, they achieve 24% strict acceptance at K=5 in stable regimes; at 1.5B, they achieve 37% strict acceptance in transition regimes. The scale-dependent finding is in regime distribution: GPT-2 124M spends 34% of training in stable regime, while Qwen 1.5B spends 64% in chaotic regime and reaches stable in only 0-2 of 40 checkpoints. Larger models are more predictable when predictable, but less often predictable -- the practical bottleneck shifts from predictor accuracy to regime availability. Cross-seed results are highly consistent (less than 1% validation loss variance), and the three-regime framework produces identical phase boundaries (plus or minus 50 steps) across seeds.

Jeremy McEntire

Neural Network Training Dynamics

Feb 23, 2026

Grokking Finite-Dimensional Algebra

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

Pascal Jr Tikeng Notsawo, Guillaume Dumas, Guillaume Rabusseau

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

Grokking Neural Network Generalization Neural Network Training Dynamics

Feb 22, 2026

Incremental Learning of Sparse Attention Patterns in Transformers

This paper studies simple transformers trained on a high-order Markov chain, where the model must incorporate information from multiple past positions, each with different statistical importance. We show that transformers learn the task incrementally, with each stage corresponding to learning how to copy information from a subset of positions via a sparse attention pattern. Notably, the learning dynamics transition from a competitive phase, where all heads focus on the statistically most important positions, to a cooperative phase, where different heads specialize in different patterns. We model these dynamics with simplified differential equations and prove stage-wise convergence of the resulting system. Functionally, these stages correspond to a sequence of increasingly expressive misspecified models, with the full model class reached only at the end. Overall, we give a theoretical account of how structured attention patterns and head specialization emerge in stages without an explicit curriculum, with implications for generalization in sequential tasks.

Oğuz Kaan Yüksel, Rodrigo Alvarez Lucendo, Nicolas Flammarion

TML Lab, EPFL

Self-Attention Transformer Training Dynamics Transformer Attention Sparse Attention Incremental Learning Attention Mechanisms+3

Feb 22, 2026

Learning from Complexity: Exploring Dynamic Sample Pruning of Spatio-Temporal Training

Spatio-temporal forecasting is fundamental to intelligent systems in transportation, climate science, and urban planning. However, training deep learning models on the massive, often redundant, datasets from these domains presents a significant computational bottleneck. Existing solutions typically focus on optimizing model architectures or optimizers, while overlooking the inherent inefficiency of the training data itself. This conventional approach of iterating over the entire static dataset each epoch wastes considerable resources on easy-to-learn or repetitive samples. In this paper, we explore a novel training-efficiency techniques, namely learning from complexity with dynamic sample pruning, ST-Prune, for spatio-temporal forecasting. Through dynamic sample pruning, we aim to intelligently identify the most informative samples based on the model's real-time learning state, thereby accelerating convergence and improving training efficiency. Extensive experiments conducted on real-world spatio-temporal datasets show that ST-Prune significantly accelerates the training speed while maintaining or even improving the model performance, and it also has scalability and universality.

Wei Chen, Junle Chen, Yuqian Wu +2

HKUST(GZ) · HKUST

Data-Efficient Learning Dataset Pruning Spatiotemporal Forecasting Neural Network Training Dynamics

Feb 20, 2026

Global Low-Rank, Local Full-Rank: The Holographic Encoding of Learned Algorithms

Grokking -- the abrupt transition from memorization to generalization after extended training -- has been linked to the emergence of low-dimensional structure in learning dynamics. Yet neural network parameters inhabit extremely high-dimensional spaces. How can a low-dimensional learning process produce solutions that resist low-dimensional compression? We investigate this question in multi-task modular arithmetic, training shared-trunk Transformers with separate heads for addition, multiplication, and a quadratic operation modulo 97. Across three model scales (315K--2.2M parameters) and five weight decay settings, we compare three reconstruction methods: per-matrix SVD, joint cross-matrix SVD, and trajectory PCA. Across all conditions, grokking trajectories are confined to a 2--6 dimensional global subspace, while individual weight matrices remain effectively full-rank. Reconstruction from 3--5 trajectory PCs recovers over 95% of final accuracy, whereas both per-matrix and joint SVD fail at sub-full rank. Even when static decompositions capture most spectral energy, they destroy task-relevant structure. These results show that learned algorithms are encoded through dynamically coordinated updates spanning all matrices, rather than localized low-rank components. We term this the holographic encoding principle: grokked solutions are globally low-rank in the space of learning directions but locally full-rank in parameter space, with implications for compression, interpretability, and understanding how neural networks encode computation.

