Neural Network Optimization

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Neural network optimization asks how to adjust a model’s parameters so it learns useful patterns efficiently and reliably. Common approaches include stochastic gradient descent and Adam, matrix-aware and spectral optimizers such as Muon, curvature-based methods, learning-rate schedules, and techniques that reduce memory or computation. Researchers also analyze loss landscapes, scaling laws, and initialization, and explore alternatives such as local or forward-only learning and analytic solutions for restricted models. A central tension is that gains depend on architecture, data, and training budget, while faster loss reduction does not necessarily improve generalization; connecting practical performance to reliable theory remains an open challenge.

Latest papers

332 papers37 papers in the last four weeks, up 28% on the four weeks before. 0.2% of all new papers.

Sep 9, 2026

Muon-C: Operator-Aligned Muon for Convolutional Kernels

Muon replaces matrix momentum with an approximately orthogonal polar direction, but its geometry depends on the matrix representation. For convolution, standard unfolding describes a local patch map rather than the convolution operator. We introduce Muon-C, an operator-aligned optimizer that represents kernel momentum as frequency-wise channel-transfer matrices, polarizes these blocks independently, and uses a critical Fourier grid to return updates exactly to the original finite kernel support. We show that the new geometry arises from combining the block partition and Fourier coordinates. The exact-polar direction is a linear minimization oracle under the critically sampled convolution norm. Its worst-case guarantee relative to the continuous convolution-operator norm is never weaker than unfolding and is strictly stronger for

3\times3

kernels. On CIFAR-10 flow matching with matched applied-update RMS, Muon-C reaches 9.87 FID at 40k iterations, compared with 22.26 for unfolded Muon and 51.31 for Adam. It reaches their final quality using

0.62\times

and

0.64\times

their model FLOPs, respectively. Under equal tuning budgets, Muon-C achieves 3.42 FID. Gains persist across data scales and transfer to classification across convolutional architectures.

Jiaxin Qing, Lexin Li

University of California, Berkeley Berkeley, CA 94720-1776, USA

Neural Network Optimization Muon Optimizer

Sep 9, 2026

Teacher Geometry Shapes Learnability in Teacher-Student Networks

Teacher-student systems, in which a teacher neural network generates training labels so that a student neural network can learn to implement the same function, are widely used as an abstract setting to study learning. However, the structure of the teachers is often overlooked by assuming randomly-generated, normally-distributed parameters. This hides substantial variation in how learnable different teachers are. We formalize learnability as the success rate of converging to the global minimum, as a function of overparameterization, learning algorithm, student initialization distribution, and teacher geometry. We both identify an easy distribution that maximizes node dissimilarity and a hard distribution that minimizes it, and show that these two distributions induce markedly different success rates across a large range of settings and for different activation functions. To explain the gap, we study the loss landscape of small neural networks that contain two distinct kinds of suboptimal local minima, out-of-bounds (OOB) minima at the edge of the data distribution and interior minima within. Assuming infinite data and a fast readout layer, we analytically reduce the loss landscape of small networks to two dimensions, showing that the region of attraction of interior minima changes as a function of teacher structure. In larger networks, maximally dissimilar teachers induce more interior minima, while minimally dissimilar teachers induce more OOB minima. Motivated by these analyses, we show that differentially increasing the learning rate of the readout layer and decreasing the learning rate of the inner biases increases success rates. These findings provide an important step in narrowing the gap between the study of teacher-student networks and more structured functions that arise in practice.

Kai J. Sandbrink, Flavio Martinelli, Alexander van Meegen +2

Department of Experimental Psychology, University of Oxford · Department of Physics, RWTH Aachen

Neural Network Optimization Neural Network Training Dynamics Teacher-Student Learning Loss Landscape Geometry

Sep 8, 2026

Good Pretraining, Bad SFT: Checkpoint Quality Across the Training Stack

Language-model checkpoints are commonly selected by pretraining loss or benchmark scores, assuming that the highest-scoring checkpoint will remain the best starting point for subsequent training. We show that this assumption can fail in a full 30B mixture-of-experts training pipeline. The checkpoints that perform better after the full downstream training stack also have higher solution density, i.e., retain downstream performance under local weight perturbations.

Sohir Maskey, Philipp Scholl, Jonas Knupp +2

Aleph Alpha

Language Model Pretraining Neural Network Optimization Language Model Post-Training

Sep 7, 2026

HyperTransfer: Understanding the Equivalence between Base Optimizer and Hyperball

Hyperball optimizers constrain parameter norms and update only their directions, establishing a distinct paradigm for neural network optimization. Although this geometry appears fundamentally different from that of conventional Base Optimizers, which update both parameter norms and directions, we show that the two paradigms are dynamically equivalent for scale-invariant networks. Building on this equivalence, we propose HyperTransfer, which constructs a Hyperball optimizer that reproduces the dynamics of a target Base Optimizer using only its initialization and learning-rate schedule, without running the target optimizer itself. We further derive the inverse mapping and extend the framework to non-scale-invariant networks. Experiments show that both HyperTransfer and the inverse mapping produce loss trajectories nearly identical to those of their targets, suggesting that Hyperball dynamics are governed primarily by the induced effective learning-rate schedule and optimizer state.

