Scaling Laws

Existing machine learning (ML) scaling laws relate predictive loss to compute, model parameters, and data. However, as models are increasingly deployed through agentic harnesses, loss alone is insufficient to characterize downstream performance: models with similar loss can exhibit different capabilities in reasoning, retrieval, planning, and adaptation. Yet, no unified framework connects such capabilities to the coupled resources available across the ML lifecycle. We bridge this gap by introducing a capability manifold, a multidimensional framework mapping downstream capabilities to pre-training, post-training, and test-time resources through bounded scaling functions. Analytical Jacobians quantify capability sensitivity to resource changes and interactions. As an initial application, we embed Kaplan- and Chinchilla-type scaling laws and test-time compute within the framework, demonstrating how existing scaling relationships can be unified as trajectories on a common capability manifold.

Clinical B-mode images are widely available as potential data sources for quantitative ultrasound (QUS) analysis for tissue characterization. However, standard clinical ultrasound devices apply unknown log-compression to RF envelope data before display and storage. Previous work has demonstrated estimation of the underlying RF envelope statistics in the presence of an unknown compression law. Using Fisher information analysis, we show that finite-offset log compression causes severe information loss when estimating the Rayleigh scale $σ$, which controls diffuse speckle. For a single image window, unknown compression raises the minimum achievable variance for unbiased estimation of $σ$ by a compression-independent factor of approximately $\FisherMinInflation$. When $M$ equal-sized windows share the same unknown compression settings, the excess variance decays as $1/M$; even in the most favorable regime, reducing the variance inflation factor below $1.1$ requires $\FisherBestCaseWindows$ windows. Our analysis treats the contrast parameter $a$ as unknown and the boundary offset $b$ as known; estimating $b$ experimentally shows even larger variance. We validate this theory using synthetic estimation experiments and demonstrate RF-scale recovery on real RF-envelope windows from the OASBUD dataset. Together, these results clarify the limitations of using routine B-mode images for QUS.

Scaling laws predict how loss decreases with increases in computation. We show, contrary to conventional wisdom, that architectural interventions can modify scaling exponents in pre-training, leading to power-law improvements in performance as computation increases. As an anchoring point, we consider the architectural formulation of looped transformers. Although not typically used in this way, looping, also known as recursive depth, provides a mechanism for model growth, by increasing the number of loops during training. Model growth, with and without shared weights, provides the biggest changes to the scaling exponents. In particular, a 7.4B model growth architecture matches GPT-3 13B on CORE with roughly $20\times$ less compute, and has compute efficiency gains that increase with scale. Moreover, simply using a boundary operator in a vanilla transformer, which normalizes and injects an earlier block, also provides an exponent increase, although to a lesser extent. In the data-constrained, multi-epoch setting, standard looping has a useful regularizing effect, where we find it is compute-optimal to increase the number of loops with scale. These results can be understood through the lens of computational depth: for a given computational budget, we wish to increase the usable depth of the transformer, which can lead to efficiency gains that increase with scale.

The standard geometric derivation of neural scaling exponents takes the intrinsic dimension of a data manifold as its input. On modular addition in $\mathbb{Z}_p$ that derivation has no input. The exact algebraic solution is an orbit of $\mathbb{Z}_p$ acting by isometries. Transitivity alone makes the ratio statistic underlying the standard dimension estimator a point mass, so the estimator is undefined, and here the two nearest neighbour distances coincide exactly. Breaking the symmetry at scale $ε$ returns a number, but one that tracks $1/ε$ with no scale free plateau. We show that the failure is general, since on any finite orbit of a group acting by isometries the estimator reports the resolution at which the set is probed rather than a dimension. What replaces the power law is exponential in hidden width, $L(h)=L_\infty+A\exp(-c\,h^α)$, with $R^2$ between 0.982 and 0.995 against 0.857 to 0.906 for a power law admitting the same floor and fitted under the same protocol. Where the data supply is sufficient the rate belongs to the regulariser rather than to the group, since weight decay moves $c$ by a factor of 47 while group order moves it by 1.10, a residual below seed to seed resolution, for every fixed $α$ between 0.75 and 2. The critical width falls with group order rather than rising, against capacity counting that assigns a fixed number of neurons to each irreducible representation.

Learning-based surrogates for AC optimal power flow (ACOPF) promise large speedups over classical solvers, but their operational value depends on physical feasibility as much as predictive accuracy. Physics-aware objectives such as the augmented Lagrangian (AL) improve constraint satisfaction at additional per-step cost, yet how this trade-off behaves with scale is uncharacterized. We sweep model and dataset sizes under both MSE and AL training, and characterize how constraint violation changes with network size across grids. Both objectives improve as power laws, but at different rates: MSE is governed primarily by model capacity, while AL is balanced across both. Violation grows roughly twice as fast with network size under MSE as under AL. On matched hardware, AL reduces violation by nearly $30\times$ for an order of magnitude more training time, with negligible added memory. The training objective determines not only where a surrogate lands but how its quality evolves with scale.

