Neural Scaling Laws

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Neural scaling laws study how performance changes as models, datasets, training time, and compute grow, often fitting power laws to loss or task metrics. Empirical methods use controlled model and data sweeps, compute-allocation curves, and scaling-law fits to forecast performance and choose training budgets; theory connects these trends to statistics, optimization, spectral structure, and learned representations. Researchers are extending the framework beyond language models to vision, scientific, medical, and sequential tasks, while studying how architecture, data composition, optimizers, and training choices alter the results. A central challenge is that simple fits can fail across data-limited regimes, task metrics, or extrapolations, so reliable predictions need to account for these dependencies and their uncertainty.

Latest papers

83 papers15 papers in the last four weeks, up 67% on the four weeks before. 0.1% of all new papers.

May 2, 2026

Prescriptive Scaling Laws for Data Constrained Training

Training compute is increasingly outpacing the availability of high-quality data. This shifts the central challenge from optimal compute allocation to extracting maximum value from limited data. The widely adopted Chinchilla scaling law assumes every training token is unique. This limits its ability to guide pretraining decisions in data-constrained regimes. We model the excess loss under repetition with a simple additive overfitting penalty and find that it accurately describes model behavior. Our scaling law yields qualitatively new compute-optimal allocation advice. Beyond a point, further repetition is counterproductive and compute is better spent on model capacity. We show that following our law's recommended configuration improves performance in data-constrained regimes. Finally, because our one-parameter form isolates overfitting in a single coefficient, it enables direct comparison across training configurations. As a case study, we show that strong weight decay (

λ=1.0

) reduces this coefficient by approximately 70%, providing a scaling-law explanation for recent findings that optimal weight decay in data-constrained regimes is an order of magnitude larger than standard practice.

Justin Lovelace, Christian Belardi, Srivatsa Kundurthy +2

Department of Computer Science Cornell University Ithaca, NY

Weight Decay Language Model Scaling Laws Neural Scaling Laws

Apr 27, 2026

A Limit Theory of Foundation Models: A Mathematical Approach to Understanding Emergent Intelligence and Scaling Laws

Emergent intelligence have played a major role in the modern AI development. While existing studies primarily rely on empirical observations to characterize this phenomenon, a rigorous theoretical framework remains underexplored. This study attempts to develop a mathematical approach to formalize emergent intelligence from the perspective of limit theory. Specifically, we introduce a performance function E(N, P, K), dependent on data size N, model size P and training steps K, to quantify intelligence behavior. We posit that intelligence emerges as a transition from finite to effectively infinite knowledge, and thus recast emergent intelligence as existence of the limit

\lim_{N,P,K \to \infty} \mathcal{E}(N,P,K)

, with emergent abilities corresponding to the limiting behavior. This limit theory helps reveal that emergent intelligence originates from the existence of a parameter-limit architecture (referred to as the limit architecture), and that emergent intelligence rationally corresponds to the learning behavior of this limit system. By introducing tools from nonlinear Lipschitz operator theory, we prove that the necessary and sufficient conditions for existence of the limit architecture. Furthermore, we derive the scaling law of foundation models by leveraging tools of Lipschitz operator and covering number. Theoretical results show that: 1) emergent intelligence is governed by three key factors-training steps, data size and the model architecture, where the properties of basic blocks play a crucial role in constructing foundation models; 2) the critical condition Lip(T)=1 for emergent intelligence provides theoretical support for existing findings. 3) emergent intelligence is determined by an infinite-dimensional system, yet can be effectively realized in practice through a finite-dimensional architecture. Our empirical results corroborate these theoretical findings.

Jun Shu, Junxiong Jia, Deyu Meng +1

School of Mathematics and Statistics, Xi’an Jiaotong University.

