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.

Feb 24, 2026

A new Uncertainty Principle in Machine Learning

Many scientific problems in the context of machine learning can be reduced to the search of polynomial answers in appropriate variables. The Hevisidization of arbitrary polynomial is actually provided by one-and-the same two-layer expression. What prevents the use of this simple idea is the fatal degeneracy of the Heaviside and sigmoid expansions, which traps the steepest-descent evolution at the bottom of canyons, close to the starting point, but far from the desired true minimum. This problem is unavoidable and can be formulated as a peculiar uncertainty principle -- the sharper the minimum, the smoother the canyons. It is a direct analogue of the usual one, which is the pertinent property of the more familiar Fourier expansion. Standard machine learning software fights with this problem empirically, for example, by testing evolutions, originated at randomly distributed starting points and then selecting the best one. Surprisingly or not, phenomena and problems, encountered in ML application to science are pure scientific and belong to physics, not to computer science. On the other hand, they sound slightly different and shed new light on the well-known phenomena -- for example, extend the uncertainty principle from Fourier and, later, wavelet analysis to a new peculiar class of nearly singular sigmoid functions.

V. Dolotin, A. Morozov

MIPT, Dolgoprudny, 141701, Russia · NRC “Kurchatov Institute", 123182, Moscow, Russia · Institute for Information Transmission Problems, Moscow 127994, Russia +1

Two-Layer Neural Networks Neural Network Optimization

Feb 23, 2026

Scale-PINN: Learning Efficient Physics-Informed Neural Networks Through Sequential Correction

Physics-informed neural networks (PINNs) have emerged as a promising mesh-free paradigm for solving partial differential equations, yet adoption in science and engineering is limited by slow training and modest accuracy relative to modern numerical solvers. We introduce the Sequential Correction Algorithm for Learning Efficient PINN (Scale-PINN), a learning strategy that bridges modern physics-informed learning with numerical algorithms. Scale-PINN incorporates the iterative residual-correction principle, a cornerstone of numerical solvers, directly into the loss formulation, marking a paradigm shift in how PINN losses can be conceived and constructed. This integration enables Scale-PINN to achieve unprecedented convergence speed across PDE problems from different physics domain, including reducing training time on a challenging fluid-dynamics problem for state-of-the-art PINN from hours to sub-2 minutes while maintaining superior accuracy, and enabling application to representative problems in aerodynamics and urban science. By uniting the rigor of numerical methods with the flexibility of deep learning, Scale-PINN marks a significant leap toward the practical adoption of PINNs in science and engineering through scalable, physics-informed learning. Codes are available at https://github.com/chiuph/SCALE-PINN.

Pao-Hsiung Chiu, Jian Cheng Wong, Chin Chun Ooi +3

Institute of High Performance Computing, A*STAR, Singapore. · Centre for Frontier AI Research, A*STAR, Singapore. · School of Mechanical Engineering, Tianjin University, China. +1

Neural PDE Solvers Physics-Informed ML Neural Network Optimization

Feb 21, 2026

Limits of Convergence-Rate Control for Open-Weight Safety

Open-weight foundation models can be fine-tuned for harmful purposes after release, yet no existing training resistance methods provide theoretical guarantees. Treating these interventions as convergence-rate control problems allows us to connect optimization speed to the spectral structure of model weights. We leverage this insight to develop a novel understanding of convergence rate control through spectral reparameterization and derive an algorithm, SpecDef, that can both provably and empirically slow first- and second-order optimization in non-adversarial settings. In adversarial settings, we establish a fundamental limit on a broad class of convergence rate control methods including our own: an attacker with sufficient knowledge can restore fast convergence at a linear increase in model size. In order to overcome this limitation, future works will need to investigate methods that are not equivalent to controlling convergence rate.

Domenic Rosati, Xijie Zeng, Hong Huang +4

Dalhousie University · Vector Institute · Indian Institute of Management Bangalore

Neural Network Optimization Spectral Methods

Feb 20, 2026

OptiRoulette Optimizer: A New Stochastic Meta-Optimizer for up to 5.3x Faster Convergence

This paper presents OptiRoulette, a stochastic meta-optimizer that selects update rules during training instead of fixing a single optimizer. The method combines warmup optimizer locking, random sampling from an active optimizer pool, compatibility-aware learning-rate scaling during optimizer transitions, and failure-aware pool replacement. OptiRoulette is implemented as a drop-in, "torch.optim.Optimizer-compatible" component and packaged for pip installation. We report completed 10-seed results on five image-classification suites: CIFAR-100, CIFAR-100-C, SVHN, Tiny ImageNet, and Caltech-256. Against a single-optimizer AdamW baseline, OptiRoulette improves mean test accuracy from 0.6734 to 0.7656 on CIFAR-100 (+9.22 percentage points), 0.2904 to 0.3355 on CIFAR-100-C (+4.52), 0.9667 to 0.9756 on SVHN (+0.89), 0.5669 to 0.6642 on Tiny ImageNet (+9.73), and 0.5946 to 0.6920 on Caltech-256 (+9.74). Its main advantage is convergence reliability at higher targets: it reaches CIFAR-100/CIFAR-100-C 0.75, SVHN 0.96, Tiny ImageNet 0.65, and Caltech-256 0.62 validation accuracy in 10/10 runs, while the AdamW baseline reaches none of these targets within budget. On shared targets, OptiRoulette also reduces time-to-target (e.g., Caltech-256 at 0.59: 25.7 vs 77.0 epochs). Paired-seed deltas are positive on all datasets; CIFAR-100-C test ROC-AUC is the only metric not statistically significant in the current 10-seed study.