Neural networks trained past memorization frequently undergo a delayed transition to generalization, a phenomenon known as grokking. Despite theoretical progress on \emph{why} this transition occurs, the quantitative structure of \emph{when} it occurs in hyperparameter space remains uncharacterized. We map the memorization-to-generalization boundary across 384 configurations of two-hidden-layer MLPs on modular arithmetic, fitting a power-law scaling relation for generalization onset time: $T_{\mathrm{grok}} \propto H^{-0.27}\, D^{-2.04}\, \eta^{-0.50}\, \lambda^{-0.64}$ ($R^2 = 0.732$; $0.821$ with interactions). The exponent hierarchy reveals that data complexity ($D^{-2.04}$) is the dominant driver of regime transition, not model capacity ($H^{-0.27}$): doubling data accelerates generalization by ${\sim}4\times$, while doubling width yields only ${\sim}1.2\times$. A sharp phase boundary at weight decay $\lambda \gtrsim 1.0$ separates grokking from non-grokking configurations, and weight norm trajectories show monotonic compression during the transition, consistent with implicit regularization selecting low-complexity solutions. These results provide a quantitative foundation for predicting and controlling regime transitions in overparameterized networks.

Mixture-of-Experts (MoE) models expand model capacity without a proportional increase in training compute, but increasing sparsity makes reliable hyperparameter transfer challenging. In this work, we show that conventional hyperparameter scaling laws are insufficient for ultra-sparse MoEs: the optimal learning rate and batch size vary with activation ratio, and these shifts cannot be explained by either total or activated parameter count alone. To characterize this dependence, we conduct 1,800 pre-training runs spanning six activated-parameter scales and models with up to 6B total non-embedding parameters, processing approximately 20 trillion tokens at a cost of 200,000 equivalent H800 GPU-hours. Our results reconcile conflicting findings in prior work by revealing two scaling regimes. At fixed sparsity, the optimal batch size follows a power-law relationship with training tokens $D$, whereas the optimal learning rate scales with training compute $C$ and remains robust to the allocation between model size and data. Across sparsity levels, the activation ratio $A$ enters both relationships as an additional multiplicative power-law factor. These observations lead to unified hyperparameter scaling laws that transfer across MoE sparsity levels. Large-scale evaluation shows that the scaling form outperforms alternative functional forms. On a held-out ultra-sparse MoE with 12B total parameters and only 1/64 of its experts activated, the predicted hyperparameters remain close to the observed optima, supporting joint extrapolation across model scale and sparsity. Further experiments demonstrate transfer across expert granularities and isolate the effect of activation ratio from that of total expert count.

We study the training dynamics of multiclass logistic regression on high-dimensional Gaussian mixture models with a large number of classes and establish precise scaling laws governing the cross-entropy risk under gradient-based optimization. We show that learning proceeds sequentially across classes, from most to least frequent. When the class priors follow a power law distribution, the risk dynamics decompose into three phases: an initial plateau until the first class is learned, a power-law decay regime during which sequential learning occurs, and a final convergence regime. We then analyze how model capacity interacts with optimization under a fixed compute budget. When the effective dimension is restricted via projection onto leading principal components, the risk decomposes into a capacity term (a power law in the retained dimension) and an optimization term (a power law in training time). Optimizing this tradeoff yields a compute-optimal scaling law for logistic regression, with explicit prescriptions for model size and training time as functions of compute. These results extend theoretical scaling laws from linear regression to multiclass classification, while connecting to empirical scaling laws observed in large-scale neural networks.

Associative Recall (AR) is the cognitive ability to learn and retrieve links between items in memory. In NLP, AR is used as a benchmark for evaluating the in-context memory capacity of architectures such as Mamba, and has been found to strongly correlate with language modeling performance. This paper explores AR from the perspective of mechanistic interpretability, aiming to reverse-engineer the exact internal algorithm used by Mamba to perform recall. Our key insight is that Mamba performs recall by implicitly learning linear hash functions, and we identify the low-level circuit that enables this behavior. Building on these findings and inspired by theoretical tools in similarity-preserving hashing, such as the Johnson-Lindenstrauss lemma, we develop a theoretical framework for analyzing AR, which we term Recall Scaling Laws. Given the vocabulary size and the number of facts in context, this framework allows us to (1) predict the embedding and state dimensions required for Mamba to achieve perfect recall, (2) predict recall success probability given the model dimensions, and (3) analyze multi-layer models and multi-head SSM patterns. Empirical results show that our theoretical findings are accurate and predictive, offering insights into how AR capacity scales with vocabulary, state, embedding size, and architecture.

Existing theories derive neural scaling from data geometry or a specified data-model spectrum, but systems trained on the same data can scale differently when architecture or optimization changes the representations they can efficiently reach. We introduce Coupled Scaling, a task-conditioned framework in which finite-budget scaling depends on the relation between task structure and the geometry accessible to an architecture-optimization system. In a solvable mode-truncation model, loss separates into target energy outside architectural support and an unresolved supported tail. For an arbitrary priority order, the residual lies between the best-N supported tail and the tail beyond the largest completed high-value prefix. If the cumulative-tail and coverage log-rates are $γ_{A,T}$ and $ρ_{A,O,T}$, the residual exponent lies in $[ρ_{A,O,T}γ_{A,T},γ_{A,T}]$. Under bounded off-prefix gain, the completed prefix is rate-determining and $α_{A,O,T}=ρ_{A,O,T}γ_{A,T}$; for $a_{A,T,j}\asymp j^{-b_{A,T}}$, this gives $α_{A,O,T}=ρ_{A,O,T}(b_{A,T}-1)$. A fixed-kernel specialization derives the training-time exponent from the near-zero tail of a task-weighted spectral measure defined independently of the loss fit. The framework separates architectural support from finite-budget acquisition and motivates two tests: static task-relevant geometry should track loss at a common budget, while multiscale geometry should track coupling-specific exponent ordering, including reversal across contrasting tasks. An audit of released emergence trajectories identifies the controls needed for a direct factorial test that measures geometry separately from the scaling fit.