Neural Scaling Laws AI Scaling Laws

Apr 20, 2026

OmniMouse: Scaling properties of multi-modal, multi-task Brain Models on 150B Neural Tokens

Scaling data and artificial neural networks has transformed AI, driving breakthroughs in language and vision. Whether similar principles apply to modeling brain activity remains unclear. Here we leveraged a dataset of 3.1 million neurons from the visual cortex of 73 mice across 323 sessions, totaling more than 150 billion neural tokens recorded during natural movies, images and parametric stimuli, and behavior. We train multi-modal, multi-task models that support three regimes flexibly at test time: neural prediction, behavioral decoding, neural forecasting, or any combination of the three. OmniMouse achieves state-of-the-art performance, outperforming specialized baselines across nearly all evaluation regimes. We find that performance scales reliably with more data, but gains from increasing model size saturate. This inverts the standard AI scaling story: in language and computer vision, massive datasets make parameter scaling the primary driver of progress, whereas in brain modeling -- even in the mouse visual cortex, a relatively simple system -- models remain data-limited despite vast recordings. The observation of systematic scaling raises the possibility of phase transitions in neural modeling, where larger and richer datasets might unlock qualitatively new capabilities, paralleling the emergent properties seen in large language models. Code available at https://github.com/enigma-brain/omnimouse.

Konstantin F. Willeke, Polina Turishcheva, Alex Gilbert +18

Department of Ophthalmology, Byers Eye Institute, Stanford University · 2Stanford Bio-X, Stanford University · 3Wu Tsai Neurosciences Institute, Stanford University +6

Neural Decoding Visual Neuroscience Neural Scaling Laws

Apr 19, 2026

How Much Data is Enough? The Zeta Law of Discoverability in Biomedical Data, featuring the enigmatic Riemann zeta function

How much data is enough to make a scientific discovery? As biomedical datasets scale to millions of samples and AI models grow in capacity, progress increasingly depends on predicting when additional data will substantially improve performance. In practice, model development often relies on empirical scaling curves measured across architectures, modalities, and dataset sizes, with limited theoretical guidance on when performance should improve, saturate, or exhibit cross-over behavior. We propose a scaling-law framework for cross-modal discoverability based on spectral structure of data covariance operators, task-aligned signal projections, and learned representations. Many performance metrics, including AUC, can be expressed in terms of cumulative signal-to-noise energy accumulated across identifiable spectral modes of an encoder and cross-modal operator. Under mild assumptions, this accumulation follows a zeta-like scaling law governed by power-law decay of covariance spectra and aligned signal energy, leading naturally to the appearance of the Riemann zeta function. Representation learning methods such as sparse models, low-rank embeddings, and multimodal contrastive objectives improve sample efficiency by concentrating useful signal into earlier stable modes, effectively steepening spectral decay and shifting scaling curves. The framework predicts cross-over regimes in which simpler models perform best at small sample sizes, while higher-capacity or multimodal encoders outperform them once sufficient data stabilizes additional degrees of freedom. Applications include multimodal disease classification, imaging genetics, functional MRI, and topological data analysis. The resulting zeta law provides a principled way to anticipate when scaling data, improving representations, or adding modalities is most likely to accelerate discovery.

Paul M. Thompson

Stevens Institute for Neuroimaging & Informatics · University of Southern California · Los Angeles, CA, USA

Neural Scaling Laws Data-Efficient Learning

Apr 10, 2026

Generalization and Scaling Laws for Mixture-of-Experts Transformers

We develop a theory of generalization and scaling for Mixture-of-Experts (MoE) Transformers that cleanly separates active per-input capacity from routing combinatorics. By conditioning on fixed routing patterns and union-bounding across them, we derive a sup-norm covering-number bound whose metric entropy scales with the active parameter budget and incurs a MoE-specific routing overhead. Combined with a standard ERM analysis for squared loss, this yields a generalization bound under a

d

-dimensional manifold data model and

C^β

targets, showing that approximation and estimation trade off as in dense networks once active parameters are accounted for appropriately. We further prove a constructive approximation theorem for MoE architectures, showing that, under the approximation construction, error can decrease either by scaling active capacity or by increasing the number of experts, depending on the dominant bottleneck. From these results we derive neural scaling laws for model size, data size, and compute-optimal tradeoffs. Overall, our results provide a transparent statistical reference point for reasoning about MoE scaling, clarifying which behaviors are certified by worst-case theory and which must arise from data-dependent routing structure or optimization dynamics.