Stamatis Mastromichalakis

Independent Researcher, tmnetworks.gr

Neural Network Optimization Stochastic Optimization Learning to Optimize

Feb 20, 2026

Hardware-Friendly Input Expansion for Accelerating Function Approximation

One-dimensional function approximation is a fundamental problem in scientific computing and engineering applications. While neural networks possess powerful universal approximation capabilities, their optimization process is often hindered by flat loss landscapes induced by parameter-space symmetries, leading to slow convergence and poor generalization, particularly for high-frequency components. Inspired by the principle of symmetry breaking in physics, this paper proposes a hardware-friendly approach for function approximation through input-space expansion. The core idea involves augmenting the original one-dimensional input (e.g.,

x

) with constant values (e.g.,

π

) to form a higher-dimensional vector (e.g.,

[π, π, x, π, π]

), effectively breaking parameter symmetries without increasing the network's parameter count. We evaluate the method on ten representative one-dimensional functions, including smooth, discontinuous, high-frequency, and non-differentiable functions. Experimental results demonstrate that input-space expansion significantly accelerates training convergence (reducing LBFGS iterations by 12% on average) and enhances approximation accuracy (reducing final MSE by 66.3% for the optimal 5D expansion). Ablation studies further reveal the effects of different expansion dimensions and constant selections, with

π

consistently outperforming other constants. Our work proposes a low-cost, efficient, and hardware-friendly technique for algorithm design.

Hu Lou, Yin-Jun Gao, Dong-Xiao Zhang +3

Northwest Institute of Nuclear Technology, Xi’an, 710000, Shaanxi, China.

Neural Network Approximation Theory Neural Network Optimization

Feb 19, 2026

Learning with Boolean threshold functions

We develop a method for training neural networks on Boolean data in which the values at all nodes are strictly

\pm 1

, and the resulting models are typically equivalent to networks whose nonzero weights are also

\pm 1

. The method replaces loss minimization with a nonconvex constraint formulation. Each node implements a Boolean threshold function (BTF), and training is expressed through a divide-and-concur decomposition into two complementary constraints: one enforces local BTF consistency between inputs, weights, and output; the other imposes architectural concurrence, equating neuron outputs with downstream inputs and enforcing weight equality across training-data instantiations of the network. The reflect-reflect-relax (RRR) projection algorithm is used to reconcile these constraints. Each BTF constraint includes a lower bound on the margin. When this bound is sufficiently large, the learned representations are provably sparse and equivalent to networks composed of simple logical gates with

\pm 1

weights. Across a range of tasks -- including multiplier-circuit discovery, binary autoencoding, logic-network inference, and cellular automata learning -- the method achieves exact solutions or strong generalization in regimes where standard gradient-based methods struggle. These results demonstrate that projection-based constraint satisfaction provides a viable and conceptually distinct foundation for learning in discrete neural systems, with implications for interpretability and efficient inference.

Veit Elser, Manish Krishan Lal

Department of Physics, Cornell · Mathematics, Independent Researcher, India

Boolean Function Learning Binary Neural Networks Neural Network Optimization

Feb 18, 2026

HAWX: A Hardware-Aware FrameWork for Fast and Scalable ApproXimation of DNNs

This work presents HAWX, a hardware-aware scalable exploration framework that employs multi-level sensitivity scoring at different DNN abstraction levels (operator, filter, layer, and model) to guide selective integration of heterogeneous AxC blocks. Supported by predictive models for accuracy, power, and area, HAWX accelerates the evaluation of candidate configurations, achieving over 23* speedup in a layer-level search with two candidate approximate blocks and more than (3106) speedup at the filter-level search only for LeNet-5, while maintaining accuracy comparable to exhaustive search. Experiments across state-of-the-art DNN benchmarks such as VGG-11, ResNet-18, and EfficientNetLite demonstrate that the efficiency benefits of HAWX scale exponentially with network size. The HAWX hardware-aware search algorithm supports both spatial and temporal accelerator architectures, leveraging either off-the-shelf approximate components or customized designs.

