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

May 14, 2026cs.NE

Advancing Direct Training for Spiking Neural Networks with Circulate-Firing Neurons and Learnable Gradients

Spiking Neural Networks (SNNs) have emerged with promising energy-efficient property, yet a substantial performance gap persists compared to Artificial Neural Networks (ANNs). This gap stems from at least two key limitations: first, conventional spiking neurons offer limited information representation capacity, underutilizing the rich dynamics of membrane potentials; second, fixed surrogate gradient (SG) functions across time steps leads to imprecise gradient propagation, impeding effective direct training. To address these two challenges, we propose a new direct training algorithm with three core innovations: first, a circulate-firing spiking neuron model that enhances information representation capacity by leveraging membrane potentials more effectively; second, a time-step-wise learnable surrogate gradient function, enabling accurate gradient estimation during backpropagation; third, a positive-negative balanced loss function to achieve equilibrium between positive and negative membrane potentials and further boost SNN performance. Extensive experiments demonstrate that our methods achieve competitive performance across multiple datasets. Our methods can generalize seamlessly to advanced architectures of Transformer, consistently outperforming existing methods. Our work highlights the effectiveness of further harnessing intrinsic membrane dynamics of SNNs for performance improvement, and thus open a new avenue for advancing high-performance spiking neural architectures.

Feifan Zhou, Xiang Wei, Yang Liu +1

School of Artificial and Intelligence, Tianjin University, Tianjin 300072, China

Spiking Neural Networks Neural Network Training Dynamics

May 14, 2026cs.LG

Slower Generalization, Faster Memorization: A Sweet Spot in Algorithmic Learning

Critical-data-size accounts of grokking suggest a natural post-threshold intuition: once training data is sufficient to identify the underlying rule, additional data should accelerate validation convergence. We show that this intuition can fail in a controlled structured-output task. In Needleman--Wunsch (NW) matrix generation, small Transformers reach high validation exact-match accuracy fastest at an intermediate dataset size, not at the largest one. Past this dataset-size sweet spot, generalization remains achievable but requires more gradient updates. Conversely, in the regime where partial validation competence first appears, larger datasets can require fewer updates to reach high training accuracy, suggesting that emerging rule structure can accelerate fitting beyond example-wise memorization. A multiplication baseline does not show the same post-threshold slowdown. These results separate the critical data size for the onset of generalization from the dataset size that optimizes update-based convergence, and identify structured-output tasks where learning the rule and completing exact-fitting can diverge.

Shin So, Kyelim Lee, Albert No

Yonsei University

Neural Network Generalization Neural Network Training Dynamics Grokking

May 14, 2026cs.LG

Dynamics of the Transformer Residual Stream: Coupling Spectral Geometry to Network Topology

Large language models are remarkably capable, yet how computation propagates through their layers remains poorly understood. A growing line of work treats depth as discrete time and the residual stream as a dynamical system, where each layer's nonlinear update has a local linear description. However, previous analyses have relied on scalar summaries or approximate linearizations, leaving the full spectral geometry of trained LLMs unknown. We perform full Jacobian eigendecomposition across three production--scale LLMs and show that training installs a monotonic spectral gradient through depth -- from non-normal, rotation-dominated early layers to near--symmetric late layers -- together with a cumulative low-rank bottleneck that funnels perturbations into a small fraction of the residual stream's effective dimensions. Our experiments reveal that this gradient and the dimensional collapse are learned rather than architectural, and is largely dissolved when structured non-normality is removed. We further show that the topological positioning of graph communities predicts whether the Jacobian amplifies or suppresses them, with the sign of the coupling determined by the local operator type, a relationship absent at initialization. These results map a learned spectral geometry in LLMs that links perturbation propagation and compression to the network's functional topology.

Jesseba Fernando, Grigori Guitchounts

Network Science Institute, Northeastern University · Flagship Pioneering

Dynamical Systems Transformer LLM Interpretability Spectral Methods Neural Network Training Dynamics

May 13, 2026cs.LG

Deep Learning as Neural Low-Degree Filtering: A Spectral Theory of Hierarchical Feature Learning

Understanding how deep neural networks learn useful internal representations from data remains a central open problem in the theory of deep learning. We introduce Neural Low-Degree Filtering (Neural LoFi), a stylized limit of gradient-based training in which hierarchical feature learning becomes an explicit iterative spectral procedure. In this limit, the dynamics at each layer decouple: given the current representation, the next layer selects directions with maximal accessible low-degree correlation to the label. This yields a tractable surrogate mechanism for deep learning, together with a natural kernel-space interpretation. Neural LoFi provides a mathematically explicit framework for studying multi-layer feature learning beyond the lazy regime. It predicts how representations are selected layer by layer, explains how emergence of concepts arises with given sample complexity,and gives a concrete mechanism by which depth progressively constructs new features from old ones through low-degree compositionality. We complement the theory with mechanistic experiments on fully connected and convolutional architectures, showing that Neural LoFi improves over lazy random-feature baselines, recovers meaningful structured filters, and predicts representations aligned with early gradient-descent feature discovery with real datasets.