Mansour Zoubeirou a Mayaki

Univ Lyon, Lyon 2, LIRIS UMR 5205 F-69676, Lyon, France

Mixture-of-Experts Models AI Scaling Laws Neural Scaling Laws

Apr 6, 2026

Towards Scaling Law Analysis For Spatiotemporal Weather Data

Compute-optimal scaling laws are relatively well studied for NLP and CV, where objectives are typically single-step and targets are comparatively homogeneous. Weather forecasting is harder to characterize in the same framework: autoregressive rollouts compound errors over long horizons, outputs couple many physical channels with disparate scales and predictability, and globally pooled test metrics can disagree sharply with per-channel, late-lead behavior implied by short-horizon training. We extend neural scaling analysis for autoregressive weather forecasting from single-step training loss to long rollouts and per-channel metrics. We quantify (1) how prediction error is distributed across channels and how its growth rate evolves with forecast horizon, (2) if power law scaling holds for test error, relative to rollout length when error is pooled globally, and (3) how that fit varies jointly with horizon and channel for parameter, data, and compute-based scaling axes. We find strong cross-channel and cross-horizon heterogeneity: pooled scaling can look favorable while many channels degrade at late leads. We discuss implications for weighted objectives, horizon-aware curricula, and resource allocation across outputs.

Alexander Kiefer, Prasanna Balaprakash, Xiao Wang

Bredesen Center, University of Tennessee, Knoxville, Tennessee · Oak Ridge National Laboratory, Oak Ridge, Tennessee

Weather Forecasting Spatiotemporal Forecasting Neural Scaling Laws Autoregressive Modeling

Mar 26, 2026

On Neural Scaling Laws for Weather Emulation through Continual Training

Neural scaling laws, which in some domains can predict the performance of large neural networks as a function of model, data, and compute scale, are the cornerstone of building foundation models in Natural Language Processing and Computer Vision. We study neural scaling in Scientific Machine Learning, focusing on models for weather forecasting. To analyze scaling behavior in as simple a setting as possible, we adopt a minimal, scalable, general-purpose Swin Transformer architecture, and we use continual training with constant learning rates and periodic cooldowns as an efficient training strategy. We show that models trained in this minimalist way follow predictable scaling trends and even outperform standard cosine learning rate schedules. Cooldown phases can be re-purposed to improve downstream performance, e.g., enabling accurate multi-step rollouts over longer forecast horizons as well as sharper predictions through spectral loss adjustments. We also systematically explore a wide range of model and dataset sizes under various compute budgets to construct IsoFLOP curves, and we identify compute-optimal training regimes. Extrapolating these trends to larger scales highlights potential performance limits, demonstrating that neural scaling can serve as an important diagnostic for efficient resource allocation. We open-source our code for reproducibility.

Shashank Subramanian, Alexander Kiefer, Arnur Nigmetov +3

Lawrence Berkeley National Laboratory · Oak Ridge National Laboratory · ICSI and UC Berkeley +1

Neural Scaling Laws Weather Forecasting

Mar 21, 2026

Problems with Chinchilla Approach 2: Systematic Biases in IsoFLOP Parabola Fits

Chinchilla Approach 2 is among the most widely used methods for fitting neural scaling laws. Its parabolic approximation introduces systematic biases in compute-optimal allocation estimates, even on noise-free synthetic data. Applied to published Llama 3 IsoFLOP data at open frontier compute scales, these biases imply a parameter underallocation corresponding to 6.5% of the

3.8\times10^{25}

FLOP training budget and $1.4M (90% CI: $412K-$2.9M) in unnecessary compute at 50% H100 MFU. Simulated multimodal model misallocations show even greater opportunity costs due to higher loss surface asymmetry. Three sources of this error are examined: IsoFLOP sampling grid width (Taylor approximation accuracy), uncentered IsoFLOP sampling, and loss surface asymmetry (

α\neq β

). Chinchilla Approach 3 largely eliminates these biases but is often regarded as less data-efficient, numerically unstable, prone to local minima, and harder to implement. Each concern is shown to be unfounded or addressable, especially when the partially linear structure of the objective is exploited via Variable Projection, enabling unbiased inference on all five loss surface parameters through a two-dimensional optimization that is well-conditioned, analytically differentiable, and amenable to dense, or even exhaustive, grid search. It may serve as a more convenient replacement for Approach 2 or a more scalable alternative for adaptations of Approach 3 to richer scaling law formulations. See https://github.com/Open-Athena/vpnls for details and https://openathena.ai/scaling-law-analysis for other results from this study.