Jinghui Yuan, Hongtao Zhang, Jade Zou +4

StepFun · University of Chinese Academy of Sciences, StepFun · University of Chinese Academy of Sciences +2

Neural Network Optimization Neural Network Training Dynamics Gradient Descent Dynamics

Sep 4, 2026

Optimizer Memory Schedules for Outscaling the Overtraining Axis

We investigate how optimizers scale across the overtraining axis and show that relative optimizer performance and optimal hyperparameters change substantially with training horizon. In particular, we study how matrix-preconditioned methods (Muon and SOAP) and a momentum-scheduled method (ADANA) scale relative to AdamW. We compare these four optimizers across models from 51M to 253M parameters and overtraining (OT) factors from 1x to 256x, sweeping the base learning rate at every setting. The preferred learning rate schedule can reverse across the overtraining axis, the best weight decay coefficient scales approximately as sqrt(OT), and longer horizons generally favor longer fixed memory. ADANA's scaling advantage over AdamW persists after tuning AdamW's fixed memory separately at each horizon. Log-time weight decay and momentum cooldown provide substantial gains for ADANA that compound as training increases. With this treatment, ADANA outscales AdamW with an exponent advantage close to that predicted by DANA theory on power-law random features. Muon and SOAP instead provide roughly constant token-efficiency advantages over AdamW across most of the measured range, although SOAP may gain further at the highest overtraining factors. ADANA begins behind both matrix-preconditioned optimizers but closes the gaps as training increases, surpassing Muon and becoming competitive with SOAP at our highest OT factors. These results establish training horizon as an essential axis for optimizer evaluation and design.

Katie Everett, Shikai Qiu

MIT CSAIL

Neural Network Optimization Muon Optimizer Learning Rate Scheduling

Sep 1, 2026

Gradient-Update Mismatch: Rethinking Conflict-Free Training of Physics-Informed Neural Networks

Training Physics-Informed Neural Networks (PINNs) requires jointly optimizing physics residual and initial/boundary condition loss terms, which often induce conflicting gradients. Gradient surgery methods mitigate this issue by constructing directions from loss-specific gradients to reduce conflict before optimizer transformation. However, even when the constructed direction is conflict-free, this property may not be preserved after optimizer transformation. Let

a_t

denote the direction constructed by gradient surgery,

u_t

the optimizer proposal, and

\mathcal{C}_t

the conflict-free cone induced by the loss-specific gradients. We show that modern optimizers can transform

a_t

through mechanisms such as historical state, adaptive scaling, preconditioning, or decoupled weight decay, so

a_t \in \mathcal{C}_t

does not generally imply

u_t \in \mathcal{C}_t

. We refer to this optimizer-induced discrepancy in conflict-freeness between

a_t

and

u_t

as Gradient-Update Mismatch (GUM). Accordingly, we propose Gradient-Update Alignment (GUA), which projects

u_t

onto

\mathcal{C}_t

to obtain the aligned update

p_t

and applies

p_t

to the parameters. When the optimizer maintains internal state, GUA further adjusts this state toward targets reconstructed from the applied update. We conduct extensive experiments and find that GUM is widespread across momentum, adaptive, and curvature-based optimizers, with conflict rates reaching up to 86.3%. Across all PINN settings, GUA achieves conflict-free applied updates and consistently improves various gradient surgery methods, reducing the relative

L_2

error by up to 98.2% in individual settings. Data and code are available at https://github.com/JingXiao10/GUA.

Jing Xiao, Xinhai Chen, Qinglin Wang +5

Laboratory of Digitizing Software for Frontier Equipment, National University of Defense Technology, Changsha 410073, China · National Key Laboratory of Parallel and Distributed Computing, National University of Defense Technology, Changsha 410073, China · School of Computer Science and Technology, National University of Defense Technology, Changsha 410073, China

Neural Network Optimization Gradient Interference Gradient Projection Physics-Informed ML

Sep 1, 2026

Subspace Levenberg Marquardt Algorithms in Training Neural Networks

The Levenberg-Marquardt (LM) algorithm is a well-known second-order method for rapid convergence and strong robustness when training small- to medium-sized neural networks (NNs). However, its computational and memory costs increase significantly as the number of parameters in an NN grows. To address this limitation, subspace methods have been proposed, such as the Krylov subspace LM (KSLM) and the hybrid subspace LM (HSLM), making second-order algorithms more efficient. In this work, we evaluate the subspace Levenberg-Marquardt algorithms for regression and classification tasks in neural networks. We compare the performance of subspace LM variants with the classical LM method, as well as other popular first-order algorithms, such as stochastic gradient descent (SGD) and Adam.

M. Duc Hoang

Department of Mathematics, University of California, Davis, Davis, CA, USA

Neural Network Optimization Krylov Subspace Methods

Sep 1, 2026

A Study of Hidden-State Optimization Order in Predictive Coding Networks

Local learning methods offer an alternative to end-to-end backpropagation, but their unstructured local objectives can produce weak feature learning in deep networks. We study whether the order of hidden-state optimization can address this limitation. We propose a boundary-first inference schedule that partitions a model into chunks, first coordinates hidden states at chunk boundaries, and then refines representations within each chunk. We instantiate this schedule in predictive coding networks (PCNs), a local-learning framework in which hidden activities and prediction errors are explicitly exposed during inference. On CIFAR-10, the resulting boundary-first predictive-coding instantiation improves accuracy over standard predictive coding by

9.77\%

under a standard parametrization and by

5.51\%

under a

μ

-parametrization. Diagnostic analyses further show more non-trivial early-layer updates, lower initial-to-final CKA, and more diverse layerwise gradients, consistent with stronger feature learning. These results support boundary-first, chunk-based inference as a practical design principle for predictive-coding training and motivate its study in broader local-learning systems.