Scaling laws in modern deep learning describe how held-out loss improves as model capacity, training data, and compute increase, often following power-law trends. We investigate whether analogous scaling regularities arise in actuarial ratemaking, where data are tabular, heterogeneous, and noisy, and where classical models such as GLMs remain strong baselines. Using a real-world motor insurance portfolio, we train models from different families across increasing fractions of the training data and multiple random seeds, evaluating out-of-sample Poisson deviance, a likelihood-based loss for Poisson count predictions in which lower values indicate better held-out fit. We find that all model families improve with additional data, but scaling exponents differ substantially: TabM exhibits markedly stronger data scaling than purely supervised tabular Transformers and standard MLP baselines. Transformer variants show weak parameter scaling unless augmented with additional inductive biases (TabM-style adaptation or self-supervision). These results provide quantitative guidance on model selection by data regime and suggest that effective scaling on actuarial tabular tasks depends on architecture and loss function objective design, with simple increases in Transformer size providing limited gains.

Optimal hyperparameter scaling laws describe how the best hyperparameters for large language model (LLM) training change with model and data scale, enabling practitioners to predict optimal configurations at production scales without expensive large-scale tuning. However, estimating these scaling laws conventionally requires exhaustive grid searches over thousands of training runs, consuming enormous computational resources. We introduce Power-Law Entropy Search (PLES), a computational cost-aware acquisition function built on multi-fidelity Bayesian optimization that efficiently estimates optimal hyperparameter scaling laws through adaptive experimentation. A key innovation in PLES is that it searches for candidates that reduce the overall uncertainty of a scaling law estimate, instead of optimizing a single objective function. At each iteration, PLES selects the candidate configuration that maximally reduces the uncertainty of the scaling law estimates per unit computational cost, naturally favoring informative small-scale experiments. We evaluate PLES on synthetic benchmarks, surrogate models fitted to real LLM training data, and actual LLM pre-training runs. Across all settings, PLES converges to accurate optimal hyperparameter scaling laws using less than one-tenth of the computational budget required by conventional grid search and other baselines.

Robotic manipulation faces a fundamental scaling challenge: robust generalization demands broad physical experience, yet action-labeled robot trajectories are expensive to collect and inherently limited in diversity. Egocentric videos offer a far more scalable source of embodied experience, capturing object interactions, contact dynamics, tool use, and long-horizon behaviors across diverse environments. The central challenge is how to convert this abundant but action-free experience into effective robot control. We introduce ZimaBlue, a scalable framework for learning generalizable World Action Models (WAMs) from large-scale video. ZimaBlue follows a three-stage training curriculum: it first performs causal embodied video pre-training on large-scale human and robot egocentric videos, then grounds the learned visual dynamics in heterogeneous robot trajectories through video-action mid-training with a unified action representation, and finally specializes the model to a target robot for deployment. To make generative WAMs practical for real-time control, ZimaBluefurther adopts an asynchronous Slow-Fast dual-system architecture, where a high-capacity Slow world model provides generalizable spatiotemporal representations and a lightweight Fast branch enables 30 Hz action prediction on NVIDIA RTX 4090. On real-robot zero-shot evaluations, scaling from target-robot data alone to over 120,000 hours of embodied video improves success from 36.1% to 77.8%. ZimaBlue further delivers strong performance across multiple benchmarks, with particularly pronounced gains on unseen tasks.

The downwash wake of a hovering quadrotor governs both the vehicle's own performance and the safe spacing of multi-rotor formations. Prior measurements have largely characterized the mean flow, using single-point anemometry, volumetric tracking, or planar cuts through part of the rotor system. Higher-order turbulent statistics of the merged wake, and how they relate to canonical jet scaling, have remained unresolved, particularly for small quadrotors at the low-Reynolds-number end of the size range. Here, we present a detailed particle image velocimetry (PIV) study of the downwash of a hovering Crazyflie 2.1 quadrotor (arm length, $l = 46$ mm), sampled along a diagonal cut, passing through rotors along the symmetry axis of the quadrotor, and a front-rotor cut, passing through adjacent rotors. The four rotor jets merge into a single column by $z/l \approx 5$, beyond which the mean velocity profiles progressively approach the canonical round-jet self-similar form, collapsing by $z/l \approx 13$ when scaled by the local centerline velocity and half-width. Centerline decay and half-width growth follow canonical scaling laws with an effective source diameter $D_\text{eff} = 2.29\,l$, effective Reynolds number $Re_{D_\text{eff}} = 3 \times 10^4$, at the low end of the range over which canonical jet scaling has been established, and spreading and decay constants nonetheless within the canonical round-jet range. Resolving both cuts shows that the turbulent normal stresses retain a bimodal, cut-dependent signature of the four-rotor source throughout the measurement domain.