Yatin Dandi, Matteo Vilucchio, Luca Arnaboldi +2

Information Learning and Physics Laboratory, École Polytechnique Fédérale de Lausanne (EPFL) · Statistical Physics of Computation Laboratory, École Polytechnique Fédérale de Lausanne (EPFL)

Hierarchical Representation Learning Neural Network Training Dynamics Statistical Learning Theory

May 13, 2026cs.LG

First-Passage Prediction of Grokking Delay: ACalibrated Law under AdamW with Causal Validation

We give the first quantitative prediction of grokking delay under AdamW. Treating the delay as a first-passage time, we derive a closed-form law T_grok - T_mem = (1 / 2 kappa_LL eta lambda) log(V_mem / V_star), where V_t = ||theta_t||^2 is the squared parameter norm, V_star is an architecture-dependent threshold, and kappa_LL absorbs the AdamW correction to the clean-SGD contraction rate 2 eta lambda. Calibrating (kappa_LL, V_star) on a single hyperparameter cell predicts grokking delays on 26 held-out runs with MAPE 17.7% over a 41x delay range; the law generalises to MLPs (MAPE 18.0%, N=34) and degrades to 23.3% on cross-task extension (N=46, 43.5x range), with a structured residual in which V_star / V_mem stays comparatively stable within architecture (CV about 14% on the 1L transformer). First-passage of V_t is necessary but not sufficient. A quantile-margin theorem establishes that positive delay requires both norm separation V_mem > V_post and angular reachability of a threshold alpha_star = arcsin(C / V_T_mem^(1/2)), where C is computable from the empirical NTK feature map and the validation-margin quantile. Calibrating C on modulus p=89 predicts alpha_star = 47.2 degrees at p=97 (observed 47.8 degrees, error 1.3%) as a prior cross-cell prediction. Causal interventions that freeze the norm or remove weight decay at memorisation eliminate grokking (0/6 vs. 3/3 baseline), trapping the angular displacement near 12 degrees. kappa_LL is empirically measured per architecture rather than derived from (beta_1, beta_2, epsilon); within-architecture CV stays at most 15% across four architectures, but values differ by about 2x between architectural variants beyond depth alone. Empirical scope is algorithmic tasks (modular arithmetic, sparse parity) under AdamW; whether the law transfers to natural-language scale models is open.

Truong Xuan Khanh, Truong Quynh Hoa, Luu Duc Trung +1

H&K Research Studio, Clevix LLC, Hanoi, Vietnam · Banking Academy of Vietnam, Hanoi, Vietnam

Neural Network Memorization Neural Network Training Dynamics Algorithmic Reasoning Grokking

May 12, 2026cs.LG

State-Space NTK Collapse Near Bifurcations

Rich feature learning in tasks that unfold over time often requires the model to pass through bifurcations, constituting qualitative changes in the underlying model dynamics. We develop a local theory of gradient descent near these transitions through the empirical state-space neural tangent kernel (sNTK). Our central finding is that bifurcations both dominate and simplify learning dynamics: near bifurcations, we can reduce sNTK to a rank-one operator corresponding to learning in a classical normal form system, providing an analytically tractable description of the local learning geometry, even for high-dimensional recurrent systems. Concretely, we give a procedure for decomposing sNTK into bifurcation-relevant and residual channels, showing that near commonly codimension-1 bifurcations the relevant channel is a rank-one operator that is highly amplified. This amplification causes the bifurcation channel to dominate the full sNTK. Thus, bifurcations locally warp the learning landscape, funneling gradient descent into a few critical dynamical directions and making the nearby kernel and loss geometry predictable from classical normal forms. We illustrate this in a student-teacher recurrent neural network: the first learned bifurcation coincides with a sharp collapse in sNTK effective rank and the emergence of a dominant parameter direction whose restricted sNTK closely matches the landscape predicted by the scalar pitchfork normal form. Finally, we show that low-rank natural gradient methods resolve the resulting learning instability near bifurcations with very little overhead over SGD.

James Hazelden, Eric Shea-Brown

University of Washington

Recurrent Neural Networks Neural Network Training Dynamics Neural Tangent Kernel

May 12, 2026cs.LG

Detecting overfitting in Neural Networks during long-horizon grokking using Random Matrix Theory

Training Neural Networks (NNs) without overfitting is difficult; detecting that overfitting is difficult as well. We present a novel Random Matrix Theory method that detects the onset of overfitting in deep learning models without access to train or test data. For each model layer, we randomize each weight matrix element-wise,

\mathbf{W} \to \mathbf{W}^{\mathrm{rand}}

, fit the randomized empirical spectral distribution with a Marchenko-Pastur distribution, and identify large outliers that violate self-averaging. We call these outliers Correlation Traps. During the onset of overfitting, which we call the "anti-grokking" phase in long-horizon grokking, Correlation Traps form and grow in number and scale as test accuracy decreases while train accuracy remains high. Traps may be benign or may harm generalization; we provide an empirical approach to distinguish between them by passing random data through the trained model and evaluating the JS divergence of output logits. Our findings show that anti-grokking is an additional grokking phase with high train accuracy and decreasing test accuracy, structurally distinct from pre-grokking through its Correlation Traps. More broadly, we find that some foundation-scale LLMs exhibit the same Correlation Traps, indicating potentially harmful overfitting.

Hari K. Prakash, Charles H Martin

University of California San Diego Data Science and Engineering · Calculation Consulting San Francisco, CA, USA

Neural Network Generalization Neural Network Training Dynamics Grokking

May 12, 2026cs.LG

Understanding Sample Efficiency in Predictive Coding

Predictive Coding (PC) is an influential account of cortical learning. Much of recent work has focused on comparing PC to Backpropagation (BP) to find whether PC offers any advantages. Small scale experiments show that PC enables learning that is more sample efficient and effective in many contexts, though a thorough theoretical understanding of the phenomena remains elusive. To address this, we quantify the efficiency of learning in BP and PC through a metric called ``target alignment'', which measures how closely the change in the output of the network is aligned to the output prediction error. We then derive and empirically validate analytical expressions for target alignment in Deep Linear Networks. We show that learning in PC is more efficient than BP, which is especially pronounced in deep, narrow and pre-trained networks. We also derive exact conditions for guaranteed optimal target alignment in PC and validate our findings through experiments. We study full training trajectories of linear and non-linear models, and find the predicted benefits of PC persist in practice even when some assumptions are violated. Overall, this work provides a mechanistic understanding of the higher learning efficiency observed for PC over BP in previous works, and can guide how PC should be parametrised to learn most effectively.