Xueyuan Li, Danilo Vasconcellos Vargas

Predictive Coding Neural Network Optimization Local Learning

Aug 29, 2026

SS-ESOAP: Self-Scaled Adaptive Preconditioning for Physics-Informed Learning

Physics-informed neural networks (PINNs) often face ill-conditioned objectives that limit high-accuracy training. Dense quasi-Newton methods improve local conditioning but require expensive optimizer state, while Kronecker-factored methods such as SOAP scale to larger networks but rely on periodic basis updates. We introduce \method, which augments SOAP-style preconditioning with a scalar secant-energy correction adapted to Kronecker geometry and an adaptive basis update followed by variance-state downscaling. We characterize the directional secant matching induced by the scalar correction and give a bound on variance-state mismatch across basis changes. Across eight PDE benchmarks, \method attains the lowest final residual on six, including Burgers and Boussinesq, while SOAP-family baselines perform better on Gray-Scott and Ginzburg-Landau. On Boussinesq, \method reaches a residual of

10^{-5}

in 4.1 hours with 9.2 GB peak VRAM, while Adam does not reach this target within 14 hours. Three-seed

L^2

and

H^1

errors on four representative PDEs support the link between lower residuals and improved solution accuracy. These results position \method as a scalable option for stiff, high-accuracy physics-informed training, rather than a uniform replacement for existing optimizers.

Guangyuan Wang, Mads Toftrup, Sebastian Loeschcke +2

McGill University · California Institute of Technology · Aarhus University +1

Neural Network Optimization Physics-Informed ML Adaptive Gradient Methods

Aug 28, 2026

From the Loss Landscape to Diverse Feature Learning in Neural Networks

Over the course of the last decade, neural networks have grown from an academic curiosity to moving the markets of nations. Despite this explosion in both research and deployment, relatively little is understood about how they achieve the solutions they do. This is both scientifically relevant, and pressing for society. When neural networks make decisions across self-driving, construction, law, hiring and health, there have been and will continue to be unintended consequences. However, attempting to generalize the failures of the largest and most important production systems makes for a very difficult task. Yet signs of these failures exist at all scales of neural networks, so we should be able to study a much more tractable setting. All neural networks must undergo an optimization process, called training, to be useful. To a great degree, understanding neural networks is understanding their optimization: through what process and exposure to which data did they arrive at their results. Yet our knowledge on this topic as a field is quite imprecise. In particular, a curious phenomenon called mode connectivity, the ability to connect neural networks in the loss surface, defies explanation entirely. This dissertation elucidates, explains and exploits this special structure in the loss landscape...

David Aram Yunis

TOYOTA TECHNOLOGICAL INSTITUTE AT CHICAGO Chicago, Illinois

Loss Landscape Mode Connectivity Neural Network Optimization Neural Network Training Dynamics Representation Learning

Aug 28, 2026

Blog: Survey of Optimizers

Neural-network optimization in 2025-2026 is no longer well described as a succession of new Adam variants. The design space has expanded from coordinates to matrices and layers, from fixed training horizons to policies over time, and from mathematical update rules to state representations that must survive sharding and low-precision computation. This survey organizes recent optimizers and training optimization methods along four largely independent axes: temporal estimation, update geometry, horizon management, and representation and systems. It connects the spectral normalization of Muon, the historical matrix statistics of Shampoo and SOAP, adaptive and hybrid matrix methods, memory-efficient optimizers, schedule-free training, small-batch corrections, and quantized optimizer states. The central empirical conclusion is deliberately non-triumphal: matrix-aware methods represent a genuine advance, but there is no context-independent replacement for AdamW. Rankings change with model scale, data-to-parameter ratio, batch size, schedule, parameter partition, tuning budget, and whether the target metric is tokens, FLOPs, wall-clock time, or memory. The practical consequence is a compositional view of optimizer design and a stricter protocol for evaluating optimizer claims.

Ruoran Xu

Neural Network Optimization Memory-Efficient Optimization

Aug 28, 2026

Residual-Guided Randomized Neural Networks

Randomized neural networks enable fast and analytically tractable training by fixing the input to hidden layer parameters at random and learning the output weights in closed form; however, their performance critically depends on a single uninformed draw of hidden units. This one shot and task uninformed feature construction often leads to redundant representations and suboptimal utilization of model capacity. To address this limitation, we propose a simple and broadly applicable residual guided procedure that greedily constructs the hidden layer using a closed form residual decrease criterion. At each stage, we (i) generate a pool of random candidate units, (ii) score each candidate by the exact reduction it induces in the ridge regularized objective, (iii) select the top k units, and (iv) refit the readout in closed form using the standard design with direct input links. This procedure yields a progressive training process with a guaranteed monotonic decrease of the training objective. The method is model agnostic: only the candidate generation is architecture specific, while the scoring selection refitting loop is shared across models. Extensive experiments on 71 benchmark datasets from the UCI repository, covering both binary and multiclass classification tasks, demonstrate that the proposed residual-guided models consistently outperform their baseline counterparts in terms of accuracy, stability, and overall ranking performance.