Proof assistants such as Lean 4 promise the paradigm of formally verified mathematics, but large-scale formalization projects have faced major barriers to entry, including the need for expertise in formal verification (as well as the underlying mathematics) and the significant time required for writing formal proofs. AI coding agents have dramatically reduced these barriers; human users can now use natural language to prompt agents to write complex proofs in Lean. This opens up the intriguing possibility of internet-scale mathematical collaboration involving both humans and AI agents, where correctness is machine-checked. To realize this possibility, we introduce Prove2Me (https://prove2.me), an open collaborative platform for formalizing mathematics. Users launch formalization "missions", to which AI agents contribute formal proofs toward completion. We designed mechanisms and a specialized harness in Prove2Me that enable large-scale collaboration so that agents can build on one another's work and freely reuse existing results. In doing so, Prove2Me aims to turn math formalization into a scalable, crowd-sourced effort open to anyone with an agent.

We study the scaling behavior of learning rate and batch size in pretraining dense large language models on English-prevalent corpora. Beyond scaling jointly optimal learning rates and batch sizes, we investigate their marginal evolution with model capacity and data scale and develop a model that captures these relationships. As we employ a Warmup-Stable-Decay learning rate schedule, we further investigate the gains from learning rate annealing over a broad range of hyperparameters settings, models and data budgets, and whether the optimal learning rate and batch size transfer between the stable and decay phases. Finally, we characterize the dependence of loss on model capacity and dataset size, evaluating recently proposed scaling forms that explicitly model their interaction. We find these approaches particularly effective at capturing both undertraining and overtraining regimes across our experiments. This study establishes a first baseline and scaling procedure for the development of future OpenEuroLLM models. We open-source the complete collection of pretraining runs used in this study.

Language model pretraining has become almost synonymous with prohibitive cost, placing it out of reach for much of the academic and open-source communities. Although strong open-source efforts already exist, including open-weight models and open-source training recipes, a cost-efficient, hardware-accessible, and open-source pretraining recipe has long been missing. Even at a small scale, training Llama-3.2-3B costs over $1.5M, and reproducing SmolLM3-3B needs over $700K. In this report, we present an open pretraining recipe designed to lower this barrier. Using this recipe, we train a collection of Puro-2B models from scratch on up to 1.4 trillion tokens with FP8 precision on consumer-grade RTX 5090 GPUs. The models in the collection differ in token budgets and selected recipe variants. Our best model is trained at a compute cost of less than $6.9K and approaches Qwen2.5-1.5B performance under our evaluation protocol. This cost efficiency is enabled by a combination of approaches, including hardware selection, low-precision training, hyperball optimization, curriculum model averaging, and the data recipe. Beyond the recipe itself, we provide two additional results. First, across the Puro-2B collection, we derive a Puro Cost Scaling Law that relates training cost to average model performance; the fitted law suggests that about $4.4K, less than $5,090, is sufficient to reach the performance of Qwen2-1.5B. Second, as an end-to-end case study, we examine how pretraining data curricula shape downstream performance after post-training. Such controlled studies are enabled by having access to the full pretraining pipeline rather than model weights alone. We release the full training recipe for Puro-2B, including data, code, and model weights under Apache 2.0 at https://huggingface.co/collections/thu-pacman/puro-2b.

Artificial Intelligence (AI) safety systems combine character shaping (e.g., Reinforcement Learning from Human Feedback [RLHF], Constitutional AI), which modifies behavioral distributions at training time, with rule enforcement (e.g., output filters, safety classifiers), which blocks harmful outputs at inference time, yet little formal analysis exists on how their optimal balance should change as deployment scales increase. We introduce a stylized comparative-statics model that parameterizes safety design as a resource allocation alpha in [0,1] between these two approaches, incorporating scale-dependent filter degradation, common-mode failures, and character fragility -- the risk that shaped behavior degrades or collapses under novel conditions. Under a multiplicative Pareto damage model, we derive closed-form expected harm and supplement it with tail-risk (CVaR) analysis via Monte Carlo simulation. Across three scenarios (optimistic, moderate, pessimistic), the optimal alpha* is interior or at the rules-only boundary and shifts weakly toward character shaping as deployment scale T grows, from negligible (Delta alpha* = +0.01) to pronounced (Delta alpha* = +0.21) depending on scenario. The dominant parameter is the baseline character fragility rate p^(0)_frag, which shifts alpha* by 0.50 across its range -- far exceeding the effect of tail severity, filter quality, or common-mode failure probability. CVaR and expected-harm optima converge at large T. These results suggest that safety architecture decisions depend less on deployment scale per se than on the reliability of character shaping under distributional shift.

Neural networks trained by gradient descent on a smooth cost function can nevertheless learn in steps: the cost holds on long plateaus and then drops abruptly. Meanwhile, training losses instead follow smooth power laws. Variants of both behaviors occur in architectures with very different microscopic structures, which is the signature of a few relevant collective variables. We show that a symmetry fixes what those variables are: a network layer is a sum over interchangeable units, so relabeling the units leaves it unchanged; given smoothness and the condition that a unit's gradient vanish at the origin, symmetry then enforces a universal leading form for the expansion about the near-zero weights present at the start of training, the quadratic $\Tr[WW^{\top}A(x)]$, in which every architectural detail is confined to a single structure matrix" $A(x)$ that we compute for each architecture. Perceptrons, attention layers, mixtures of experts, and convolutions become one model at different $A$. Its training dynamics then close on the order parameter" $M=WW^{\top}$ and, whenever the data matrices share an eigenbasis, reduce to a Lotka--Volterra equation whose modes switch on one after another. The smaller the initial weights, the further apart the switch-on times, and the plateaus appear as a singular limit of a smooth flow; when many modes are unresolved the same events merge into a power law in training time whose exponent the theory predicts. We confirm both numerically across training methods and architectures.