Gaspard Oliviers, Elene Lominadze, Rafal Bogacz

Nuffield Department of Clinical Neurosciences, University of Oxford, United Kingdom · MRC Centre of Research Excellence in Restorative Neural Dynamics, United Kingdom

Backpropagation Predictive Coding Deep Linear Networks Neural Network Training Dynamics Data-Efficient Learning

May 12, 2026cs.LG

OUI as a Structural Observable: Towards an Activation-Centric View of Neural Network Training

Activation functions are what make deep networks expressive: without them, the model collapses to a linear map. Yet we still evaluate training mostly from the outside, through loss, accuracy, return, or final calibration, while the internal structural evolution of the network remains largely unobserved. In this paper, we argue that the Overfitting--Underfitting Indicator (OUI) should be understood as a first practical observable of that internal structure. Across our recent results, OUI consistently appears as an early, label-free, activation-based signal that reveals whether a network is entering a poor or promising training regime before convergence. In supervised learning, it anticipates weight decay regimes; in reinforcement learning, it discriminates learning-rate regimes early in PPO actor--critic; and in online control, it can drive layer-wise weight decay adaptation. Read together with recent evidence that activation patterns tend to stabilize earlier than parameters, these results suggest a broader research direction: an activation-centric theory of training dynamics. OUI is becoming an empirical foothold toward this theory.

Alberto Fernández-Hernández, Jose I. Mestre, Cristian Pérez-Corral +3

Universitat Politècnica de València, Valencia, Spain · Universitat Jaume I, Castelló de la Plana, Spain · Openchip & Software Technologies S.L., Spain

Neural Network Generalization Neural Network Training Dynamics

May 11, 2026cs.LG

Error whitening: Why Gauss-Newton outperforms Newton

The Gauss-Newton matrix is widely viewed as a positive semidefinite approximation of the Hessian, yet mounting empirical evidence shows that Gauss-Newton descent outperforms Newton's method. We adopt a function space perspective to analyze this phenomenon. We show that the generalized Gauss-Newton (GGN) matrix projects the Newton direction in function space onto the model's tangent space, while a Jacobian-only variant obtained by applying the least squares Gauss-Newton matrix to non-least squares losses projects the function space loss gradient onto this same tangent space. Both projections eliminate distortions from the model's parameterization. Specifically, the evolution of the prediction-target mismatch depends on the model's parameterization through the matrix

JJ^\top

where

J

is the Jacobian of the model with respect to its parameters. The projections effectively replace

JJ^\top

with the identity. We call this effect error whitening. Once the parameterization is removed, the prediction-target mismatch evolves according to dynamics dictated by the structure of the loss and the projection produced by the optimizer. Error whitening is a special property of Gauss-Newton descent that rigorously distinguishes it from Newton's method. We empirically demonstrate that Gauss-Newton optimizers follow the theoretically predicted function space dynamics and outperforms Newton's method, Adam, and Muon across case studies spanning supervised learning, physics-informed deep learning, and approximate dynamic programming.

Maricela Best McKay, Nathan P. Lawrence, Brian Wetton +1

University of British Columbia · University of California, Berkeley

Gauss-Newton Optimization Deep Learning Optimization Second-Order Optimization Neural Network Training Dynamics

May 11, 2026stat.ML

Uniform Scaling Limits in AdamW-Trained Transformers

We study the large-depth limit of transformers trained with AdamW, by modelling the hidden-state dynamics as an interacting particle system (IPS) coupled through the attention mechanism. Under appropriate scaling of the attention heads, we prove that the joint dynamics of the hidden states and backpropagated variables converge in

L^2

, uniformly over the initial condition, to the solution of a forward--backward system of ODEs at rate

\mathcal O(L^{-1}+L^{-1/3}H^{-1/2})

. Here,

L

and

H

denote the depth and number of heads of the transformer, respectively. The limiting system of ODEs can be identified with a McKean--Vlasov ODE (MVODE) when the attention heads do not incorporate causal masking. By using the flow maps associated with this MVODE and applying concentration of measure techniques, we obtain bounds on the difference between the discrete and continuous models that are uniform over compact sets of initial conditions. As this is achieved without resorting to a covering argument, the constants in our bounds are independent of the number of tokens. Furthermore, under a suitable adaptation to AdamW, the bounds become independent of the token embedding dimension.

William Gibson, Christoph Reisinger

Mathematical Institute · University of Oxford · Oxford, OX2 6GG, UK

Transformer Self-Attention Neural Network Training Dynamics

May 11, 2026math.OC

On the global convergence of gradient descent for wide shallow models with bounded nonlinearities

A surprising phenomenon in the training of neural networks is the ability of gradient descent to find global minimizers of the training loss despite its non-convexity. Following earlier works, we investigate this behavior for wide shallow networks. Existing results essentially cover the case of ReLU activations and the case of sigmoid activations with scalar output weights. We study a large class of models that includes multi-head attention layers and two-layer sigmoid networks with vector output weights. Building upon [Chizat and Bach, 2018], we prove that all non-global minimizers of the training loss are unstable under gradient descent dynamics. Thus, when the initial distribution of the parameters has full support (which includes the popular Gaussian case), and in the many hidden neurons or attention heads limit, continuous-time gradient descent can only converge to global minimizers. Establishing the instability of non-global minimizers corresponds to the construction of an ``escaping active set'' -- we complete the proof of [Chizat and Bach, 2018] to construct this set for models with bounded nonlinearities and scalar output weights. We also extend this construction to new cases for models with vector output weights. Finally, we show the well-posedness and the stability with respect to discretization of the mean field training dynamic for sub-Gaussian initializations.