Mushir Akhtar, M. Tanveer, Mohd. Arshad

Department of Mathematics Indian Institute of Technology Indore

Neural Network Optimization

Aug 27, 2026

Physics-Informed Stochastic Configuration Machine: A Backpropagation-Free Neural Network with Fast Training for Nonlinear Differential Equations

While Physics-Informed Neural Networks (PINNs) have emerged as a transformative paradigm for solving complex differential equations, their reliance on backpropagation-based gradient descent and automatic differentiation (AD) imposes significant computational bottlenecks and severe non-convex optimization challenges. To overcome these fundamental limitations, we propose the Physics-Informed Stochastic Configuration Machine (PI-SCM), a novel backpropagation-free framework for both forward and inverse problems in differential equations. The core mathematical contribution lies in the analytical evaluation of local Jacobians for nonlinear differential operators, which facilitates a linearized representation of the physical loss and projects it into a unified, linearized algebraic subspace. This reformulation allows for the explicit determination of optimal network weights via a sequence of generalized linear least squares solvers, effectively bypassing the iterative traps of traditional nonlinear optimizers. We develop a progressive algorithmic suite comprising localized construction (PI-SC-I), sliding-window updating (PI-SC-II), and global updating (PI-SC-III), and rigorously establish their universal approximation properties. Extensive experiments demonstrate that PI-SCM achieves high-fidelity predictive accuracy and robust parameter identification while accelerating the training process by orders of magnitude compared to standard PINNs. Our work provides a highly efficient and scalable foundation for next-generation, real-time Scientific Machine Learning applications.

Yuehao Song, Zhong Chen, Lihui Cen +2

School of Automation, Central South University, Changsha, China · School of Automation, Central South University, 932 South Lushan Road, Changsha 410083, China · Johns Hopkins University, Baltimore, USA +3

Physics-Informed ML Neural Network Optimization PDE Inverse Problems

Aug 25, 2026

Delayed Optimizer-State Transport Shapes Short-Horizon Training Decisions

Adaptive optimizers retain gradient history in moment variables, allowing a local change in loss weighting to alter later updates. We examine whether this delayed transport is large enough to change prospective short-horizon decisions. On committed future-minibatch sequences, we differentiate eight-step AdamW trajectories through the complete model--optimizer state and select exposure-matched Math--Code loss schedules before independent evaluation. Across 12 unused 0.3M Transformer histories, full transport lowers token-disjoint loss relative to an optimizer-aware immediate derivative in 10/12 histories (mean benefit

4.71\times10^{-4}

; exact one-sided sign test,

p=0.0193

). The two controllers act equally often but select different schedules in 60/96 windows. Crossed checkpoint--future-path tests attribute this reordering to the interaction between optimizer state and near-future data, while an independent Ising--CNN experiment shows that deleting moment-state transport destroys accurate response prediction. Full-transport scores also concentrate exact-rollout winners in larger candidate libraries, focusing finite-amplitude evaluation on a shortlist. On these committed short paths, optimizer memory and near-future data order are therefore actionable components of the training state, providing a mechanism-based criterion for when finite-horizon rather than one-step intervention is required.

Jinhui Guo

School of Physics, Beihang University, Beijing 100191, China

Neural Network Optimization Neural Network Training Dynamics Learning to Optimize Adaptive Loss Weighting

Aug 25, 2026

Physics-Informed Error Field Learning: A Post-Training Optimization Framework for Physics-Informed Neural Networks

Physics-Informed Neural Networks (PINNs) have emerged as an important class of numerical methods for solving partial differential equations (PDEs). However, during the late-stage optimization process, further parameter updates often yield diminishing accuracy improvements while increasing computational costs. To address this issue, this paper proposes a Physics-Informed Error Field Learning (PIEFL) framework for PINNs. Unlike conventional approaches that continuously approximate the solution field using a single network, PIEFL introduces an auxiliary error network after the primary network achieves satisfactory accuracy and shifts the learning objective from the solution field to the error field. By deriving error control equations under physical constraints, the error network learns the discrepancy between the current approximation and the exact solution, and the learned error correction is combined with the primary prediction to improve solution accuracy. The proposed framework avoids continuous optimization of the entire solution space and focuses computational resources on correcting existing prediction errors. Moreover, PIEFL requires no modification to the primary network architecture, making it compatible with existing PINN models and applicable as a general post-training optimization strategy. Numerical experiments on representative PDEs demonstrate that PIEFL achieves higher solution accuracy under the same computational budget, validating its effectiveness in improving the performance of PINNs.

Jiuyun Sun, Yong Zhang

College of Mathematics and Systems Science, Shandong University of Science and Technology, Qianwangang Road 579, Qingdao, 266590, Shandong, China

Neural PDE Solvers Physics-Informed ML Neural Network Optimization

Aug 24, 2026

A Physical Response-and-Memory Model for Muon Optimization

Training large language models is costly. How low a loss the same compute can ultimately reach depends on how each step's gradient is converted into a weight update; the rule that performs this conversion is the optimizer. From SGD and AdamW to the recent Muon, effective update rules have mostly been shaped by engineering intuition and then selected on benchmarks. Muon semi-orthogonalizes the momentum matrix before applying the update and has kept breaking records on public training benchmarks; yet why the semi-orthogonalized direction works, and over how long a history the momentum should average, are two questions at present answered mainly by experience. Here we treat the weight matrix during training as a responsive medium with memory and build a physical model for it, in which both questions find answers: the semi-orthogonalized direction is the maximally dissipative response under an output-side safety budget, which explains why it works; momentum is the internal stress accumulated by the medium; how long it should average is set by the relaxation of this stress, and a real medium relaxes on more than one timescale, the simplest form being one fast and one slow. On this basis we propose the Bi-Maxwell optimizer. The framework further yields a testable consequence: gradient directions change fast early in training and more slowly later, so the optimal memory length should grow with training stage; step-by-step measurements of a proxy for it by a read-only probe across 8 independent training trajectories are consistent with this consequence. Replacing the memory kernel alone, from a single timescale to two, brings training to the target loss in noticeably fewer steps on a public large-language-model optimizer benchmark.