Scaling laws promised cost-effective experiments; six years later, they have yet to fully deliver. Instead, researchers have found them unreliable at small scales (starting at 4M parameters) and concluded that sizable models cannot be avoided. We show this is not the case: the confounding factor is hyperparameters. Small models are highly sensitive, but hyperparameter sensitivity fades with scale. This small-scale sensitivity makes scaling laws easy to miss because they only emerge on the fully tuned frontier, and reaching that frontier requires an extensive search far beyond what most ever run. By ablating the basic scaling law recipe, we show well-tuned hyperparameters matter more than any other ingredient. Further, we reveal why those hyperparameters become easier to find: as scale increases, the hyperparameter loss surface becomes lower dimensional. Nevertheless while scaling laws exist in small models, extrapolation hits statistical limitations. A holistic approach is required. Synthesizing our insights with the recent literature, we develop a new methodology for model-centric research and demonstrate it on a question that once took the field years to settle: where to place normalization layers in the transformer architecture. From small-scale experiments, we recover the large scale result: pre-normalization works better as models grow in size. With the right tools and a better understanding, small-scale experiments can deliver on scaling laws' long-awaited promise.

In a multi-agent system, there are often stubborn followers of specific opinions or beliefs. Motivated by this observation, in this paper, we aim to understand how stubborn agents affect the distribution of opinions in a network where both stubborn and non-stubborn agents interact with each other. To do so, we assume that all agents have an opinion in the set $\{0,1\}$ and each non-stubborn agent updates its opinion according to the $2k$\textit{-choices rule}, where the agent samples $2k$ neighbours (including both stubborn and non-stubborn neighbours) uniformly at random and adopts the majority opinion among the sampled group of neighbours and itself. We assume that a proportion of agents, $γ_i$, are stubborn followers of opinion $i\in \{0,1\}$. It is natural to expect that the steady-state distribution of the opinions in the network will be dominated by the opinion with the larger proportion of stubborn followers. We show that while this is true, the time to reach steady-state depends heavily on the values of the parameters $γ_0$ and $γ_1$. When the individual values of these parameters, as well as their difference, are small, it can take an exponentially long time (in the network size) to reach the steady-state. In sharp contrast, when at least one of the parameters $γ_0$ and $γ_1$ is large, the network reaches the steady-state in a time that is only logarithmic in the network size. Hence, there exists a sharp phase transition in the network dynamics based on the proportions of stubborn agents. We also characterise the behaviour of the system when the parameters $γ_0$ and $γ_1$ lie on the boundary of the phase transition. In this boundary region, we show using Stein's method that the dynamics are driven by a diffusion process which takes polynomial time to mix.

Accurate pre-deployment estimation of CNN inference cost--energy, latency, and peak memory--is increasingly critical as models are deployed on resource-constrained GPU platforms. Existing approaches rely on FLOPs, latency measurements, or single-device profiling as energy proxies, overlooking the non-linear interactions between architectural design and hardware load. We present a workload characterization study of 13 419 CNN configurations on two GPU platforms (RTX 5090 and RTX 3080) under GPU telemetry, revealing that energy, latency, and memory exhibit fundamentally distinct scaling behaviors: energy and latency diverge by 3x under high computational demand, and cross-GPU transferability differs by target--energy and latency require platform-specific models while memory transfers well across the two tested platforms. Building on these characterization findings, we develop CARB, a cascade-blended ensemble that jointly predicts all three targets with R2 ~0.99, and a two-stage deployment screening workflow that eliminates over 90% of candidates in seconds, reducing large design spaces to a Pareto-prioritized shortlist validated against real hardware.

Neural scaling laws are foundational for language model development, yet standard formulations systematically under- and overestimate loss at data-scarce and overtraining extremes. This failure originates in the underlying assumption that model size and training data impact the loss independently. To address this, we introduce the Skaling law, a generalized functional form that couples model capacity and data through a single interaction exponent. This simple extension reduces the Mean Absolute Percentage Error (MAPE) by 1.5-3x across both interpolation and extrapolation regimes. When paired with a sparse grid strategy restricted to low-compute regimes, the Skaling law achieves accurate full-grid extrapolation using approximately 10x less compute than uniform sweeps. By enabling reliable performance prediction from small-scale experiments, the Skaling law provides a more robust and resource-efficient framework for allocating compute budgets in next-generation model training.

Diffusion language models (dLLMs) offer an alternative to autoregressive (AR) language modeling, yet the scaling behavior of Mixture-of-Experts (MoE) dLLMs remains poorly understood. We systematically characterize how optimization hyperparameters, compute allocation, and architecture scale for MoE dLLMs, identifying quantitative differences from scaling trends previously reported for AR models. Specifically, for optimization, the optimal nominal batch size grows faster, while the optimal learning rate decays more rapidly with compute. For model--data allocation, IsoFLOP analysis reveals a slight data-side tilt: the optimal token budget grows faster than activated model-side computation. For MoE architecture, larger scales increasingly favor larger expert pools at fixed activated capacity, while moderate expert granularity remains consistently effective and the preferred fraction of activated capacity assigned to shared experts remains stable across scales. Guided by these findings, we train LLaDA MoE v2, a 30B-A3B dLLM, from scratch on 23.5T tokens. With approximately 65% as many pretraining tokens as Qwen3, LLaDA MoE v2 approaches Qwen3 on several knowledge, reasoning, and coding benchmarks. After supervised fine-tuning alone, it outperforms SDAR Chat on seven of eight reasoning and coding benchmarks and remains close to Qwen3 on several tasks. These results establish practical scaling laws and design principles for MoE dLLMs.