Samira Nazari, Mohammad Saeed Almasi, Mahdi Taheri +4

University of Zanjan, Zanjan, Iran · Brandenburg Technical University, Cottbus, Germany · Tallinn University of Technology, Tallinn, Estonia +1

Neural Network Hardware Acceleration Neural Network Optimization Efficient Neural Network Inference AI Accelerator Inference Approximate Computing

Feb 18, 2026

Conjugate Learning Theory: Uncovering the Mechanisms of Trainability and Generalization in Deep Neural Networks

In this work, we propose a notion of practical learnability grounded in finite sample settings, and develop a conjugate learning theoretical framework based on convex conjugate duality to characterize this learnability property. Building on this foundation, we demonstrate that training deep neural networks (DNNs) with mini-batch stochastic gradient descent (SGD) achieves global optima of empirical risk by jointly controlling the extreme eigenvalues of a structure matrix and the gradient energy, and we establish a corresponding convergence theorem. We further elucidate the impact of batch size and model architecture (including depth, parameter count, sparsity, skip connections, and other characteristics) on non-convex optimization. Additionally, we derive a model-agnostic lower bound for the achievable empirical risk, theoretically demonstrating that data determines the fundamental limit of trainability. On the generalization front, we derive deterministic and probabilistic bounds on generalization error based on generalized conditional entropy measures. The former explicitly delineates the range of generalization error, while the latter characterizes the distribution of generalization error relative to the deterministic bounds under independent and identically distributed (i.i.d.) sampling conditions. Furthermore, these bounds explicitly quantify the influence of three key factors: (i) information loss induced by irreversibility in the model, (ii) the maximum attainable loss value, and (iii) the generalized conditional entropy of features with respect to labels. Moreover, they offer a unified theoretical lens for understanding the roles of regularization, irreversible transformations, and network depth in shaping the generalization behavior of deep neural networks. Extensive experiments validate all theoretical predictions, confirming the framework's correctness and consistency.

Binchuan Qi

Tongji University, Siping Street, 200092 Shanghai, China · Zhejiang Yuying College of Vocational Technology, Sihao Street, Hangzhou, 310018, Zhejiang, China

Neural Network Generalization Neural Network Optimization Information-Theoretic Generalization Bounds Empirical Risk Minimization Stochastic Gradient Descent

Feb 16, 2026

Algorithmic Simplification of Neural Networks with Mosaic-of-Motifs

Large-scale deep learning models are well-suited for compression. Across a variety of tasks, methods like pruning, quantization, and knowledge distillation have been used to achieve massive reductions in model parameters with only marginal performance drops. This raises the central question: Why are deep neural networks suited for compression? In this work, we take up the perspective of algorithmic complexity to explain this behavior. We hypothesize that the parameters of trained models have more structure and, hence, exhibit lower algorithmic complexity compared to the weights at (random) initialization. Furthermore, model compression methods harness this reduced algorithmic complexity to compress models. Although an unconstrained parameterization of model weights,

\mathbf{w} \in \mathbb{R}^n

, can represent arbitrary weight assignments, the solutions found during training exhibit repeatability and structure, making them simpler to implement than a trivial program. To this end, we formalize the Kolmogorov complexity of

\mathbf{w}

by

\mathcal{K}(\mathbf{w})

. We introduce a constrained parameterization

\widehat{\mathbf{w}}

that partitions parameters into blocks of size

s

and restricts each block to be selected from a set of

k

reusable motifs, specified by a reuse pattern (or mosaic). The resulting method,

\mathit{Mosaic\text{-}of\text{-}Motifs}

(MoMos), provides a theoretically justified parameterization that biases optimization toward algorithmically simpler solutions. Empirical evidence from multiple experiments shows that MoMos consistently lowers the algorithmic complexity of neural networks during training while preserving the performance of unconstrained models. These results suggest that parameter compressibility is not only observed after training, but can be induced from the optimization domain.

Pedram Bakhtiarifard, Tong Chen, Jonathan Wenshøj +2

Department of Computer Science, University of Copenhagen, Denmark

Kolmogorov Complexity Neural Network Compression Neural Network Optimization

Feb 16, 2026

Scale redundancy and soft gauge fixing in positively homogeneous neural networks

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

Rodrigo Carmo Terin

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

Neural Network Optimization Neural Network Training Dynamics

Feb 12, 2026

Learning to Control: The iUzawa-Net for Nonsmooth Optimal Control of Linear PDEs

We propose an optimization-informed deep neural network approach, named iUzawa-Net, aiming for the first solver that enables real-time solutions for a class of nonsmooth optimal control problems of linear partial differential equations (PDEs). The iUzawa-Net unrolls an inexact Uzawa method for saddle point problems, replacing classical preconditioners and PDE solvers with specifically designed learnable neural networks. We prove universal approximation properties and establish the asymptotic

\varepsilon

-optimality for the iUzawa-Net, and validate its promising numerical efficiency through nonsmooth elliptic and parabolic optimal control problems. Our techniques offer a versatile framework for designing and analyzing various optimization-informed deep learning approaches to optimal control and other PDE-constrained optimization problems. The proposed learning-to-control approach synergizes model-based optimization algorithms and data-driven deep learning techniques, inheriting the merits of both methodologies.

Yongcun Song, Xiaoming Yuan, Hangrui Yue +1

Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, 637371, Singapore. · Department of Mathematics, The University of Hong Kong, Pok Fu Lam, Hong Kong SAR, China. · School of Mathematical Sciences, Nankai University, Tianjin 300071, China.