Yinze Hu, Hongjun Xiang, Xingao Gong +1

Key Laboratory of Computational Physical Sciences (Ministry of Education), Institute of Computational Physical Sciences, State Key Laboratory of Surface Physics, and Department of Physics, Fudan University, Shanghai 200433, China · School of Physics Science and Engineering, Tongji University, Shanghai 200092, China

Neural Network Optimization Muon Optimizer Neural Network Training Dynamics LLM Training

Aug 24, 2026

A Commutator Framework for Selective Spectral Alignment in Deep Neural Networks

We develop a finite-width geometric framework describing how learned feature geometries are organized, transported, and selectively aligned in deep neural networks. Incompatibility among weight-generated covariance, gates, and backward sensitivities is quantified through three families of commutators: between gates and covariance, between sensitivities and covariance, and between average gradient outer products (AGOPs) and neural feature matrices (NFMs). An exact layerwise identity decomposes the sensitivity-covariance commutator into four sources: downstream transport, adjacent-layer imbalance, pointwise sensitivity fluctuations, and nonlinear gate-covariance interactions. The AGOP-NFM commutator is a singular-value-weighted transport of the internal commutator, explaining why observed feature-side alignment alone does not determine the internal geometry from which it emerges. Buffered localized energies resolve mixing between separated covariance subspaces. We establish spectral-gap, projector-evolution, and stabilization estimates, and formulate conditional Lyapunov principles that yield decay under explicit geometric error-bound or intrinsic-damping assumptions. These criteria do not follow from gradient flow alone and clarify why risk reduction need not imply commutator collapse. Analytic examples and numerical experiments exhibit factorization of spectral and activation geometry, transient growth, and cancellation among nonzero sources. In tested finite-time regimes, cancellation dominated by a negative transport-imbalance interaction persists across depths, widths, and two regression benchmarks. Spectral alignment therefore appears as a layer- and scale-dependent compatibility phenomenon governed by transport, interaction, cancellation, and possible damping, rather than a universal consequence of training.

Kaj Nyström

Neural Representation Geometry Neural Network Training Dynamics Neural Network Optimization

Aug 22, 2026

Blockwise Stabilized Adaptive Cubic Regularization with Subsolvers via Recurrence

Overparameterized networks often have ill-conditioned, saddle rich loss landscapes. They are usually trained with first-order methods that struggle on those landscapes, while many second-order methods stay infeasible at scale. Some scalable cubic methods replace the Hessian by a computationally cheap approximation and drop the exact block curvature. This paper introduces ARC-block, a blockwise adaptive regularization with cubics (ARC) optimizer. For each parameter tensor it minimizes a cubic model over the true block Hessian, with a per-block adaptive cubic constant and a monotone guard on the full loss. Large tensors are handled matrix-free in a Lanczos-built Krylov subspace. ARC-block's CubicKrylov subsolver's descent is monotone with a per-block cubic decrease bound. For exact steps, and for Krylov steps that meet the Cartis--Gould--Toint residual test, we provide an

\mathcal{O}(\sqrt{B}\varepsilon^{-3/2})

sweep bound above a cross-block coupling floor. Another subsolver, ARC-

\varphi_1

, uses a trust-controlled exponential relaxation step. Among the cubic Newton methods evaluated here, these are the only ones whose steps stay exact on every parameter block (within the Krylov subspace), including on a

91.4

M-parameter implicit neural representation (INR). Run to convergence on FINER 2D image-fitting INR network, ARC-

\varphi_1

reaches

133.5

dB peak signal-to-noise ratio (PSNR), with no post peak collapse. Tuned Adam peaks at

78.2

dB at

2{,}629

s (step

{\sim}625

k, about

44

minutes), further steps degrade its PSNR. On the same benchmark, SOAP, L-BFGS, and the scalable diagonal cubic method AdaCubic stay at or below Adam's

78.2

dB peak. Over the benchmark's 20 images, run to a plateau rule, ARC-

\varphi_1

's per-image PSNR has median

132.4

and mean

131.5\pm3.3

dB against tuned Adam's

72.8\pm4.5

dB.

Rodion Podorozhny

Texas State University

Second-Order Optimization Neural Network Optimization

Aug 21, 2026

RODE: A Radial-Orthogonal Decoupled Engine for Optimization

Modern neural network training increasingly uses matrix-aware optimizers, yet their conditioned matrix step is typically added directly to the weight, jointly changing its norm and direction. This interaction matters because the current norm determines angular motion, while directional learning can drive norm growth and thereby alter later steps. We introduce RODE, which gives the radial and directional components separate update rules and step sizes. RODE explicitly updates the matrix Frobenius norm through a scalar radial rule, while its directional channel performs Newton--Schulz-conditioned updates in the tangent space. Controlled GPT-2 interventions show gains from both direct norm control and RODE's directional update. Across two language-modeling and two image-classification tasks, RODE outperforms both Muon variants in every direct comparison and ends with lower full-model norms. At 1.5B scale, using the learning rate transferred directly from the Qwen2-style LM sweep, RODE lowers loss from 4.145 to 3.346 and final global norm from 11964 to 2183 relative to Muon RMS, with fixed-radius RODE improving further. For Qwen3.5-9B full-parameter fine-tuning, all six optimizers use the same tuning budget and the same formal-training and evaluation settings; RODE outperforms both Muon variants on all four evaluation tasks and attains the highest mean on GSM8K and MATH-500. Thus, decoupling radial and directional dynamics offers a more effective and controllable approach to matrix optimization.