Yongzhong Xu

Neural Network Training Dynamics Grokking Low-Rank Approximation

Feb 19, 2026

The Geometry of Multi-Task Grokking: Transverse Instability, Superposition, and Weight Decay Phase Structure

Grokking -- the abrupt transition from memorization to generalization long after near-zero training loss -- has been studied mainly in single-task settings. We extend geometric analysis to multi-task modular arithmetic, training shared-trunk Transformers on dual-task (mod-add + mod-mul) and tri-task (mod-add + mod-mul + mod-sq) objectives across a systematic weight decay sweep. Five consistent phenomena emerge. (1) Staggered grokking order: multiplication generalizes first, followed by squaring, then addition, with consistent delays across seeds. (2) Universal integrability: optimization trajectories remain confined to an empirically invariant low-dimensional execution manifold; commutator defects orthogonal to this manifold reliably precede generalization. (3) Weight decay phase structure: grokking timescale, curvature depth, reconstruction threshold, and defect lead covary systematically with weight decay, revealing distinct dynamical regimes and a sharp no-decay failure mode. (4) Holographic incompressibility: final solutions occupy only 4--8 principal trajectory directions yet are distributed across full-rank weights and destroyed by minimal perturbations; SVD truncation, magnitude pruning, and uniform scaling all fail to preserve performance. (5) Transverse fragility and redundancy: removing less than 10% of orthogonal gradient components eliminates grokking, yet dual-task models exhibit partial recovery under extreme deletion, suggesting redundant center manifolds enabled by overparameterization. Together, these results support a dynamical picture in which multi-task grokking constructs a compact superposition subspace in parameter space, with weight decay acting as compression pressure and excess parameters supplying geometric redundancy in optimization pathways.

Yongzhong Xu

Grokking Neural Network Training Dynamics Weight Decay Multi-Task Learning

Feb 19, 2026

Convergence Analysis of Two-Layer Neural Networks under Gaussian Input Masking

We investigate the convergence guarantee of two-layer neural network training with Gaussian randomly masked inputs. This scenario corresponds to Gaussian dropout at the input level, or noisy input training common in sensor networks, privacy-preserving training, and federated learning, where each user may have access to partial or corrupted features. Using a Neural Tangent Kernel (NTK) analysis, we demonstrate that training a two-layer ReLU network with Gaussian randomly masked inputs achieves linear convergence up to an error region proportional to the mask's variance. A key technical contribution is resolving the randomness within the non-linear activation, a problem of independent interest.

Afroditi Kolomvaki, Fangshuo Liao, Evan Dramko +2

Computer Science Dept., Rice University, Houston, TX, USA

Neural Network Training Dynamics ReLU Neural Networks

Feb 18, 2026

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

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

Jianliang He, Leda Wang, Siyu Chen +1

Department of Statistics and Data Science, Yale University

Lottery Ticket Hypothesis Modular Addition Neural Network Training Dynamics Frequency-Domain Feature Learning Neural Network Generalization Grokking

Feb 18, 2026

Learning under noisy supervision is governed by a feedback-truth gap

When feedback is absorbed faster than task structure can be evaluated, the learner will favor feedback over truth. A two-timescale model shows this feedback-truth gap is inevitable whenever the two rates differ and vanishes only when they match. We test this prediction across neural networks trained with noisy labels (30 datasets, 2,700 runs), human probabilistic reversal learning (N = 292), and human reward/punishment learning with concurrent EEG (N = 25). In each system, truth is defined operationally: held-out labels, the objectively correct option, or the participant's pre-feedback expectation - the only non-circular reference decodable from post-feedback EEG. The gap appeared universally but was regulated differently: dense networks accumulated it as memorization; sparse-residual scaffolding suppressed it; humans generated transient over-commitment that was actively recovered. Neural over-commitment (~0.04-0.10) was amplified tenfold into behavioral commitment (d = 3.3-3.9). The gap is a fundamental constraint on learning under noisy supervision; its consequences depend on the regulation each system employs.