Optimal Control Constrained Optimization Neural Network Optimization Algorithm Unrolling

Feb 11, 2026

$μ$ pscaling small models: Principled warm starts and hyperparameter transfer

Modern large-scale neural networks are often trained and released in multiple sizes to accommodate diverse inference budgets. To improve efficiency, recent work has explored model upscaling: initializing larger models from trained smaller ones to accelerate convergence. However, this method can be sensitive to hyperparameters that need to be tuned at the target upscaled model size, which is prohibitively costly to do directly. It remains unclear whether tuning hyperparameters on smaller models and extrapolating via scaling laws is sound in this setting. We address this with principled approaches to width-based upscaling and efficient hyperparameter tuning in this setting. Motivated by

μ

P and any-dimensional architectures, we introduce a general upscaling method that, like Net2Net, copies and perturbs weights, but uses theoretically grounded, width-dependent scalings for the perturbation noise and optimizer hyperparameters. First, we prove that under zero perturbation, the upscaled model is functionally equivalent to the base model throughout training. Second, we extend the

μ

P theory to enable infinite-width limit analysis and establish hyperparameter transfer for upscaled models, greatly reducing the tuning cost. We empirically demonstrate that this method is effective on realistic datasets and architectures.

Yuxin Ma, Nan Chen, Mateo Díaz +3

Department of Applied Mathematics and Statistics Johns Hopkins University

Neural Network Optimization Hyperparameter Transfer Neural Scaling Laws Hyperparameter Optimization

Feb 9, 2026

ANCRe: Adaptive Neural Connection Reassignment for Efficient Depth Scaling

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

<1\%

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

Yilang Zhang, Bingcong Li, Niao He +1

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

Residual Learning Neural Network Optimization Neural Network Training Dynamics

Jan 24, 2026

Unrolled Neural Networks for Constrained Optimization

We develop unrolled neural networks to solve constrained optimization problems, offering accelerated, learnable counterparts to dual ascent (DA) algorithms. Our framework, termed constrained dual unrolling (CDU), comprises two coupled neural networks that jointly approximate the saddle point of the Lagrangian. The primal network emulates an iterative optimizer that finds a stationary point of the Lagrangian for a given dual multiplier, sampled from an unknown distribution. The dual network generates trajectories towards the optimal multipliers across its layers while querying the primal network at each layer. Unlike standard unrolling, which maps the updates of a prescribed iterative solver directly to network layers, CDU preserves the underlying optimization dynamics while learning the update mappings. The desired primal-dual dynamics are therefore induced through constrained learning by enforcing primal-descent and dual-ascent constraints. We formulate training the two networks as a nested optimization problem and propose an alternating procedure that updates the primal and dual networks in turn, mitigating uncertainty in the multiplier distribution required for primal network training. We establish theoretical guarantees on the feasibility and optimality of the coupled networks. We numerically evaluate the framework on quadratic programs (QPs) with linear constraints and power allocation in wireless networks.

Samar Hadou, Alejandro Ribeiro

Department of Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, PA 19104 USA

Constrained Optimization Learning to Optimize Primal-Dual Optimization Neural Network Optimization

Jan 20, 2026

Linearized subspace refinement framework to expose hidden accuracy in trained neural networks

Neural networks trained by gradient-based methods often exhibit optimization-induced accuracy plateaus in scientific machine learning tasks. We present Linearized Subspace Refinement (LSR), an architecture-agnostic post-training framework that exploits the local linearized model at a fixed trained state. By solving a reduced direct least-squares problem in a Jacobian-defined low-dimensional space, LSR computes a subspace-optimal linearized correction and yields a refined predictor with markedly improved accuracy. Across function approximation, data-driven operator learning, physics-informed operator fine-tuning, and noisy inverse problems, LSR shows that standard nonlinear training can remain far above this subspace-attainable error level. Similar accuracy plateaus persist even for the convex quadratic problem from local linearization when solved with standard iterative optimizers, identifying numerical ill-conditioning as a primary bottleneck. LSR frequently delivers order-of-magnitude error reductions, while the subspace rank provides an explicit capacity-control mechanism that balances correction strength, numerical stability, and noise sensitivity. Together, LSR exposes conditioning-limited attainable accuracy in trained-state linearized models and provides direct access to it.

Wenbo Cao, Weiwei Zhang

Institute of AI for Industries, Chinese Academy of Sciences, Nanjing 211135, China · Institute of Computing Technology, Chinese Academy of Sciences, Beijing 100190, China · School of Aeronautics, Northwestern Polytechnical University, Xi’an 710072, China +2

Subspace Optimization Neural Network Optimization Neural Network Approximation Theory Neural Network Training Dynamics

Jan 20, 2026

Performance and Complexity Trade-off Optimization of Speech Models During Training