Guoxiang Xu, Bince Qu, Qi Sun +1

Zhejiang University, Hangzhou, China

Neural Network Optimization LLM Training

Aug 20, 2026

Enhanced Artificial Neural Networks Using QHAdamW in Air Quality Forecasting

The study employed an Artificial Neural Network in combination with the optimized Adaptive Moment Estimation (Adam) algorithm, currently the only AQI forecasting model available in the Philippines. The modified QHAdamW - Quasi-Hyperbolic Momentum (QHAdam) and Adam with decoupled weight decay (AdamW) were both extensions of the Adam optimizer, and both offer unique advantages for training ANN. The proposed QHAdamW optimizer addresses the issues on convergence, generalization, and forecasting performance of Adam. Hyperparameter tuning results revealed that 0.01 and 0.001 were the most effective optimal values for the generalization performance of QHAdamW. The comparative analysis results using seven evaluation metrics revealed that the error value range is lower, and the regression coefficient, having a value approximately equal to 1, improved the model accuracy performance. Likewise, the model converges to a satisfactory level of performance with the convergence performance results of lower loss values as obtained from training and validation losses. Based on data from a real-time air quality tracking station in Manila, a feed-forward neural network is used to predict the AQI of PM2.5 and PM10 separately. This model can be used to forecast Particulate Matter (PM), to help the Department of Environment and Natural Resources-Environmental Monitoring Bureau (DENR-EMB) implement a comprehensive air quality management.

Mary Joy Daniel Vinas

Computer Studies Department, Technological University of the Philippines, City of Manila, Philippines.

Air Quality Prediction Neural Network Optimization Momentum Methods

Aug 20, 2026

Finite-Horizon Input-Output Dynamics of Minibatch Perturbations in AdamW

A minibatch can influence training beyond the update at which it is observed because AdamW stores past gradient information in its optimizer states. We study this delayed effect through paired trajectories that differ only in one gradient update and share the same subsequent training sequence. We formulate AdamW as a finite-horizon input--state--output (ISO) system whose state contains the model parameters and first- and second-moment estimates. Linearizing the joint dynamics yields a signed response operator that maps a localized gradient perturbation to its future loss effects, revealing how optimizer memory shapes their magnitude, timing, and sign. We further derive an exact multistep error decomposition and establish first-order finite-horizon accuracy under local smoothness and controlled activation switching. Experiments validate the response mechanism and optimizer-state effects, while repeated-future analyses reveal substantial prospective structure in delayed influence that can be partially recovered from ISO approximations. Code is available at https://github.com/Kanyooo/Loss_ISO.

Kang Liu, Suyan Li

School of Future Technology, Xi’an Jiaotong University, Xi’an, China, 710049 · Department of Electrical and Computer Engineering, National University of Singapore, Singapore, 119077.

Neural Network Optimization Neural Network Training Dynamics Adaptive Gradient Methods Gradient Descent Dynamics

Aug 20, 2026

Kähler landscapes for complex neural network descents and guarantees including a search and destroy of the Calabi-Yau manifold

We study landscapes for complex-parameterized networks. Our approach is motivated with an information-theoretic manifold perspective of the parameter and via classical optimization guarantees although of complex geometric variety such as through Dolbeault asymptotics. The descent path admits a Kähler information metric under a cross-entropy via the Wirtinger Hessian on the log-likelihood potential. We restrict attention to a descent update rule with natural gradient descent via a differentiated loss scaled by the inverse metric, so the descent path remains in the holomorphic tangent bundle. We emphasize Calabi-Yau information manifolds which profane theoretical guarantees via an ill-curvature-conditioned, or lack thereof, landscape. We focus on Calabi-Yau metrics specifically in a non-compact setting with a global potential, so defined geometrically rather than invoking the topological requirements of the Calabi conjecture. In non-compact settings, we can write the metric determinant with respect to a background in terms of a function, pluriharmonic or real. Using Yau's Liouville theorem, we get a partial eigenvalue blow-up effect. In an empirical setting, a Ricci-flat metric will not form, but a blow-up can be a local condition and can hold empirically on open sets even partially in generalized Kähler cases due to lower bounds on the Ricci potential. Moreover, it has been discovered that negative curvature subverts the loss landscape, specifically sectional curvature, so we expand on this and draw interconnections to negative-definite Ricci curvature. Our arguments primarily exist via geometric modality, although we establish roots in deep learning theory such as through asymptotics at initialization and connections through failure modes of neural network guarantees under vanishing and negative Ricci curvature.