Eric Czech, Zhiwei Xu, Yael Elmatad +2

Open Athena AI Foundation · Department of Statistics, University of Michigan, Ann Arbor

Scaling Laws Neural Scaling Laws Language Model Scaling Laws

Mar 13, 2026

Scaling Laws and Pathologies of Single-Layer PINNs: Network Width and PDE Nonlinearity

We establish empirical scaling laws for Single-Layer Physics-Informed Neural Networks on canonical nonlinear PDEs. We identify a dual optimization failure: (i) a baseline pathology, where the solution error fails to decrease with network width, even at fixed nonlinearity, falling short of theoretical approximation bounds, and (ii) a compounding pathology, where this failure is exacerbated by nonlinearity. We provide quantitative evidence that a simple separable power law is insufficient, and that the scaling behavior is governed by a more complex, non-separable relationship. This failure is consistent with the concept of spectral bias, where networks struggle to learn the high-frequency solution components that intensify with nonlinearity. We show that optimization, not approximation capacity, is the primary bottleneck, and propose a methodology to empirically measure these complex scaling effects.

Faris Chaudhry

Department of Computer Science, Imperial College London

Physics-Informed ML Neural Scaling Laws Neural Network Optimization Shallow Neural Networks

Mar 12, 2026

Pruning-induced phases in fully-connected neural networks: the eumentia, the dementia, and the amentia

Modern neural networks are heavily overparameterized, and pruning, which removes redundant neurons or connections, has emerged as a key approach to compressing them without sacrificing performance. However, while practical pruning methods are well developed, whether pruning induces sharp phase transitions in the neural networks and, if so, to what universality class they belong, remain open questions. To address this, we study fully-connected neural networks trained on MNIST, independently varying the dropout (i.e., removing neurons) rate at both the training and evaluation stages to map the phase diagram. We identify three distinct phases: eumentia (the network learns), dementia (the network has forgotten), and amentia (the network cannot learn), sharply distinguished by the power-law scaling of the cross-entropy loss with the training dataset size. {In the eumentia phase, the algebraic decay of the loss, as documented in the machine learning literature as neural scaling laws, is from the perspective of statistical mechanics the hallmark of quasi-long-range order.} We demonstrate that the transition between the eumentia and dementia phases is accompanied by scale invariance, with a diverging length scale that exhibits hallmarks of a Berezinskii-Kosterlitz-Thouless-like transition; the phase structure is robust across different network widths and depths. Our results establish that dropout-induced pruning provides a concrete setting in which neural network behavior can be understood through the lens of statistical mechanics.

Haining Pan, Nakul Aggarwal, J. H. Pixley

Department of Physics and Astronomy, Center for Materials Theory, Rutgers University, Piscataway, NJ 08854 USA · Center for Computational Quantum Physics, Flatiron Institute, New York, New York 10010, USA

Phase Transitions in Neural Networks Neural Network Pruning Statistical Physics of Learning Neural Scaling Laws

Mar 11, 2026

Optimal Expert-Attention Allocation in Mixture-of-Experts: A Scalable Law for Dynamic Model Design

This paper presents a novel extension of neural scaling laws to Mixture-of-Experts (MoE) models, focusing on the optimal allocation of compute between expert and attention sub-layers. As MoE architectures have emerged as an efficient method for scaling model capacity without proportionally increasing computation, determining the optimal expert-attention compute ratio becomes critical. We define the ratio

r

as the fraction of total FLOPs per token dedicated to the expert layers versus the attention layers, and explore how this ratio interacts with the overall compute budget and model sparsity. Through extensive experiments with GPT-style MoE Transformers, we empirically find that the optimal ratio

r^*

follows a power-law relationship with total compute and varies with sparsity. Our analysis leads to an explicit formula for

r^*

, enabling precise control over the expert-attention compute allocation. We generalize the Chinchilla scaling law by incorporating this architectural parameter, providing a new framework for tuning MoE models beyond size and data. Our findings offer practical guidelines for designing efficient MoE models, optimizing performance while respecting fixed compute budgets.