In speech machine learning, neural network models are typically designed by choosing an architecture with fixed layer sizes and structure. These models are then trained to maximize performance on metrics aligned with the task's objective. While the overall architecture is usually guided by prior knowledge of the task, the sizes of individual layers are often chosen heuristically. However, this approach does not guarantee an optimal trade-off between performance and computational complexity; consequently, post hoc methods such as weight quantization or model pruning are typically employed to reduce computational cost. This occurs because stochastic gradient descent (SGD) methods can only optimize differentiable functions, while factors influencing computational complexity, such as layer sizes and floating-point operations per second (FLOP/s), are non-differentiable and require modifying the model structure during training. We propose a reparameterization technique based on feature noise injection that enables joint optimization of performance and computational complexity during training using SGD-based methods. Unlike traditional pruning methods, our approach allows the model size to be dynamically optimized for a target performance-complexity trade-off, without relying on heuristic criteria to select which weights or structures to remove. We demonstrate the effectiveness of our method through three case studies, including a synthetic example and two practical real-world applications: voice activity detection and audio anti-spoofing. The code related to our work is publicly available to encourage further research.

Esteban Gómez, Tom Bäckström

Department of Information and Communications Engineering, Aalto University, Espoo, Finland

Model Compression Neural Network Optimization Speech Processing Voice Activity Detection

Dec 4, 2025

Turbo-Muon: Almost-Orthogonal Pre-Conditioning for Fast Muon Updates

Orthogonality-based optimizers, such as Muon, have recently shown strong performance across large-scale training and community-driven efficiency challenges. However, these methods rely on a costly gradient orthogonalization step. Even efficient iterative approximations such as Newton-Schulz remain expensive, typically requiring dozens of matrix multiplications to converge. We introduce a pre-conditioning procedure that improves the initialization of the Newton--Schulz iterations while incurring negligible overhead. Furthermore, our pre-conditioning reduces the initial polar error and enables the removal of one Newton-Schulz iteration (out of the five iterations usually used in practice). The resulting implementation significantly reduces Muon's overhead. At the end-to-end training level, we observe consistent runtime improvements across speed-run and standard benchmarks, including

\sim

3% reductions in training time on multiple fast training benchmarks, while matching reference performance on both language and vision tasks. Crucially, these improvements require no hyperparameter tuning and can be adopted as a simple drop-in replacement. Beyond empirical gains, we provide theoretical insight into the geometry of the update and its potential robustness against feature collapse. Our code is publicly available on github, in optax and huggingface kernels.

Thibaut Boissin, Thomas Massena, Franck Mamalet +1

IRIT-MISFIT · Institut de Recherche Technologique Saint-Exupery, France · DTIPG - SNCF, IRIT-MISFIT +2

Newton-Schulz Iteration Orthogonal Optimization Neural Network Optimization Muon Optimizer Second-Order Optimization

Nov 26, 2025

Mean-Field Model for Two-Layer Neural Networks Trained with Consensus-Based Optimization

We study Consensus-Based Optimization (CBO) for two-layer neural network training. We compare the performance of CBO against Adam on two test cases and demonstrate how a hybrid approach, combining CBO with Adam, provides faster convergence than CBO. Additionally, in the context of multi-task learning, we recast CBO into a formulation that offers less memory overhead. The CBO method allows for a mean-field model formulation, which we couple with the mean-field model of the neural network. To this end, we first reformulate CBO within the optimal transport framework. As the number of particles tends to infinity, we lift the corresponding dynamics to the Wasserstein-over-Wasserstein space and show that the variance decreases monotonically. We confirm numerically that both mean-field models converge.

William De Deyn, Michael Herty, Giovanni Samaey

Institut für Geometrie und Praktische Mathematik, RWTH Aachen University, Germany · Department of Computer Science, KU Leuven, Belgium · Extraordinary Professor, Department of Mathematics and Applied Mathematics, University of Pretoria, South Africa

Neural Network Optimization Mean-Field Theory Neural Network Training Dynamics

Nov 7, 2025

Gradient Prediction with Control Variates in the Cheap-Forward Regime

We study whether otherwise-idle inference resources could reduce the scarce-GPU cost of training. Our analysis uses a simulated compute ledger in which fleet work is billed at a fraction of a scarce-GPU forward; all experiments run on a regular GPU. Our algorithm predicts gradients with a reduced-precision, inference-style reverse-mode program and combines many predictions with a few exact gradients through a control variate, so approximation error becomes variance rather than bias. On a 124M-parameter language model and selected short training windows, the method can lower simulated ledger cost relative to the tested baselines when fleet work is sufficiently cheap. Experiments spanning 10M-774M parameters show both transfers and failures. We do not test inference-only hardware, end-to-end distributed latency, or a full optimizer-by-batch-size baseline sweep.

Kamil Ciosek, Nicolò Felicioni, Juan Elenter +1

Spotify

Neural Network Optimization

Oct 29, 2025

Nonlinear Dynamics In Optimization Landscape of Shallow Neural Networks with Tunable Leaky ReLU

In this work, we study the nonlinear dynamics of a shallow neural network trained with mean-squared loss and leaky ReLU activation. Under Gaussian inputs and equal layer width k, (1) we establish, based on the equivariant gradient degree, a theoretical framework, applicable to any number of neurons k>= 4, to detect bifurcation of critical points with associated symmetries from global minimum as leaky parameter

α

varies. Typically, our analysis reveals that a multi-mode degeneracy consistently occurs at the critical number 0, independent of k. (2) As a by-product, we further show that such bifurcations are width-independent, arise only for nonnegative

α

and that the global minimum undergoes no further symmetry-breaking instability throughout the engineering regime

α

in range (0,1). An explicit example with k=5 is presented to illustrate the framework and exhibit the resulting bifurcation together with their symmetries.