Andrew Gracyk

Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA

Riemannian Optimization Gradient Descent Neural Network Optimization Complex-Valued Neural Networks Loss Landscape Geometry

Aug 19, 2026

Activation-Keyed Momentum: An Anisotropic Momentum Update via the Delta Rule

Most modern optimizers form their momentum as an exponential moving average (EMA) of past gradients, forgetting every direction at one fixed rate. However, the inputs a deep network sees during training can be highly anisotropic, with a few directions queried frequently while most are seen rarely. Preconditioning methods address this anisotropy by wrapping extra processing around this buffer and leave the momentum update itself unchanged. We propose Activation-Keyed Momentum (AK-Momentum), which builds direction-awareness into the momentum update rule. The gradient of a linear layer splits into an input activation that acts as a key and an output-side error that acts as a value. Keying on that activation, AK-Momentum updates the momentum buffer by the canonical delta rule, so each direction is forgotten at a rate set by how often it appears. We prove that it is a valid momentum, that it applies the input-side curvature correction without matrix inversion, and that it clears stale directions faster than EMA under both a fixed and a drifting optimum. It is a drop-in replacement for the momentum buffer of any optimizer, its coefficient transfers across widths under

μ

P, and its extra compute stays between

22.2\%

and

25.0\%

of a gated-MLP block's linear cost with no persistent memory. In FineWeb-Edu pretraining, AdamW with AK-Momentum (AK-AdamW) reaches AdamW's validation loss in up to

46.39 \pm 4.32\%

fewer steps at 67M and

22.12 \pm 0.80\%

at 370M over three seeds, and the gain persists at 1B on a Chinchilla-optimal budget. A Muon baseline tuned under the same protocol sits above AK-AdamW at both language-model scales, and the gain holds for SGD, ResNet-18, and ViT-Tiny on CIFAR-10. Training-time diagnostics confirm the predicted mechanism, better gradient tracking and healthier input directions.

Euijin Hong, Guannan Qu

Electrical and Computer Engineering, Carnegie Mellon University

Deep Learning Optimization Momentum Methods Neural Network Optimization Accelerated Gradient Methods

Aug 19, 2026

Quantum Tensor Network Learning with DMRG

Tensor Networks are a relatively new machine learning approach. The architectures proposed initially are inspired by approaches from quantum many-body physics simulations. One common layout is the matrix product state (MPS) also known as a tensor train optimized with gradient descent techniques. We introduce a global normalization condition, so that the MPS represents a quantum state. We investigate two optimization methods that find the locally optimal tensors and compare them regarding their effectiveness. One is based on gradient descent and the other on an adaptation of DMRG.

Gustav J L Jäger, Martin B Plenio, Hans-Martin Rieser

Deutsches Zentrum für Luft- und Raumfahrt e.V. – Institut für KI-Sicherheit Wilhelm-Runge-Straße 10, 89081 Ulm, Germany · Universität Ulm – Institut für Theoretische Physik and IQST, Albert-Einstein-Allee 11, 89081 Ulm, Germany

Tensor Networks Quantum Machine Learning TT Decomposition Neural Network Optimization

Aug 19, 2026

Many Optimizers But Only One Training Path: Repeated Resampling for Adaptive Optimizer Selection

An optimizer is usually chosen before training a deep neural network and then kept fixed. Treating optimizer choice as a hyperparameter could boost performance, but it requires several complete training runs and discards all but the winner. Repeated Optimizer Resampling (ROR) instead searches during one evolving run. Every

b

epochs, each candidate optimizer scouts from the current model weights for

s

epochs. The best scout continues for the remaining

b-s

epochs, and that completed segment becomes the new incumbent if it improves the validation objective. This design allows the preferred optimizer to change as training progresses. We compare two variants of ROR on MNIST, Fashion-MNIST, and two motor insurance claim-count models. Nine fixed optimizers and both ROR variants are evaluated with the same ten seeds. One-epoch ROR uses 24% to 35% of the aggregate training needed to identify the best fixed optimizer exhaustively and remains close to that optimizer on all four tasks. These results support short scouting as a practical way to search over optimizers without completing every candidate run.

Ronald Richman, Mario V. Wüthrich

insureAI · ETH Zürich

Deep Learning Optimization Neural Network Optimization Hyperparameter Optimization

Aug 17, 2026

On the Principles Behind Neural Network Optimizers

Reliable optimization is central to neural network (NN) training, yet Adam, the default optimizer for modern LLMs, rests on a fragile foundation. This thesis develops a principled grounding for Adam and motivates new designs. First, we revisit Adam's divergence--convergence debate and show the existence of a problem-dependent phase transition: with properly chosen, batch-size-dependent hyperparameters, Adam converges, whereas under small-

β_2

regimes it can diverge. Second, we investigate why Adam substantially outperforms SGD on Transformers through Hessian structure. We find that the Hessian evolves toward a near-block-diagonal form along training, accompanied by strong block heterogeneity. We prove that this structure makes Adam's diagonal preconditioner effective. We further show that this special Hessian structure originates from consecutive multiplications of large matrix variables, and we provide a rigorous analysis based on random matrix theory. Finally, these insights motivate Adam-mini, a new optimizer that reduces Adam's memory footprint by 50% while preserving its performance. Our results also have broader implications beyond Adam: they reveal new local structures in matrix-based nonconvex problems, and also help understand and improve recent NN optimizers, such as Muon.