Junzhuo Li, Peijie Jiang, Changxin Tian +3

The Hong Kong University of Science and Technology (Guangzhou) · The Hong Kong University of Science and Technology · Ant Group

Mixture-of-Experts Models AI Scaling Laws Neural Scaling Laws

Mar 7, 2026

Scaling Laws in the Tiny Regime: How Small Models Change Their Mistakes

Neural scaling laws describe how model performance improves as a power law with size, but existing work focuses on models above 100M parameters. The sub-20M regime -- where TinyML and edge AI operate -- remains unexamined. We train 90 models (22K--19.8M parameters) across two architectures (plain ConvNet, MobileNetV2) on CIFAR-100, varying width while holding depth and training fixed. Both follow approximate power laws in error rate:

α= 0.156 \pm 0.002

(ScaleCNN) and

α= 0.106 \pm 0.001

(MobileNetV2) across five seeds. Since prior work fit cross-entropy loss rather than error rate, direct exponent comparison is approximate; with that caveat, these are 1.4--2x steeper than

α\approx 0.076

for large language models. The power law does not hold uniformly: local exponents decay with scale, and MobileNetV2 saturates at 19.8M parameters (

α_{\mathrm{local}} = 0.006

). Error structure also changes. Jaccard overlap between error sets of the smallest and largest ScaleCNN is only 0.35 (25 seed pairs,

\pm 0.004

) -- compression changes which inputs are misclassified, not merely how many. Small models concentrate capacity on easy classes (Gini: 0.26 at 22K vs. 0.09 at 4.7M) while abandoning the hardest (bottom-5 accuracy: 10% vs. 53%). Counter to expectation, the smallest models are best calibrated (ECE = 0.013 vs. peak 0.110 at mid-size). Aggregate accuracy is therefore misleading for edge deployment; validation must happen at the target model size.

Mohammed Alnemari, Rizwan Qureshi, Nader Begrazadah

Faculty of Computer Science and Information Technology, University of Tabuk, Tabuk, Saudi Arabia · AIST Research Center, University of Tabuk, Tabuk, Saudi Arabia · Department of Computer Science, Salim Habib University, Karachi, Pakistan +1

Neural Scaling Laws Tiny ML Image Classification

Mar 2, 2026

When Bigger is Worse: A Practitioner's Guide to Model Selection Under Data Scarcity

Scaling laws assume larger models trained on more data consistently outperform smaller ones -- an assumption that drives model selection in computer vision but remains untested in resource-constrained Earth observation (EO). We conduct a systematic efficiency analysis across three scaling dimensions: model size, dataset size, and input resolution, on rooftop photovoltaic (PV) detection in Madagascar, yielding 180 training runs across 60 configurations. Optimizing for model efficiency (mAP

_{50}

per unit of model size), we find a consistent efficiency inversion: YOLO11N achieves the highest efficiency (

22\times

higher than YOLO11X) with no accuracy penalty: it reaches the second highest absolute mAP

_{50}

(0.459), outperforming all models except YOLO11S by a margin smaller than run-to-run variance, directly contradicting the scaling prior. Resolution is the dominant resource allocation lever: moving from 416 px to 1280 px at 10 % of the data matches the efficiency gain of collecting the full dataset at low resolution. These findings are robust to the deployment objective: small high-resolution configurations are Pareto-dominant across all 60 experimental setups in the joint accuracy-throughput space, leaving no tradeoff to resolve. In data-scarce EO, the scaling prior does not just fail: it inverts.