Jingzhou Liu

Shallow Neural Networks Dynamical Systems Neural Network Optimization Neural Network Activation Functions Neural Network Training Dynamics

Oct 25, 2025

SmartMixed: A Two-Phase Training Strategy for Adaptive Activation Function Learning in Neural Networks

The choice of activation function plays a critical role in neural networks, yet most architectures still rely on fixed, uniform activation functions across all neurons. We introduce SmartMixed, a novel two-phase training strategy that allows networks to learn optimal per-neuron activation functions while preserving computational efficiency at inference. In the first phase, neurons adaptively select from a pool of candidate activation functions (ReLU, Sigmoid, Tanh, Leaky_ReLU, ELU, SELU) using a differentiable hard mixture mechanism. In the second phase, each neuron's activation function is fixed according to the learned selection, resulting in a computationally efficient network that supports continued training with optimized vectorized operations. We evaluate SmartMixed on the MNIST dataset using feedforward neural networks of different architectures. Our analysis reveals that neurons in different layers exhibit distinct preferences for activation functions, providing insights into the functional diversity within neural architectures. We also demonstrated that SmartMixed effectively trains the network by allowing neurons to select their preferred activation functions, competing against models using a single fixed state-of-the-art activation function.

Amin Omidvar

Independent Researcher Toronto, Ontario, Canada

Neural Network Optimization Neural Network Activation Functions

Sep 22, 2025

Development of Deep Learning Optimizers: Approaches, Concepts, and Update Rules

Deep learning optimizers are optimization algorithms that enable deep neural networks to learn. The effectiveness of learning is highly dependent on the optimizer employed in the training process. Alongside the rapid advancement of deep learning, a wide range of optimizers with different approaches have been developed. This study aims to provide a review of various optimizers that have been proposed and received attention in the literature. From Stochastic gradient descent to the most recent ones such as Momentum, AdamW, Sophia, and Muon in chronological order, optimizers are examined individually, and their distinctive features are highlighted in the study. The update rule of each optimizer is presented in detail, with an explanation of the associated concepts and variables. The techniques applied by these optimizers, their contributions to the optimization process, and their default hyperparameter settings are also discussed. In addition, insights are offered into the open challenges encountered in the optimization of deep learning models. Thus, a comprehensive resource is provided both for understanding the current state of optimizers and for identifying potential areas of future development.

Doğay Altınel

Istanbul Medeniyet University

Deep Learning Optimization Neural Network Optimization Adaptive Gradient Methods Adaptive Learning Rate Stochastic Gradient Descent

Sep 15, 2025

Low-rank Orthogonalization for Large-scale Matrix Optimization with Applications to Foundation Model Training

Neural network (NN) training is inherently a large-scale matrix optimization problem, yet the matrix structure of NN parameters has long been overlooked. Recently, the optimizer Muon \citep{jordanmuon}, which explicitly exploits this structure, has gained significant attention for its strong performance in foundation model training. A key component contributing to Muon's success is matrix orthogonalization. In this paper, we propose low-rank orthogonalization, which performs orthogonalization by leveraging the low-rank nature of gradients during NN training. Building on this, we introduce low-rank matrix-signed gradient descent (MSGD) and a low-rank variant of Muon. %Numerical experiments demonstrate the superior performance of low-rank orthogonalization, with low-rank Muon achieving promising results in GPT-2 and LLaMA pretraining---surpassing the carefully tuned vanilla Muon on tasks with large model sizes. {Numerical experiments demonstrate the advantages of low-rank orthogonalization: low-rank Muon generally matches or improves upon vanilla Muon on the GPT-2 and LLaMA pretraining tasks, with clearer improvements observed for relatively larger models.} Theoretically, we establish the iteration complexity of low-rank MSGD for finding an approximate stationary solution, and the iteration complexity of low-rank Muon for finding an approximate stochastic stationary solution under heavy-tailed noise. The code to reproduce our numerical experiments is available at https://github.com/dengzhanwang/Low-rank-Muon.