We study a self-supervised generation task for single-cell gene expression vectors: given a set of vectors from a cell type, we aim to generate additional gene expression vectors of that cell type. For this task we characterize both the biological fidelity of the generated gene expression vectors and the scaling behavior of the pretraining loss. The model is a causal transformer paired with a learned quantized VAE tokenizer, trained with a cross-entropy loss. To evaluate the model, we condition it on held-out gene expression vectors of a cell type and generate vectors of gene expression, comparing the resulting distribution over gene expression vectors to the ground truth distribution of that cell type. We study the scaling properties of the proposed architecture by varying the number of trained parameters and the amount of training data. To our knowledge, we find the first jointly-fit two-exponent scaling law and compute-optimal frontier for a single-cell foundation model. Finally, we discuss how this pretrained model could be finetuned for perturbation response prediction.

Detecting that a stream of high-dimensional embeddings has changed is usually framed as a choice of statistic. We give a scale law that constrains any moment-based choice and test it against topological alternatives. The law: certifying a feature of spatial scale eps carrying mass fraction f requires polynomial tests of degree N* >= log(1/f)/(2 eps), proved via the Chebyshev extremal problem; a Gauss-quadrature construction gives N* >= 4b-1 for a b-scale topology, so cost is set by feature fineness, not feature count. The law is one-sided: we exhibit an annulus whose mean, covariance and all fourth-order moments equal those of a filled disk, yet H_1 is nonzero. Its practical content is a calibration rule. The upper bound is attained by Gaussian test functions, the RKHS witness of an RBF kernel, so the law predicts which bandwidth an MMD test should use: the feature scale. On real embedding streams we measure sigma*/eps with median 1.12 (IQR 1.01-1.52, n=26) over three settings and three scales, and a data-driven bandwidth reaches AUC >= 0.95. Against an adversary optimised against the defender's statistics (mean, covariance, k-NN, kurtosis), only a bandwidth-matched kernel test still detects. For persistent homology the verdict is mixed and depends on choices usually left implicit. The summary matters more than the filtration: total persistence attains recall 0.75 at FPR 1% where the first persistence landscape attains 0.00. What survives is a cost gap, not a power gap: where persistence works it costs 116x kurtosis, which works at least as well. We conclude not that topological summaries are useless, but that on this task a kernel test whose bandwidth the law sets dominates them.

The discovery of scaling laws has highlighted the extraordinary potential of AI systems with a striking empirical pattern: as AI systems scale, their capabilities tend to improve predictably. Yet, in real-world applications, AI rarely operates in isolation; instead, it often works alongside humans, raising the question of whether these gains persist in human-AI collaboration. In this work, we develop an analytical model to examine when the empirical scaling benefits of AI translate into improved human-AI joint system performance. We demonstrate that the performance of a human-AI system can scale positively as the AI scales up-provided that humans have an accurate perception of the AI's capabilities. Human misperception, however, can fundamentally alter this relationship: i) when humans over-perceive the AI's capabilities, a scaling paradox may arise, in which greater AI scale reduces overall system performance and amplifies firm-level profit losses, and (ii) when humans under-perceive the AI's capabilities, performance still improves with scale but at a substantially slower rate. We further show that firms can actively manage these distortions through operational policies such as cost internalization and perception alignment, whose effectiveness depends on the economics of AI deployment and the direction of human misperception. These findings suggest that organizations may benefit more from managing the human-AI interface than from simply investing in larger, more expensive AI systems. More broadly, our results suggest that AI scaling should be viewed not only as a technological challenge, but also as a behavioral and operational one, and caution against the view that larger AI systems will automatically lead to better operational outcomes. Whether AI scaling creates value ultimately depends on how increased AI capabilities shape human beliefs and collaborative efforts.

We study empirical scaling properties for text conditioning in visual generation. Such properties have rarely been measured because diffusion loss does not scale with the number of tokens in natural-language prompts. Surprisingly, we find that the converged diffusion loss scales with the amount of structured language in the prompt. To quantify structured language, we adapt two complementary measures: a white-box likelihood metric (GPG) and a black-box attribute metric (ED). Across controlled training runs, the converged diffusion loss decreases approximately linearly with GPG and follows a power law with ED. Guided by these scaling properties, we improve \emph{diffusability} by constructing structured prompts with semantic and geometric annotations derived from images, and improve \emph{promptability} by training a prompter through supervised fine-tuning, cold-start, and verifier-gated on-policy distillation. The resulting system outperforms all evaluated open-weight models on nearly every compositional, reasoning, and world-knowledge benchmark, while matching or surpassing the strongest closed-weight models on most evaluations.

Modern deep learning typically keeps the batch size static throughout training, thus overlooking the joint effect of learning rate and batch size on the training dynamics. In this paper, we study the deep learning dynamics through the lens of convex optimization and derive a joint characterization of loss in terms of both schedules, applicable to general optimizers and model architectures. This characterization yields a closed-form optimal batch size schedule for any prescribed learning rate schedule, and further leads to joint scaling laws that consistently outperform static batch size baselines, highlighting the significance of dynamic batch size schedule in large language model training.