Kwame Mbobda-Kuate, Gabriel Kasmi

ENSAE Paris, Palaiseau, 91120, France. · Centre O.I.E., Mines Paris – PSL University, Sophia-Antipolis, 06904, France.

Model Selection Efficient Object Detection Neural Scaling Laws Data-Efficient Learning

Feb 28, 2026

Efficient Decoder Scaling Strategy for Neural Routing Solvers

Construction-based neural routing solvers, typically composed of an encoder and a decoder, have emerged as a promising approach for solving vehicle routing problems. While recent studies suggest that shifting parameters from the encoder to the decoder enhances performance, most works restrict the decoder size to 1-3M parameters, leaving the effects of scaling largely unexplored. To address this gap, we conduct a systematic study comparing two distinct strategies: scaling depth versus scaling width. We synthesize these strategies to construct a suite of 12 model configurations, spanning a parameter range from 1M to ~150M, and extensively evaluate their scaling behaviors across three critical dimensions: parameter efficiency, data efficiency, and compute efficiency. Our empirical results reveal that parameter count is insufficient to accurately predict the model performance, highlighting the critical and distinct roles of model depth (layer count) and width (embedding dimension). Crucially, we demonstrate that scaling depth yields superior performance gains to scaling width. Based on these findings, we provide and experimentally validate a set of design principles for the efficient allocation of parameters and compute resources to enhance the model performance.

Qing Luo, Fu Luo, Ke Li +1

Guangdong Provincial Key Laboratory of Fully Actuated System Control Theory and Technology, School of Automation and Intelligent Manufacturing, Southern University of Science and Technology, Shenzhen, China. · Department of Computer Science, City University of Hong Kong, Hong Kong SAR, China.

Neural Combinatorial Optimization Neural Scaling Laws Vehicle Routing

Feb 24, 2026

Scaling View Synthesis Transformers

Geometry-free view synthesis transformers have recently achieved state-of-the-art performance in Novel View Synthesis (NVS), outperforming traditional approaches that rely on explicit geometry modeling. Yet the factors governing their scaling with compute remain unclear. We present a systematic study of scaling laws for view synthesis transformers and derive design principles for training compute-optimal NVS models. Contrary to prior findings, we show that encoder-decoder architectures can be compute-optimal; we trace earlier negative results to suboptimal architectural choices and comparisons across unequal training compute budgets. Across several compute levels, we demonstrate that our encoder-decoder architecture, which we call the Scalable View Synthesis Model (SVSM), scales as effectively as decoder-only models, achieves a superior performance-compute Pareto frontier, and surpasses the previous state-of-the-art on real-world NVS benchmarks with substantially reduced training compute.

Evan Kim, Hyunwoo Ryu, Thomas W. Mitchel +1

MIT · Adobe

Novel View Synthesis Neural Scaling Laws

Feb 23, 2026

Scaling Law of Neural Koopman Operators

Data-driven neural Koopman operator theory has emerged as a powerful tool for linearizing and controlling nonlinear robotic systems. However, the performance of these data-driven models fundamentally depends on the trade-off between sample size and model dimensions, a relationship for which the scaling laws have remained unclear. This paper establishes a rigorous framework to address this challenge by deriving and empirically validating scaling laws that connect sample size, latent space dimension, and downstream control quality. We derive a theoretical upper bound on the Koopman approximation error, explicitly decomposing it into sampling error and projection error. We show that these terms decay at specific rates relative to dataset size and latent dimension, providing a rigorous basis for the scaling law. Based on the theoretical results, we introduce two lightweight regularizers for the neural Koopman operator: a covariance loss to help stabilize the learned latent features and an inverse control loss to ensure the model aligns with physical actuation. The results from systematic experiments across six robotic environments confirm that model fitting error follows the derived scaling laws, and the regularizers improve dynamic model fitting fidelity, with enhanced closed-loop control performance. Together, our results provide a simple recipe for allocating effort between data collection and model capacity when learning Koopman dynamics for control.

Abulikemu Abuduweili, Yuyang Pang, Feihan Li +1

Authors are with the Robotics Institute, Carnegie Mellon University, Pittsburgh, PA 15213, USA.