Chuan He, Zhanwang Deng, Zhaosong Lu

Department of Mathematics Linköping University, Sweden · Academy for Advanced Interdisciplinary Studies Peking University, Beijing, People’s Republic of China · Department of Industrial and Systems Engineering University of Minnesota, USA

Neural Network Optimization Low-Rank Approximation Matrix Optimization

Sep 3, 2025

Understanding and Improving Shampoo and SOAP via Kullback-Leibler Minimization

Shampoo and its efficient variant, SOAP, employ structured second-moment estimations and have shown strong performance for training neural networks (NNs). In practice, however, Shampoo typically requires step-size grafting with Adam to be competitive, and SOAP mitigates this by applying Adam in Shampoo's eigenbasis -- at the cost of additional memory overhead from Adam in both methods. Prior analyses have largely relied on the Frobenius norm to motivate these estimation schemes. We instead recast their estimation procedures as covariance estimation under Kullback-Leibler (KL) divergence minimization, revealing a previously overlooked theoretical limitation and motivating principled redesigns. Building on this perspective, we develop

\textbf{KL-Shampoo}

and

\textbf{KL-SOAP}

, practical schemes that match or exceed the performance of Shampoo and SOAP in NN pre-training while achieving SOAP-level per-iteration runtime. Notably, KL-Shampoo does not rely on Adam to attain competitive performance, eliminating the memory overhead introduced by Adam. Across our experiments, KL-Shampoo consistently outperforms SOAP, Shampoo, and even KL-SOAP, establishing the KL-based approach as a promising foundation for designing structured methods in NN optimization. An implementation of KL-Shampoo/KL-SOAP is available at https://github.com/yorkerlin/KL-Methods

Wu Lin, Scott C. Lowe, Felix Dangel +3

1Vector Institute, Toronto, Canada · University of Cambridge, Cambridge, United Kingdom · 3Microsoft, United States +1

Deep Learning Optimization Neural Network Optimization Second-Order Optimization Adaptive Gradient Methods

May 28, 2025

Favorability of Loss Landscape with Weight Decay Requires Both Large Overparametrization and Initialization

The optimization of neural networks under weight decay remains poorly understood from a theoretical standpoint. While weight decay is standard practice in modern training procedures, most theoretical analyses focus on unregularized settings. In this work, we investigate the loss landscape of the

\ell_2

-regularized training loss for two-layer ReLU networks. We show that the landscape becomes benign -- i.e., free of spurious local minima -- under large overparametrization, specifically when the network width

m

satisfies

m \gtrsim \min(n^d, 2^n)

, where

n

is the number of data points and

d

the input dimension. More precisely in this regime, almost all constant activation regions contain a global minimum and no spurious local minima. We further show that this level of overparametrization is not only sufficient but also necessary via the example of orthogonal data. Finally, we demonstrate that such loss landscape results primarily hold relevance in the large initialization regime. In contrast, for small initializations -- corresponding to the feature learning regime -- optimization can still converge to spurious local minima, despite the global benignity of the landscape.

Etienne Boursier, Matthew Bowditch, Matthias Englert +1

INRIA LMO, Université Paris-Saclay Orsay, France · Mathematics Institute University of Warwick Coventry, UK · Department of Computer Science University of Warwick Coventry, UK

Loss Landscape Overparameterized Neural Network Weight Decay Neural Network Initialization Neural Network Optimization Global Optimization+2

May 20, 2025

KO: Kinetics-inspired Neural Optimizer with PDE Simulation Approaches

The design of effective optimization algorithms for neural networks remains a fundamental challenge, and most existing methods rely on heuristic extensions of gradient-based updates. We introduce KO (Kinetics-inspired Optimizer), a plug-and-play optimization module grounded in kinetic theory and partial differential equations. KO models parameter dynamics as a particle system, augmenting standard gradient updates with stochastic interactions induced by a discretization of the Boltzmann transport equation. This mechanism naturally promotes parameter diversity and mitigates weight condensation, the tendency of parameters to collapse into low-dimensional subspaces, a phenomenon closely associated with degraded generalization. We provide both a rigorous theoretical analysis and a physical interpretation, showing that KO provably increases parameter diversity while preserving convergence guarantees. Extensive experiments on image classification benchmarks (CIFAR-10/100, ImageNet) and large-scale language model pretraining demonstrate that KO consistently improves accuracy over competitive baselines with negligible additional computational cost.

Mingquan Feng, Yixin Huang, Yifan Fu +2

Shanghai Jiao Tong University

Neural Network Optimization Neural Network Training Dynamics

May 20, 2025

Forward Target Propagation: A Forward-Only Approach to Global Error Credit Assignment via Local Losses

Training neural networks has traditionally relied on backpropagation (BP), a gradient-based algorithm that, despite its widespread success, suffers from key limitations in both biological and hardware perspectives. These include backward error propagation by symmetric weights, non-local credit assignment, and frozen activity during backward passes. We propose Forward Target Propagation (FTP), a biologically plausible and computationally efficient alternative that replaces the backward pass with a second forward pass. FTP estimates layerwise targets using only feedforward computations, eliminating the need for symmetric feedback weights or learnable inverse functions, hence enabling modular and local learning. We evaluate FTP on fully connected networks, CNNs, and RNNs, demonstrating accuracies competitive with BP on MNIST, CIFAR10, and CIFAR100, as well as effective modeling of long-term dependencies in sequential tasks. Moreover, FTP outperforms BP under quantized low-precision and emerging hardware constraints while also demonstrating substantial efficiency gains over other biologically inspired methods such as target propagation variants and forward-only learning algorithms. With its minimal computational overhead, forward-only nature, and hardware compatibility, FTP provides a promising direction for energy-efficient on-device learning and neuromorphic computing.