Romain Petit, Clarice Poon, Gabriel Peyré

CNRS and ENS, PSL Universit´e · Mathematics Institute, University of Warwick

Shallow Neural Networks Gradient Descent Neural Network Training Dynamics Gradient Descent Dynamics

May 11, 2026cs.LG

Simply Stabilizing the Loop via Fully Looped Transformer

Scaling model performance typically requires increasing model size. Looped Transformer offers a compelling alternative by iteratively reusing the same Transformer blocks, trading additional computation for improved performance without increasing parameter count or context length. Because the number of loop iterations can be adjusted at inference, it also provides a natural mechanism for balancing performance and test-time compute. However, Looped Transformer still suffers from training instability when the number of loop iterations increases. Our analysis reveals that this instability stems from two sources: gradient oscillation and residual explosion. To address these two problems, we propose the Fully Looped Transformer, which introduces two parameter-free modifications: (1) Fully Looped Architecture, which distributes inter-loop signals across all layers to mitigate residual explosion; (2) Attention Injection, which reuses the existing attention block to suppress gradient oscillation. These modifications stabilize training dynamics, enabling the Fully Looped Transformer to be trained stably up to 12 loop iterations, whereas other baseline looped models collapse in this regime. In milder settings where Looped Transformer does not collapse, Fully Looped Transformer still improves average downstream-task performance by up to 13.2%. Overall, our experiments demonstrate that Fully Looped Transformer improves training stability, enhances downstream performance, and provides preliminary adaptability under different test-time compute budgets by varying loop iterations at inference.

Rao Fu, Zixuan Yang, Jiankun Zhang +4

Hong Kong Baptist University · Jilin University

Test-Time Scaling Recurrent Transformers Neural Network Training Dynamics

May 11, 2026cs.LG

The two clocks and the innovation window: When and how generative models learn rules

Generative models trained on finite data face a fundamental tension: their score-matching or next-token objective converges to the empirical training distribution rather than the population distribution we seek to learn. Using rule-valid synthetic tasks, we trace this tension across two training timescales:

τ_{\mathrm{rule}}

, the step at which generations first become rule-valid, and

τ_{\mathrm{mem}}

, the step at which models begin reproducing training samples. Focusing on parity and extending to other binary rules and combinatorial puzzles, we characterize how these two clocks,

τ_{\mathrm{rule}}

and

τ_{\mathrm{mem}}

, depend on key aspects of the learning setup. Specifically, we show that

τ_{\mathrm{rule}}

increases with rule complexity and decreases with model capacity, while

τ_{\mathrm{mem}}

is approximately invariant to the rule and scales nearly linearly with dataset size

N

. We define the \emph{innovation window} as the interval

[τ_{\mathrm{rule}}, τ_{\mathrm{mem}}]

. This window widens with increasing

N

and narrows with rule complexity, and may vanish entirely when

τ_{\mathrm{rule}} \geq τ_{\mathrm{mem}}

. The same two-clock structure arises in both diffusion (DiT) and autoregressive (GPT) models, with architecture-dependent offsets. Dissecting the learned score of DiT models reveals a corresponding evolution of the optimization landscapes, where rule-valid samples' basins expand substantially around

τ_{\mathrm{rule}}

, while training samples' basins begin to dominate around

τ_{\mathrm{mem}}

. Together, these results yield a unified and predictive account of when and how generative models exhibit genuine innovation.

Binxu Wang, Emma Lucia Byrnes Finn, Bingbin Liu

Kempner Institute at Harvard University

Memorization in Generative Models Generative Modeling Autoregressive Generation Neural Network Training Dynamics Diffusion Models

May 11, 2026cs.LG

Flag Varieties: A Geometric Framework for Deep Network Alignment

Alignment, the tendency of adjacent weight matrices in deep networks to develop compatible subspace orientations, underlies gradient flow, Neural Collapse, and representation similarity across architectures. Despite extensive empirical documentation, these phenomena have resisted unified theoretical treatment: existing explanations are post-hoc, each fitted to a specific observation with whatever mathematics is at hand. We reverse this direction by deriving the mathematical structure that layerwise alignment inherently demands. Using geometric invariant theory, we prove that alignment geometry has a canonical closed, polystable stratum given by a flag variety, and that subspace intersection dimension is its unique reparameterization-invariant observable, establishing that subspace metrics are not empirical conventions but mathematical necessities. This unified framework yields two dynamical consequences: ridge regularization drives subspace alignment at an exponential rate set by weight decay, whereas nonlinear activations induce a commutator obstruction to exact basis alignment, generically present in nonlinear networks and absent in linear ones. Together these give a geometric explanation of the Level-2/3 hierarchy in Neural Collapse from first principles rather than post-hoc analysis. The commutator magnitude and head subspace overlap further serve as weight-space windows into internal alignment structure, requiring no forward passes. Experiments on multilayer perceptrons, residual networks, and pretrained language models support the proposed diagnostics and delineate their scope.