Classical compute-optimal scaling laws assume an unbounded supply of fresh pretraining data, yet pretraining is increasingly entering a regime in which compute grows faster than the availability of high-quality data. We propose Compute-Data (CD) scaling laws, a unified framework that bridges compute-optimal scaling, where data scales freely with compute, and data-optimal scaling, where the corpus is fixed while compute can grow without bound. CD scaling extends classical scaling laws by introducing a token-effectiveness function, $η$, which quantifies the value of a derived token-produced, for example, through multi-epoch repetition or paraphrasing-relative to a fresh token, ranging from a perfect substitute to having no value. We fit $η$ for two data-expansion strategies, multi-epoch repetition and paraphrasing, across model sizes from 14M to 600M parameters using the Dolma-3 corpus. We find that token effectiveness is far from constant: it depends jointly on model size, the tokens-per-parameter ratio, and the amount of derived data, and it saturates as the corpus is expanded. The functional form of $η$ implies diminishing returns when substituting compute for data as either model size or data availability increases. It also partitions training into three operational regimes---compute-bound, data-bound, and model-bound---and shows that classical compute-optimal allocation is suboptimal across most practically relevant settings.

Human language exhibits lawful structure at the level of words (frequency, vocabulary growth) and word pairs (co-occurrence across distance). Here we show that the arrangement of words in sequence -- a central determinant of meaning -- obeys a comparable law. Using large language models as probabilistic probes, we measured the reduction in target perplexity conferred by prior context at distance d beyond that of the same words scrambled; this difference, the contextual persistence function P(d), isolates the influence of arrangement. Across ten corpora spanning six language families and written and spoken modalities, P(d) decayed approximately as 1/d ($P(d) \propto d^{-α}$, mean $α= 1.04$; median $r^2 = 0.96$). The effect vanished in scrambled and synthetic controls, replicated across independent probes, and did not appear in genomic or protein sequences under domain-native models. An exponent near 1 distributes contextual influence approximately uniformly across logarithmic timescales. The results establish a scaling law of contextual persistence in human language.

The discovery of scaling laws has motivated training neural networks on ever increasing quantities of data. This is typically done with a constant decoupled weight decay which causes the network weights to shrink steadily over the course of training. Taking inspiration from the Robbins--Monro conditions, we propose to scale weight decay by the fraction of the peak learning rate $η/η_{\max}$. We prove that this scaled weight decay preserves the asymptotic stationarity guarantees of the corresponding unregularized methods for both stochastic gradient descent and the non-Euclidean spectral optimizer Muon, thereby avoiding the additional asymptotic bias introduced by constant decoupled weight decay. This retains the stability benefits of weight decay without changing the asymptotic optimization target. Using a steady-state analysis, we explain why under standard weight decay the weight norm shrinks steadily as training proceeds, whereas under scaled weight decay it settles to a roughly constant value. When applied to the training of mixture-of-experts models, Muon with scaled weight decay (Muon-SW) consistently outpaces Muon with identical hyperparameters, reaching the same validation loss $\mathbf{30\%}$ faster at our largest scale across models from $72 - 930$ million parameters trained at $\sim 600$ tokens per active parameter. If this trend continues to hold, the method promises to substantially accelerate the pre-training of frontier models while requiring only a few lines of code to implement.

The largest machine learning models in particle physics are also the most expensive to train, yet the return on scaling a given architecture cannot be estimated before that compute is spent. Scaling laws have been fit for jets, but none has yet been shown to predict the performance of models it was not fit on. We show that, for a generic transformer pretrained on collider jets, it can be forecast. Fitting a joint model-and-data scaling law on small models alone, spanning three orders of magnitude of training compute, we predict the loss of models trained afterward with more than one hundred times more compute to within one percent. We then connect the forecast to downstream physics performance: across two standard tagging benchmarks, lower pretraining loss yields systematically lower fine-tuning loss and higher background rejection after fine-tuning. Within this model family and these tasks, a compute budget can therefore be translated into expected physics performance before any large model is trained. The final frontier model is consistent with the published numbers for current state-of-the-art physics-aware foundation models trained on the same corpus, on accuracy, AUC, and quark/gluon rejection, with a residual edge for the physics-aware model only in the high-purity tail of top tagging. We release five pretrained models spanning multiple sizes, together with the complete training recipe and code.

While scaling laws for imitation learning have primarily focused on generalization in open-world settings, the relationship between data and precision in closed-world tasks like robotic assembly remains largely unexplored. This paper systematically investigates this relationship and introduces a novel scaling law. We find that to achieve a fixed success rate, the required number of demonstrations $N$ grows super-exponentially as the target precision $P$ approaches a limit $c$. This relationship is accurately captured by the model $\log(N) \propto 1/(P-c)$. Crucially, we reveal that the limit precision $c$ is not a static physical constant of the task but an emergent property of the entire agent system, including its sensors and expert policy. Through experiments on canonical manipulation tasks, we validate this law and demonstrate that improving system components, such as adding a wrist camera or using a more effective expert, measurably lowers $c$, thus expanding the system's achievable precision. Our work provides a new theoretical framework for precision in robotics and a quantitative metric to evaluate system capabilities. Furthermore, these findings provide a practical methodology for guiding the development and debugging of high-precision manipulation systems.