Elan Schonfeld, Elias Wisnia

Department of Biology, Columbia University, New York, NY, USA · Department of Chemistry, Columbia University, New York, NY, USA

Learning with Noisy Labels Neural Network Training Dynamics

Feb 18, 2026

Optimizer choice matters for the emergence of Neural Collapse

Neural Collapse (NC) refers to the emergence of highly symmetric geometric structures in the representations of deep neural networks during the terminal phase of training. Despite its prevalence, the theoretical understanding of NC remains limited. Existing analyses largely ignore the role of the optimizer, thereby suggesting that NC is universal across optimization methods. In this work, we challenge this assumption and demonstrate that the choice of optimizer plays a critical role in the emergence of NC. The phenomenon is typically quantified through NC metrics, which, however, are difficult to track and analyze theoretically. To overcome this limitation, we introduce a novel diagnostic metric, NC0, whose convergence to zero is a necessary condition for NC. Using NC0, we provide theoretical evidence that NC cannot emerge under decoupled weight decay in adaptive optimizers, as implemented in AdamW. Concretely, we prove that SGD, SignGD with coupled weight decay (a special case of Adam), and SignGD with decoupled weight decay (a special case of AdamW) exhibit qualitatively different NC0 dynamics. Also, we show the accelerating effect of momentum on NC (beyond convergence of train loss) when trained with SGD, being the first result concerning momentum in the context of NC. Finally, we conduct extensive empirical experiments consisting of 3,900 training runs across various datasets, architectures, optimizers, and hyperparameters, confirming our theoretical results. This work provides the first theoretical explanation for optimizer-dependent emergence of NC and highlights the overlooked role of weight-decay coupling in shaping the implicit biases of optimizers.

Jim Zhao, Tin Sum Cheng, Wojciech Masarczyk +1

University of Basel · Warsaw University of Technology IDEAS Research Institute

Neural Collapse Neural Network Training Dynamics Adaptive Gradient Methods Weight Decay Stochastic Gradient Descent

Feb 18, 2026

Low-Dimensional and Transversely Curved Optimization Dynamics in Grokking

Grokking -- the delayed transition from memorization to generalization in small algorithmic tasks -- remains poorly understood. We present a geometric analysis of optimization dynamics in transformers trained on modular arithmetic. PCA of attention weight trajectories reveals that training evolves predominantly within a low-dimensional execution subspace, with a single principal component capturing 68-83% of trajectory variance. To probe loss-landscape geometry, we measure commutator defects -- the non-commutativity of successive gradient steps -- and project them onto this learned subspace. We find that curvature grows sharply in directions orthogonal to the execution subspace while the trajectory remains largely confined to it. Importantly, curvature growth consistently precedes generalization across learning rates and hyperparameter regimes, with the lead time obeying a power law in the grokking timescale. Causal intervention experiments show that motion along the learned subspace is necessary for grokking, while artificially increasing curvature is insufficient. Together, these results support a geometric account in which grokking reflects escape from a metastable regime characterized by low-dimensional confinement and transverse curvature accumulation. All findings replicate across this learning-rate range, a qualitatively different slow regime (lr=5e-5, wd=0.1, 3 layers), and three random seeds, though alignment dynamics differ quantitatively between regimes. Causal intervention experiments establish that orthogonal gradient flow is necessary but not sufficient for grokking: suppressing it prevents generalization with a monotonic dose-response across four operations, while artificially boosting curvature defects has no effect.

Yongzhong Xu

Grokking Neural Network Training Dynamics Gradient Descent Dynamics Loss Landscape Geometry

Feb 16, 2026

On the Emergence of Implicit Curriculum in RLVR Learning Dynamics

Reinforcement learning with verifiable rewards (RLVR) has been a main driver of recent breakthroughs in large reasoning models. Yet it remains a mystery how rewards based solely on final outcomes can help overcome the long-horizon barrier to extended reasoning. To understand this, we develop a theory of the training dynamics of RLVR for transformers on compositional reasoning tasks. Our theory shows that mixed-difficulty training naturally induces an implicit curriculum: without any explicit schedule, easier problems become learnable first and shape the frontier for harder ones, creating a learning progression from easy to hard during optimization. The effectiveness of this curriculum is governed by the smoothness of the difficulty spectrum. When the spectrum is smooth, training dynamics enter a well-behaved relay regime, in which persistent gradient signals on easier problems make slightly harder ones tractable and keep training at the edge of competence. When the spectrum contains abrupt discontinuities, training undergoes grokking-type phase transitions with prolonged plateaus before progress recurs. As a technical contribution, our analysis develops and adapts techniques from Fourier analysis on finite groups to our setting. We validate the predicted mechanisms empirically via controlled synthetic experiments and real-model RLVR runs.