Koopman Operator Learning Robot Dynamics Learning Neural Scaling Laws Robotic Control

Feb 17, 2026

Neural Scaling Laws for Boosted Jet Tagging

The success of Large Language Models (LLMs) has established that scaling compute, through joint increases in model capacity and dataset size, is the primary driver of performance in modern machine learning. While machine learning has long been an integral component of High Energy Physics (HEP) data analysis workflows, the compute used to train state-of-the-art HEP models remains orders of magnitude below that of industry foundation models. With scaling laws only beginning to be studied in the field, we investigate neural scaling laws for boosted jet classification using the public JetClass dataset. We derive compute optimal scaling laws and identify an effective performance limit that can be consistently approached through increased compute. We study how data repetition, common in HEP where simulation is expensive, modifies the scaling yielding a quantifiable effective dataset size gain. We then study how the scaling coefficients and asymptotic performance limits vary with the choice of input features and particle multiplicity, demonstrating that increased compute reliably drives performance toward an asymptotic limit, and that more expressive, lower-level features can raise the performance limit and improve results at fixed dataset size.

Matthias Vigl, Nicole Hartman, Michael Kagan +1

Technical University of Munich · SLAC National Accelerator Laboratory

Neural Scaling Laws AI Scaling Laws Jet Tagging High-Energy Physics

Feb 16, 2026

Scaling Laws for Masked-Reconstruction Transformers on Single-Cell Transcriptomics

Neural scaling laws -- power-law relationships between loss, model size, and data -- have been extensively documented for language and vision transformers, yet their existence in single-cell genomics remains largely unexplored. We present the first systematic study of scaling behaviour for masked-reconstruction transformers trained on single-cell RNA sequencing (scRNA-seq) data. Using expression profiles from the CELLxGENE Census, we construct two experimental regimes: a data-rich regime (512 highly variable genes, 200,000 cells) and a data-limited regime (1,024 genes, 10,000 cells). Across seven model sizes spanning three orders of magnitude in parameter count (533 to 3.4 x 10^8 parameters), we fit the parametric scaling law to validation mean squared error (MSE). The data-rich regime exhibits clear power-law scaling with an irreducible loss floor of c ~ 1.44, while the data-limited regime shows negligible scaling, indicating that model capacity is not the binding constraint when data are scarce. These results establish that scaling laws analogous to those observed in natural language processing do emerge in single-cell transcriptomics when sufficient data are available, and they identify the data-to-parameter ratio as a critical determinant of scaling behaviour. A preliminary conversion of the data-rich asymptotic floor to information-theoretic units yields an estimate of approximately 2.30 bits of entropy per masked gene position. We discuss implications for the design of single-cell foundation models and outline the additional measurements needed to refine this entropy estimate.

Ihor Kendiukhov

Department of Computer Science, University of Tübingen

Neural Scaling Laws Transformer Single-Cell Transcriptomics

Feb 16, 2026

Divine Benevolence is an $x^2$ : GLUs scale asymptotically faster than MLPs

Scaling laws can be understood from ground-up numerical analysis, where traditional function approximation theory can explain shifts in model architecture choices. GLU variants now dominate frontier LLMs and similar outer-product architectures are prevalent in ranking models. The success of these architectures has mostly been left as an empirical discovery. In this paper, we apply the tools of numerical analysis to expose a key factor: these models have an

x^2

which enables asymptotically faster scaling than MLPs. GLUs have piecewise quadratic functional forms that are sufficient to exhibit quadratic order of approximation. Our key contribution is to demonstrate that the

L(P)

scaling slope is

L(P)\propto P^{-3}

for GLUs but only

L(P)=P^{-2}

for MLPs on function reconstruction problems. We provide a parameter construction and empirical verification of these slopes for 1D function approximation. From the first principles we discover, we make one stride and propose the ``Gated Quadratic Unit'' which has an even steeper

L(P)

slope than the GLU and MLP. This opens the possibility of architecture design from first principles numerical theory to unlock superior scaling in large models. Replication code is available at https://github.com/afqueiruga/divine_scaling.