Jingchuan Xiao, Xinyi Sui, Cihan Ruan

Department of Mathematics and Computer Studies, Mary Immaculate College, Ireland · Department of Computer Science and Engineering, Santa Clara University, USA

Neural Representation Geometry Neural Collapse Neural Network Training Dynamics

May 9, 2026cs.LG

Predicting Plasticity in Deep Continual Learning: A Theoretical Perspective

Deep continual learning requires models to adapt to new tasks without retraining from scratch. However, neural networks can lose their ability to adapt to new tasks after training on previous ones, a phenomenon known as loss of plasticity. There have been several explanations and diagnostics proposed for plasticity loss. Motivated by the philosophy that "all models are wrong, but some are useful", we ask: can existing diagnostics predict a neural network's plasticity? In this work, we take a practical view to interpret plasticity as trainability, i.e., a neural network's future optimization gain on a target task. We first take a theoretical approach, showing, by constructing a few counterexamples, that some widely adopted diagnostics of plasticity, including representation rank and neural tangent kernel rank, can fail to predict the loss of trainability in both regression and classification settings. We instead propose a novel metric, called optimization readiness, which combines gradient strength and gradient reliability. We prove that optimization readiness lower bounds one-step optimization gain under standard smoothness assumptions, providing a theoretical guarantee for its predictive power. Empirically, we show that across commonly used deep continual learning settings, such as Slowly-Changing Regression and Permuted MNIST, optimization readiness more reliably ranks checkpoints by trainability than prior diagnostics, even with substantially fewer samples.

Jiuqi Wang, Jayanth Srinivasa, Claire Chen +3

University of Virginia · Cisco Research · California Institute of Technology +1

Deep Learning Optimization Loss of Plasticity Continual Learning Neural Network Training Dynamics

May 9, 2026cs.LG

Spherical Boltzmann machines: a solvable theory of learning and generation in energy-based models

Energy-based models (EBMs) are flexible generative architectures inspired by statistical physics, but their learning and generative properties remain poorly understood. Here, we analyze a solvable EBM in the high-dimensional limit: the spherical Boltzmann machine (SBM). Combining tools from random matrix theory and dynamical mean-field theory, we: solve exact equations describing the training dynamics of the SBM; compute the Bayesian evidence, which acts as a partition function in parameter space and encodes global properties of the trained model; and uncover cascades of phase transitions that occur both during training and as a function of hyperparameters, related to successive alignment and condensation of the top modes of the coupling matrix to the data. We connect these transitions to sampling-time generative phenomena in a teacher-student scenario, including: sampling temperature tuning, double descent as a function of regularization strength, tempered posterior effects, and out-of-equilibrium effects during training that induce biases in the trained model. We provide numerical evidence demonstrating that all these phenomena appear in standard generative architectures, beyond the SBM.

Thomas Tulinski, Simona Cocco, Rémi Monasson +1

Laboratoire de Physique de l’École Normale Supérieure, PSL, CNRS UMR8023, Sorbonne Université, 24 rue Lhomond, 75005 Paris, France · Institut de Physique Théorique, Université Paris-Saclay,2026 CNRS UMR3681, CEA, Gif-sur-Yvette, France

Statistical Physics of Learning Phase Transitions Energy-Based Models Generative Modeling Neural Network Training Dynamics

May 8, 2026cs.LG

The Propagation Field: A Geometric Substrate Theory of Deep Learning

Modern deep learning treats neural networks primarily as endpoint functions from inputs to outputs. Inspired by the shift from force to geometry in physics, we ask whether a network should instead be understood through the geometry of its internal propagation. We define a neural propagation field as the collection of hidden-state trajectories and local Jacobian operators across depth. Endpoint losses constrain only the boundary behavior of this field, leaving its interior geometry underdetermined. We show that endpoint-equivalent models can differ by orders of magnitude in trajectory and Jacobian structure, and introduce observable field metrics such as path sensitivity, solver consistency, and trajectory/Jacobian retention. In controlled teacher-flow and PDE systems, endpoint fitting fails to recover the underlying propagation law. In real multi-path tasks, field-aware objectives improve unseen-path generalization, OOD robustness, and calibration when aligned with the observation structure, but can collapse when over-constrained. In continual learning, field-preservation regularization complements replay and distillation: on Split CIFAR-100, DER++ with field preservation improves average accuracy, backward transfer, and field-retention metrics. These results identify propagation-field quality as a measurable and trainable property of neural networks beyond endpoint performance.

Xingrui Gu

1EECS, University of California, Berkeley, Berkeley, CA.

Neural Network Generalization Neural Representation Geometry Jacobian Regularization Neural Network Training Dynamics

May 8, 2026cs.LG

Convergence Analysis of Newton's Method for Neural Networks in the Overparameterized Limit

A convergence analysis is developed for the regularized Newton method for training neural networks (NNs) in the overparameterized limit. As the number of hidden units tends to infinity, the NN training dynamics converge in probability to the solution of a deterministic limit equation involving a ``Newton neural tangent kernel'' (NNTK). Explicit rates characterizing this convergence are provided and, in the infinite-width limit, we prove that the NN converges exponentially fast to the target data (i.e., a global minimizer with zero loss). We show that this convergence is uniform across the frequency spectrum, addressing the spectral bias inherent in gradient descent. The eigenvalues of the NTK for gradient descent accumulate at zero, leading to slow convergence for target data with high-frequency components. In contrast, the NNTK has uniformly lower bounded eigenvalues if the regularization parameter is selected appropriately, allowing Newton's method to converge more quickly for data with high-frequency components. Mathematical challenges that need to be addressed in our analysis include the implicit parameter update of the Newton method with a potentially indefinite Hessian matrix and the fact that the dimension of this linear system of equations tends to infinity as the NN width grows. This complicates deriving the training dynamics in the overparameterized limit as well as proving the convergence of the finite-width dynamics thereto. The analysis identifies a scaling formula for selecting the regularization parameter, which we show can vanish at a suitable rate as the number of hidden units becomes larger. We prove that, for sufficiently large numbers of hidden units, the regularized Hessian remains positive definite during training and the Newton updates for individual NN parameters converge to zero, showing that the model behaves as a linearization around the initialization.