Yushun Zhang

School of Data Science, The Chinese University of Hong Kong, Shenzhen, China

Neural Network Optimization Adaptive Gradient Methods

Aug 14, 2026

GradRepair-ODE: Certified Gradient Repair for Neural ODE Training

Neural ordinary differential equations use numerical solvers inside the training loop. The solver determines the forward trajectory and also affects the gradient passed to the optimizer. That coupling creates a reliability problem for scientific machine learning and continuous-time generative modeling, including diffusion probability-flow ordinary differential equations and flow-matching models. Under loose step sizes, stiff dynamics, chaotic sensitivity, or event discontinuities, a differentiable ODE pipeline can return a finite gradient whose direction is numerically suspect. We introduce GradRepair-ODE, a reliability framework for checking, repairing, and rejecting ODE gradients at the optimizer step. The method computes several gradient candidates, compares them with directional finite-difference checks and solver diagnostics, diagnoses likely numerical failure modes, repairs selected gradients through path switching or stricter recomputation, and rejects steps whose descent direction cannot be certified. In six synthetic ODE systems, GradRepair-ODE leaves low-risk systems unchanged, repairs Robertson and Lorenz gradients to cosine similarity 1.000 against a strict reference, reduces unsafe accepted steps from 37 to 0, and rejects an event-discontinuous case instead of applying an uncertified update. The paper argues for a simple change in the training contract: an ODE gradient should reach the optimizer with numerical evidence attached.

Ziqian Bi, Xin Liang Chia

Purdue University, West Lafayette, IN, USA · Rice University, Houston, TX, USA

Neural ODEs Ordinary Differential Equations Neural Network Optimization

Aug 14, 2026

Where the Cost Falls: A Deployment-Aware Adoption Order for Stability Enhancements to Cycle-Consistent Adversarial Networks

Teams that adopt cycle-consistent adversarial networks for unpaired image-to-image translation meet the same obstacles: adversarial training oscillates or collapses, cycle consistency preserves coarse layout while finer texture drifts, and a single discriminator judging global realism misses local artifacts. Four enhancements address these failures, and they are usually compared on output quality alone. We show that they also divide sharply by where their cost falls, and that this division, which follows from the architecture and not from any particular run, yields an adoption order for teams under a compute or latency budget. A Wasserstein objective with gradient penalty, a VGG19 perceptual loss on the cycle reconstruction, and multi-scale discriminators change training only, so a team can adopt or drop them without altering what ships. Self-attention alone persists into the deployed generator, with memory growing as the square of the feature-map size, which makes it the one component a resource-constrained team should defer. We integrate all four onto a lightly tuned baseline for horse-to-zebra translation, introduced one at a time on a fixed control and then combined, and for each we give the failure mode it targets and how it integrates. We document the collapse and reconstruction-artifact modes the baseline produced, report what visual inspection of saved samples showed for each variant, and report Fréchet Inception Distance and Kernel Inception Distance for the combined model. We specify the protocol still needed, covering the individual variants, perceptual similarity, and downstream segmentation, to rank these enhancements on measured evidence.

Rowan Hussein, Mohamed Ouf

University of Ottawa, Ottawa, Canada · Queen’s University, Kingston, Canada

Generative Adversarial Networks Unpaired Image-to-Image Translation Neural Network Optimization

Aug 13, 2026

SAGE: Surrogate-gradient Adaptation via Attention-Guided Entropy for Spiking Transformers

Spiking neural networks (SNNs) offer an energy-efficient alternative to conventional deep neural networks by exploiting sparse event-driven computation, but their training remains challenging because the non-differentiable spike function requires surrogate gradients whose fixed shape may be suboptimal across layers and training stages. In this work, we introduce SAGE, an uncertainty-modulated surrogate-gradient mechanism for Transformer-based SNNs. SAGE estimates block-level uncertainty from normalized self-attention entropy and uses this signal to adapt the surrogate-gradient slope during training while leaving the inference model unchanged. By modulating only the training-time surrogate parameter, the proposed method preserves the original architecture and deployment cost while improving optimization flexibility. Experiments on CIFAR-10/100 demonstrate that SAGE achieves improved accuracy over fixed-surrogate baselines, with results up to 1-2% consistent gains across multiple simulation time steps. These results highlight the potential of attention-derived uncertainty as a lightweight training signal for adaptive surrogate-gradient learning in transformer-based SNNs.

Kiran Nair, Rodrigue Rizk, KC Santosh

USD Artificial Intelligence Research Lab Department of Computer Science University of South Dakota Vermillion, SD 57069, USA

Spiking Neural Networks Neural Network Optimization Spiking Transformers

Aug 12, 2026

Training Under Challenge: Executable Certificates and Challenge-Closed Optimality for Neural Networks

A flat training curve does not reveal whether a neural network has reached a global optimum, is locally trapped, is representation-limited, or is mismatched to its trainer. We introduce Training Under Challenge, an executable-certificate framework in which predeclared, architecture-valid procedures construct complete alternatives in the same certified class and reevaluate the same objective. Any lower-valued candidate is a replayable witness that lower-bounds the checkpoint's empirical global-optimality gap. Passing a finite suite is only suite-relative; global-gap conclusions require a separately justified coverage mechanism. We define a resource-indexed challenge-power modulus that characterizes the largest gap compatible with passage. For squared loss, current block-decrease operators make coverage checkable and yield uniform and realized-residual bounds. We prove the converse frontier: without coverage, a first-order ReLU trainer can reach infinitely many exact conditional head optima while converging to a non-global point. On a channel-gated ResNet-18 distillation problem with known optimum, eight internal challenges cover all 240 audited output directions, and realized-residual bounds lie within factors of 1.74--3.02 of the true gap. Paired predictive certificates separate decoder under-use from representation insufficiency, while quantized-denoising studies demonstrate diagnosis, repair, and current-state recertification.

Farhang Yeganegi, Arian Eamaz, Mojtaba Soltanalian

Department of Electrical and Computer Engineering University of Illinois Chicago Chicago, Illinois, USA

Neural Network Verification Neural Network Optimization