Nazmus Saadat As-Saquib, A N M Nafiz Abeer, Hung-Ta Chien +3

Texas A&M University, College Station, TX 77843 · Sandia National Laboratories, Livermore, CA 94550

Neural Network Optimization Local Learning

Apr 2, 2025

AYLA: Architecting a loss landscape in shallow neural networks to accelerate feature recovery

Feature learning in shallow neural networks exhibits rich yet fragile dynamics, including prolonged plateaus, abrupt phase transitions, and sensitivity to optimization hyperparameters. While recent theoretical work has characterized these behaviors through the geometry of loss landscapes, saddle escape mechanisms, and emergent scaling laws, practical methods for actively shaping these dynamics remain limited. In this paper, we introduce AYLA, a principled loss reparameterization framework that dynamically modulates gradient magnitudes during training without altering the location of stationary points or optimal solutions. AYLA applies a smooth, sigmoid-controlled power-law transformation to empirical loss, yielding a state-dependent effective learning rate that accelerates descent in flat or saddle-dominated regions while stabilizing late-stage optimization. Crucially, AYLA preserves all critical points of the original objective, acting solely as a monotone transformation that reshapes optimization trajectories rather than objectives. We evaluate AYLA in controlled teacher student settings using two-layer tanh networks trained on synthetic Gaussian data. Across stochastic gradient descent and multiple loss-exponent schedules, AYLA consistently improves feature recovery. This evidence is observed in terms of weight alignment, per-neuron cosine similarity, hidden-activation correlation, and spectral properties of learned representations, while AYLA maintains competitive or faster loss convergence. Spectral analyses further demonstrate that AYLA mitigates rank collapse and promotes richer internal representations, signaling a transition from lazy to active feature-learning regimes. AYLA offers a lightweight, theoretically grounded way to improve shallow-network optimization, especially in resource-limited or noise-sensitive settings.

Behnam Gheshlaghi, Shahin Atakishiyev

Shallow Neural Networks Representation Learning Neural Network Optimization Neural Network Training Dynamics

Feb 26, 2025

Fourier Multi-Component and Multi-Layer Neural Networks: Unlocking High-Frequency Potential

The architecture of a neural network and the choice of its activation function are both fundamental to its performance. Equally important is ensuring that these two elements are well matched, as their alignment is key to effective representation and learning. In this paper, we introduce the Fourier Multi-Component and Multi-Layer Neural Network (FMMNN), a model that combines sine-type activations with the multi-component and multi-layer structure of MMNNs. In an FMMNN, each component is represented as a trainable linear combination of fixed random sine-type basis functions, while multi-layer composition generates more complex and adaptive high-frequency features. We establish that FMMNNs retain exponential expressive power for function approximation even under a low-rank architectural structure. We also analyze the optimization landscape of FMMNNs and find it to be substantially more favorable than that of standard fully connected neural networks, especially for high-frequency targets. In addition, we propose a scaled random initialization method for the first-layer weights in FMMNNs, which accelerates training and improves final performance when sufficient samples are available. Extensive numerical experiments support our theoretical insights, showing that FMMNNs achieve strong accuracy and favorable convergence behavior on oscillatory function-approximation benchmarks.

Shijun Zhang, Hongkai Zhao, Yimin Zhong +1

Department of Applied Mathematics Hong Kong Polytechnic University · Department of Mathematics Duke University · Department of Mathematics and Statistics Auburn University +1

Fourier Feature Embeddings Neural Network Optimization Neural Network Activation Functions Neural Network Approximation Theory Neural Network Expressivity Modular Neural Networks

Oct 30, 2024

Koopman-informed recurrent neural networks

Recurrent neural networks are a successful neural architecture for many time-dependent problems, including time series analysis, forecasting, and modeling of dynamical systems. In the context of dynamical systems, training with backpropagation through time can lead to challenges arising from exploding or vanishing gradients. In this contribution, we introduce Koopman-informed recurrent neural networks, a computational approach to construct all weights and biases of a recurrent neural network without using gradient-based methods. The approach is based on a combination of random feature networks and Koopman operator theory for dynamical systems. The hidden parameters of a single recurrent block are sampled at random, while the outer weights are constructed using extended dynamic mode decomposition. This approach alleviates some problems with backpropagation commonly related to recurrent networks. The connection to Koopman operator theory also allows us to start using results in this area to analyze recurrent neural networks. In computational experiments on time series, forecasting for chaotic dynamical systems, control problems, and on real-world data, we observe that with comparable forecasting accuracy, the training time of the Koopman-informed recurrent neural networks is significantly improved when compared to models trained with commonly used gradient-based methods.

Erik Lien Bolager, Ana Čukarska, Iryna Burak +2

Munich Data Science Institute · School of Computation, Information and Technology, Technical University of Munich, Germany · Interdisciplinary Center for Scientific Computing (IWR) +1

Neural Network Optimization Time Series Forecasting Recurrent Neural Networks