Konstantin Riedl, Konstantinos Spiliopoulos, Justin Sirignano

Mathematical Institute University of Oxford · Department of Mathematics & Statistics Boston University

Neural Network Optimization Spectral Bias Second-Order Optimization Neural Network Training Dynamics Neural Tangent Kernel

May 8, 2026cs.LG

Convergent Stochastic Training of Attention and Understanding LoRA

Transformers have revolutionized machine learning and deploying attention layers in the model is increasingly standard across a myriad of applications. Further, for large models, it is common to implement Low Rank Adaptation (LoRA), whereby a factorized parameterization of them is trained, to achieve a surprisingly beneficial accuracy-size trade-off. In this work, via a unified framework we rigorously establish trainability of such models under stochastic methods. We prove that for any mild regularization, the empirical regression loss on a attention layer and LoRA on a shallow neural net, both induce Poincaré inequality for the corresponding Gibbs' measure. Then it follows via invoking recent results that a certain SDE, which mimics the SGD, minimizes the corresponding losses. In both the cases, our first-of-its-kind results of trainability on attention and nets, do not rely on any assumptions on the data or the size of the architecture.

Zhengkai Sun, Dibyakanti Kumar, Alejandro F Frangi +2

Stochastic Optimization Low-Rank Adaptation Transformer Attention Neural Network Training Dynamics

May 8, 2026cond-mat.dis-nn

Spectral Dynamics in Deep Networks: Feature Learning, Outlier Escape, and Learning Rate Transfer

We study the evolution of hidden-weight spectra in wide neural networks trained by (stochastic) gradient descent. We develop a two-level dynamical mean-field theory (DMFT) that jointly tracks bulk and outlier spectral dynamics for spiked ensembles whose spike directions remain statistically dependent on the random bulk. We apply this framework to two settings: (1) infinite-width nonlinear networks in mean-field/

μ

P scaling and (2) deep linear networks in the proportional high-dimensional limit, where width, input dimension, and sample size diverge with fixed ratios. Our theory predicts how outliers evolve with training time, width, output scale, and initialization variance. In deep linear networks,

μ

P yields width-consistent outlier dynamics and hyperparameter transfer, including width-stable growth of the leading NTK mode toward the edge of stability (EoS). In contrast, NTK parameterization exhibits strongly width-dependent outlier dynamics, despite converging to a stable large-width limit. We show that this bulk+outlier picture is descriptive of simple tasks with small output channels, but that tasks involving large numbers of outputs (ImageNet classification or GPT language modeling) are better described by a restructuring of the spectral bulk. We develop a toy model with extensive output channels that recapitulates this phenomenon and show that edge of the spectrum still converges for sufficiently wide networks.

Clarissa Lauditi, Cengiz Pehlevan, Blake Bordelon

John A. Paulson School of Engineering and Applied Sciences, Harvard University · Kempner Institute for the Study of Natural and Artificial Intelligence, Harvard University · Center for Mathematical Sciences and Applications, Harvard University +1

Deep Linear Networks Neural Network Training Dynamics Neural Tangent Kernel

May 8, 2026cs.LG

Training-Induced Escape from Token Clustering in a Mean-Field Formulation of Transformers

Transformers perform inference by iteratively transforming token representations across layers. This layerwise computation has been studied empirically, and recent mean-field theories of Transformer dynamics explain how attention can drive token distributions toward clustering. However, existing mean-field analyses largely treat model parameters as prescribed, leaving open how training reshapes this clustering picture. We study this question in a noisy mean-field Transformer in which only a parameter-linear FFN is trained under

L^2

regularization. We find and analyze a training-induced phase in the dynamics: after initially following attention-driven clustering, the token distribution can leave the clustered regime near the final layers. Our mathematical analysis is based on an entropy-regularized interaction energy that captures the clustering bias of attention. More broadly, our results point toward a training-aware mean-field theory of Transformer dynamics, in which training and inference dynamics are treated together.

Noboru Isobe, Daisuke Inoue, Masaaki Imaizumi

1RIKEN Center for Advanced Intelligence Project · 2Imperial College London · 3The University of Tokyo +1

Transformer Clustering Transformer Attention Mean-Field Theory Neural Network Training Dynamics

May 8, 2026cs.LG

Bilevel Graph Structure Learning, Revisited: Inner-Channel Origins of the Reported Gain

Bilevel graph structure learning is widely understood to improve graph neural networks by jointly optimizing model parameters and a learned graph structure, with the resulting performance gain attributed to the rewired adjacency. We find that this attribution may be overstated: training-dynamics effects in the inner loop, rather than the rewiring itself, capture a substantial share of the gain. To establish this, we introduce frozen-

φ

, a control that freezes the graph while retaining the inner-loop training schedule. This decomposes the bilevel gain into an inner channel of

T

-step training dynamics with implicit gradient regularization and a graph channel of the graph rewiring itself. On spatio-temporal flow forecasting the inner channel matches or exceeds the full bilevel pipeline, accounting for 78-101% of the gain; on node classification it accounts for 37-44% under a Bernoulli edge-level parameterization. We also verify that classical spectral diagnostics can dissociate from task gain. We propose frozen-

φ

as a standardized diagnostic for bilevel graph structure learning, with graph distillation as a method-agnostic complement. A three-precondition framework further predicts the sign of the bilevel gain on all six benchmarks.