Alejandro Francisco Queiruga

Neural Scaling Laws Neural Network Approximation Theory Multilayer Perceptrons

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 7, 2026

Deriving Neural Scaling Laws from the statistics of natural language

Despite the fact that experimental neural scaling laws have substantially guided empirical progress in large-scale machine learning, no existing theory can quantitatively predict the exponents of these important laws for any modern LLM trained on any natural language dataset. We provide the first such theory in the case of data-limited scaling laws. We isolate two key statistical properties of language that alone can predict neural scaling exponents: (i) the decay of pairwise token correlations with time separation between token pairs, and (ii) the decay of the next-token conditional entropy with the length of the conditioning context. We further derive a simple formula in terms of these statistics that predicts data-limited neural scaling exponents from first principles without any free parameters or synthetic data models. Our theory exhibits a remarkable match with experimentally measured neural scaling laws obtained from training GPT-2 and LLaMA style models from scratch on two qualitatively different benchmarks, TinyStories and WikiText.

Francesco Cagnetta, Allan Raventós, Surya Ganguli +1

Theoretical and Scientific Data Science, International School for Advanced Studies (SISSA), Trieste, Italy · Department of Applied Physics, Stanford University, Stanford, California · Department of Physics and Astronomy, Johns Hopkins University, Baltimore, Maryland +1

Language Model Scaling Laws Neural Scaling Laws

Dec 15, 2025

From Zipf's Law to Neural Scaling through Heaps' Law and Hilberg's Hypothesis

We inspect the deductive connection between the neural scaling law and Zipf's law -- two statements discussed in machine learning and quantitative linguistics. The neural scaling law describes how the cross entropy rate of a foundation model -- such as a large language model -- changes with respect to the amount of training tokens, parameters, and compute. By contrast, Zipf's law posits that the distribution of tokens exhibits a power law tail. Whereas similar claims have been made in more specific settings, we show that the neural scaling law is a consequence of Zipf's law under certain broad assumptions that we reveal systematically. The derivation steps are as follows: We derive Heaps' law on the vocabulary growth from Zipf's law, Hilberg's hypothesis on the entropy scaling from Heaps' law, and the neural scaling from Hilberg's hypothesis. We illustrate these inference steps by a toy example of the Santa Fe process that satisfies all four statistical laws.

Łukasz Dębowski

Institute of Computer Science, Polish Academy of Sciences, ul. Jana Kazimierza 5, 01-248 Warszawa, Poland

Zipf's Law Neural Scaling Laws

Jun 16, 2025

Random Matrix Theory for Deep Learning: Beyond Eigenvalues of Linear Models

Modern Machine Learning (ML) and Deep Neural Networks (DNNs) often operate on high-dimensional data and rely on overparameterized models, where classical low-dimensional intuitions break down. In particular, the proportional regime where the data dimension, sample size, and number of model parameters are all large and comparable, gives rise to novel and sometimes counterintuitive behaviors. This paper extends traditional Random Matrix Theory (RMT) beyond eigenvalue-based analysis of linear models to address the challenges posed by nonlinear ML models such as DNNs in this regime. We introduce the concept of High-dimensional Equivalent, which unifies and generalizes both Deterministic Equivalent and Linear Equivalent, to systematically address three technical challenges: high dimensionality, nonlinearity, and the need to analyze generic eigenspectral functionals. Leveraging this framework, we provide precise characterizations of the training and generalization performance of linear models, nonlinear shallow networks, and deep networks. Our results capture rich phenomena, including scaling laws, double descent, and nonlinear learning dynamics, offering a unified perspective on the theoretical understanding of deep learning in high dimensions.

Zhenyu Liao, Michael W. Mahoney

School of Electronic Information and Communications, Huazhong University of Science and Technology, Wuhan, China · ICSI, LBNL, and Department of Statistics, University of California, Berkeley, USA

Random Matrix Theory Neural Network Generalization Neural Network Training Dynamics Neural Scaling Laws High-Dimensional Asymptotics