Yu Huang, Zixin Wen, Yuejie Chi +4

Department of Statistics and Data Science, Wharton School, University of Pennsylvania · Machine Learning Department, Carnegie Mellon University · Department of Statistics and Data Science, Yale University +1

RL for Language Model Reasoning Reinforcement Learning with Verifiable Rewards Curriculum Learning Neural Network Training Dynamics

Feb 16, 2026

Scale redundancy and soft gauge fixing in positively homogeneous neural networks

Neural networks with positively homogeneous activations exhibit an exact continuous reparametrization symmetry: neuron-wise rescalings generate parameter-space orbits along which the input--output function is invariant. We interpret this symmetry as a gauge redundancy and introduce gauge-adapted coordinates that separate invariant and scale-imbalance directions. Inspired by gauge fixing in field theory, we introduce a soft orbit-selection (norm-balancing) functional acting only on redundant scale coordinates. We show analytically that it induces dissipative relaxation of imbalance modes to preserve the realized function. In controlled experiments, this orbit-selection penalty expands the stable learning-rate regime and suppresses scale drift without changing expressivity. These results establish a structural link between gauge-orbit geometry and optimization conditioning, providing a concrete connection between gauge-theoretic concepts and machine learning.

Rodrigo Carmo Terin

King Juan Carlos University, Facuty of Experimental Sciences and Technology, Department of Applied Physics, Av. del Alcalde de Móstoles, 28933, Madrid, Spain

Neural Network Optimization Neural Network Training Dynamics

Feb 9, 2026

ANCRe: Adaptive Neural Connection Reassignment for Efficient Depth Scaling

Scaling network depth has been a central driver behind the success of modern foundation models, yet recent investigations suggest that deep layers are often underutilized. This paper revisits the default mechanism for deepening neural networks, namely residual connections, from an optimization perspective. Rigorous analysis proves that the layout of residual connections can fundamentally shape convergence behavior, and even induces an exponential gap in convergence rates. Prompted by this insight, we introduce adaptive neural connection reassignment (ANCRe), a principled and lightweight framework that parameterizes and learns residual connectivities from the data. ANCRe adaptively reassigns residual connections with negligible computational and memory overhead (

<1\%

), while enabling more effective utilization of network depth. Extensive numerical tests across pre-training of large language models, diffusion models, and deep ResNets demonstrate consistently accelerated convergence, boosted performance, and enhanced depth efficiency over conventional residual connections.

Yilang Zhang, Bingcong Li, Niao He +1

ML Research, Morgan Stanley, New York, NY 10019 · Department of CS, ETH Zürich, 8092 Zürich, Switzerland · Department of ECE, University of Minnesota, Minneapolis, MN 55455

Residual Learning Neural Network Optimization Neural Network Training Dynamics

Jan 30, 2026

Adaptive Momentum and Nonlinear Damping for Neural Network Training

Momentum Stochastic Gradient Descent (mSGD) relies on a fixed momentum coefficient shared across all parameters, failing to account for the heterogeneous structure of modern loss landscapes. In this work, we adopt a continuous-time formulation to introduce individual, adaptive momentum coefficients regulated by the kinetic energy of each model parameter. This mechanism automatically adjusts to evolving training dynamics to maintain stability without sacrificing convergence speed. We demonstrate that this adaptive friction is inextricably linked to cubic damping, a suppression mechanism from structural dynamics. We additionally introduce two optimization schemes by augmenting the continuous dynamics of mSGD and Adam with a cubic damping term. Empirically, our methods demonstrate robustness and match or outperform Adam on training ViT, BERT, and GPT2 tasks where mSGD typically struggles. We further provide theoretical results establishing the exponential convergence of the proposed schemes.

Aikaterini Karoni, Rajit Rajpal, Benedict Leimkuhler +1

School of Mathematics University of Bristol · School of Mathematics University of Edinburgh · CERMICS, CNRS, École des Ponts, Institut Polytechnique de Paris; Inria Paris, France

Momentum Methods Stochastic Optimization Convergence Neural Network Training Dynamics Stochastic Gradient Descent