Minkyoung Kim, Beakcheol Jang

Graduate School of Information, Yonsei University

Graph Structure Learning Graph Neural Networks Bilevel Optimization Neural Network Training Dynamics

May 7, 2026cs.LG

On the Implicit Reward Overfitting and the Low-rank Dynamics in RLVR

Recent extensive research has demonstrated that the enhanced reasoning capabilities acquired by models through Reinforcement Learning with Verifiable Rewards (RLVR) are primarily concentrated within the rank-1 components. Predicated on this observation, we employed Periodic Rank-1 Substitution and identified a counterintuitive phenomenon: RLVR may exhibit implicit reward overfitting to the training dataset. Specifically, the model can achieve satisfactory performance on the test set even when its rewards remain relatively low during the training process. Furthermore, we characterize three distinct properties of RL training: (1) The effective rank-1 component in RLVR don't maintain other model knowledge except mathematical reasoning capability. (2) RLVR fundamentally functions by optimizing a specific singular spectrum. The distribution of singular values of almost all linear layers in RLVR-trained model behaves like heavy-tailed distribution. (3) the left singular vectors associated with rank-1 components demonstrate a stronger alignment tendency during training, which echoes the discovery that RLVR is optimizing sampling efficiency in essence. Taken together, our findings and analysis further reveal how RLVR shapes model parameters and offer potential insights for improving existing RL paradigms or other training paradigms to implement continual learning.

Hao Ye, Jisheng Dang, Junfeng Fang +7

Lanzhou University · National University of Singapore · Beijing Institute of Technology

Reinforcement Learning Reinforcement Learning with Verifiable Rewards Neural Network Training Dynamics

May 7, 2026cs.LG

MinMax Recurrent Neural Cascades

We introduce MinMax Recurrent Neural Cascades (MinMax RNCs), a class of recurrent neural networks built from a novel form of recurrence over the MinMax algebra. We show that MinMax RNCs enjoy key properties that are difficult to obtain simultaneously: strong formal expressivity, efficient evaluation, stable dynamics, and non-vanishing state gradients. First, their formal expressivity corresponds to the regular languages, arguably the maximal expressivity for finite-memory systems. Second, in addition to evaluation in recurrent form, they also admit parallel-scan evaluation with logarithmic depth and linear work in the input length. Third, their states and activations are uniformly bounded for all sequence lengths. Fourth, their loss gradients exist almost everywhere and are uniformly bounded for all sequence lengths. Fifth, they do not exhibit vanishing state gradients: the gradient of a state with respect to a past state can retain norm one independently of the temporal distance between the states. Empirically, we find that these theoretical properties translate into strong practical performance. MinMax RNCs solve the considered synthetic tasks perfectly, generalise to long sequences, and outperform the recurrent baselines considered in our experiments. We also train a 112M-parameter MinMax RNC for next-token prediction, obtaining competitive performance for its size and providing initial evidence that MinMax recurrence can scale to real-world sequence-modelling tasks.

Alessandro Ronca

IRIS-AI

Recurrent Neural Networks Neural Network Training Dynamics

May 7, 2026cs.LG

Region Seeding via Pre-Activation Regularization: A Geometric View of Piecewise Affine Neural Networks

Deep networks with continuous piecewise affine activations induce polyhedral partitions of the input space, making the number of realized affine regions a natural measure of expressive capacity and a key determinant of how well the model can approximate nonlinear target functions. In practice, standard training realizes far fewer region refinements in data-visited neighborhoods than the architecture could in principle support, while existing region-count theory is primarily architectural and offers little guidance on how optimization shapes the realized partition near the data. Our theory provides a sufficient condition under which bringing neuron switching surfaces sufficiently close to data points ensures their intersection with local neighborhoods, which in turn implies a strict increase in the local affine-region count, yielding a principled training-time handle for seeding data-relevant partitions early in optimization. Guided by these results, we propose a plug-and-play region-seeding regularizer that encourages early partitioning while allowing task-driven refinement to dominate later in training. Experiments show that the regularizer increases the number of realized affine regions via exact enumeration and improves overall performance on toy datasets, while also improving early-stage accuracy and achieving comparable (or slightly improved) final accuracy on ImageNet-1k for classical models.

Yi Wei, Xuan Qi, Furao Shen

DITEN University of Genoa, Genoa, Italy · State Key Laboratory of Novel Software Technology School of Artificial Intelligence Nanjing University, Jiangsu, China

Neural Network Activation Functions Neural Network Approximation Theory Neural Network Training Dynamics Neural Network Expressivity

May 7, 2026cs.LG

The Weight Gram Matrix Captures Sequential Feature Linearization in Deep Networks

Understanding how deep neural networks learn representations remains a central challenge in machine learning theory. In this work, we propose a feature-centric framework for analyzing neural network training by relating weight updates to feature evolution. We introduce a simple identity, the Feature Learning Equation, which identifies the weight Gram matrix as the key object capturing feature dynamics. This enables us to interpret gradient descent as implicitly inducing a hypothetical evolution of features, whose covariance structure - termed the Virtual Covariance - characterizes how representations evolve during training. Building on this perspective, we introduce Target Linearity, a measure quantifying the linear alignment between features and targets. By analyzing the training and layer-wise dynamics, we show that deep networks learn to sequentially transform representations toward target-linear structure. This linearization perspective provides a unified interpretation of several empirical phenomena, including Neural Collapse and linear interpolation in generative models.

Taehun Cha, Daniel Beaglehole, Adityanarayanan Radhakrishnan +1

Department of Mathematics Korea University · Computer Science and Engineering UC San Diego · MIT Mathematics Broad Institute of MIT and Harvard

Neural Network Training Dynamics Gradient Descent Dynamics

May 7, 2026cs.LG

Cumulative-Goodness Free-Riding in Forward-Forward Networks: Real, Repairable, but Not Accuracy-Dominant

Forward-Forward (FF) training lets each layer learn from a local goodness criterion. In cumulative-goodness variants, later layers can inherit a task that earlier layers have partly separated. We formalize this as layer free-riding: under the softplus FF criterion, the class-discrimination gradient reaching block

d

decays exponentially with the positive margin accumulated by earlier blocks (Theorem 3.1); a squared-hinge barrier reproduces the pathology. History-free and hardness-gated repairs raise deeper-layer separation by up to

5\times

on CIFAR-100 and

45\times

in an 8-block CIFAR-10 model, yet these and a depth-scaled auxiliary term move accuracy by under one percentage point among non-degenerate variants; on Tiny ImageNet, a harder cross-dataset check of the selected configuration, layer health likewise does not track accuracy. A theory-derived attenuation-compensated objective raises deepest-block separation

5.9\times

at a consistent cost in the FF classifier's own accuracy that a probe trained on the frozen features recovers; in a smaller CIFAR-10 control, most of its separation gain comes from dropping the depth-scaled auxiliary term it replaces. Accuracy invariance is a property of the deployed sum-based scoring rule under redistribution (Proposition 3.2) and, empirically, of the structured trained readouts we test, not of what the network learns: retrained variants disagree on 13-26% of test predictions. Architecture and augmentation matter far more than the non-degenerate training rules studied. Cumulative free-riding is therefore a real and repairable pathology, but for the softplus and squared-hinge objectives, architectures, and datasets we study, it is not the dominant factor limiting accuracy.

Amirhossein Yousefiramandi

Neural Network Training Dynamics Local Learning Forward-Forward Learning

May 7, 2026cs.LG

Grokking or Glitching? How Low-Precision Drives Slingshot Loss Spikes

Deep neural networks exhibit periodic loss spikes during unregularized long-term training, a phenomenon known as the "Slingshot Mechanism." Existing work usually attributes this to intrinsic optimization dynamics, but its triggering mechanism remains unclear. This paper proves that this phenomenon is a result of floating-point arithmetic precision limits. As training enters a high-confidence stage, the difference between the correct-class logit and the other logits may exceed the absorption-error threshold. Then during backpropagation, the gradient of the correct class is rounded exactly to zero, while the gradients of the incorrect classes remain nonzero. This breaks the zero-sum constraint of gradients across classes and introduces a systematic drift in the parameter update of the classifier layer. We prove that this drift forms a positive feedback loop with the feature, causing the global classifier mean and the global feature mean to grow exponentially. We call this mechanism Numerical Feature Inflation (NFI). This mechanism explains the rapid norm growth before a Slingshot spike, the subsequent reappearance of gradients, and the resulting loss spike. We further show that NFI is not equivalent to an observed loss spike: in more practical tasks, partial absorption may not produce visible spikes, but it can still break the zero-sum constraint and drive rapid growth of parameter norms. Our results reinterpret Slingshot as a numerical dynamic of finite-precision training, and provide a testable explanation for abnormal parameter growth and logit divergence in late-stage training.

Hanqing Liu, Jianjun Cao, Yuanze Li +1

Tsinghua University · The University of Tokyo

Neural Network Training Dynamics

May 7, 2026cs.CL

Navigating by Old Maps: The Pitfalls of Static Mechanistic Localization in LLM Post-Training

The "Locate-then-Update" paradigm has become a predominant approach in the post-training of large language models (LLMs), identifying critical components via mechanistic interpretability for targeted parameter updates. However, this paradigm rests on a fundamental yet unverified assumption: can mechanisms derived from current static parameters reliably guide future dynamic parameter updates? To investigate this, we systematically track the structural evolution of Transformer circuits throughout the supervised fine-tuning (SFT) process, revealing the underlying dynamics of task mechanisms. We introduce three novel metrics-Circuit Distance, Circuit Stability, and Circuit Conflict-to analyze circuit evolution across three dimensions: neural migration, semantic stability, and cross-task interference. Our empirical results reveal that circuits inherently exhibit "Free Evolution" during parameter updates. Consequently, static mechanisms extracted from current states inevitably suffer from temporal latency, making them fundamentally inadequate for guiding future states. Moreover, by deconstructing the "illusion of effectiveness" in existing methods, this work underscores the necessity of "foresight" in mechanistic localization and proposes a predictive framework for future research.

Hang Chen, Jiaying Zhu, Hongyang Chen +3

School of Computer Science and Technology Xi’an Jiaotong University · School of Computer Science and Engineering The Chinese University of Hong Kong · Shaanxi Co., Ltd(Xi’an 710077, China) China Mobile Group +1

Supervised Fine-Tuning Transformer Interpretability LLM Interpretability Mechanistic Interpretability Circuits in Language Models LLM Post-Training+1