Prior classical-ML learning-curve work fits power laws to tree, linear, and kernel models on tabular data, but at small scale: typically one curve, one team, a handful of cells. We present a distributed classroom-scale replication: 127 graduate students each ran a fixed protocol on 3 assigned datasets, drawn from 18 tabular classification and regression datasets and 6 model families (Boosting, Random Forest, SVM, Linear/Logistic, Ridge, Lasso), yielding 11,536 training runs and 1,648 fitted power-law curves of the form error(N) = a N^(-b) + c. Three findings. (1) Power laws fit: R^2 > 0.8 on 77.7% of cells, with tree ensembles dominating at full data (Boosting 50% of datasets, RandomForest 33%; linear models underperform on classification). (2) Approximate shared exponents within a model family: for 5 of 6 families, a single family-level exponent predicts each family's cross-dataset curves nearly as well as per-dataset exponents (R^2 gap < 0.011), though AIC favors the unconstrained fit and curve collapse is partial (32-58% of points within +/-0.5 dex). We frame this as approximate predictive compressibility, not dataset-independent universality; Lasso fails outright (negative control) and Ridge is fragile under leave-one-dataset-out. (3) Replicator-implementation variance: with random_state=42 fixed, independent re-implementations of the same protocol still differ by mean CV(b) = 0.144 on the fitted exponent -- not seed variance, but the spread induced by unconstrained parts of the protocol (preprocessing, encoding, missing-value handling). We release the aggregated curves, per-cell fits, and a practical data-requirement table for N* to reach target error 0.15.

Injecting factual knowledge into large language models (LLMs) reliably and at scale remains an open challenge. Hypernetworks provide a promising solution to large-scale knowledge injection. Although hypernetworks are typically applied for test-time adaptation, we explore their use in train-time knowledge injection, where, given a large corpus of facts, we train a hypernetwork to generate a fixed LoRA adapter that, when inserted into the target model, enable the model to answer questions about those facts. In this work, we investigate whether hypernetworks can be used to perform train-time knowledge injection and how this ability varies with scale. The scaling behavior of hypernetworks remains largely unstudied. Our design decouples the hypernetwork's injection capacity from the target model's general capability, enabling, for the first time, a rigorous study of scaling laws for hypernetwork architectures. We characterize how loss, reasoning accuracy, and out-of-distribution (OOD) generalization vary with hypernetwork depth, width, and target network size. We construct a large-scale dataset, called MegaWikiQA, containing tens of millions of multi-hop question-answer examples across 39 domains constructed from examples in Wikidata5M. Our results reveal: (i) hypernetwork-based injection exhibits broadly predictive power law scaling along all architecture axes; and (ii) hypernetworks are capable of reliable OOD generalization at increasing scales, suggesting that hypernetwork provides a promising alternative to other train-time adaptation methods such as LoRA finetuning and full fine-tuning, exhibiting steeper scaling exponents in all OOD evaluations. Together, these results establish hypernetworks as a principled and scalable substrate for train-time adaptation, and provide the first empirically grounded scaling laws to guide hypernetworks for factual reasoning in large language models.

The Internet taught us that the value of a network depends on \emph{how} its nodes connect: broadcast stars scale as $V\!\propto\!N$ (Sarnoff), fully-connected meshes as $N^2$ (Metcalfe), and group-forming networks as $2^{N}$ (Reed). We ask the analogous question for networks of AI agents. We model the net value of connection as a function of coordination-group size, derive from it the properties an optimal collaboration protocol must have, and introduce ANet Patu-1 -- a self-organizing consensus protocol in which the network continuously re-forms its own coalitions, adaptively riding the upper envelope of all three regimes at $O(1)$ parallel consensus rounds. To measure value without opinion-grading, we score an emergent protocol by formally specifying it and deriving its complexity, the way distributed algorithms are analyzed. Two results follow. (i)~Emergence -- a crowd of the \emph{cheapest} model, when heterogeneous, starts weak but its collective value compounds with $N$ and \emph{overtakes} a crowd of a far \emph{stronger} model that is homogeneous: a crossover that marks a scaling law for collaboration rather than for scale. (ii)~Reflexivity -- a heterogeneous network, given only its own problem and no design hints, converges on ANet Patu-1 itself, reconstructing the high-dimensional law that governs its own connective value.

Quantization is a powerful strategy to build capable and resource-efficient large language models (LLMs) by reducing the bitwidth of the parameters. While quantized LLMs achieve state-of-the-art performance on unperturbed inputs using standard predictive metrics, their performance on perturbed inputs, measured using reliability metrics, remains underexplored, despite its importance for reliable deployment. To address this gap, we first conduct a comprehensive reliability evaluation of quantized LLMs consisting of three key components: (1) Uncertainty: We assess the trustworthiness of LLMs quantized to 2, 3, 4, and 8 bits using six different quantization methods, employing established uncertainty metrics. (2) Calibration: We assess how well-calibrated the uncertainty estimates of quantized models are across model scales and bit precisions. (3) Robustness: We design character-level and word-level input perturbations to evaluate the reliability of quantized models under semantically-preserving variations in the inputs that arise in real-world applications. Second, we characterize how reliability scales with the total number of model bits. Our study reveals that while the performance scales monotonically with the total number of bits, the reliability scalings are nonlinear. A reliability peak occurs for 4-bit quantized models, indicating that quantizing moderately sized models offers the best reliability-efficiency trade-off. Additionally, our empirical findings reveal that quantization enhances the robustness of LLMs to natural input perturbations.

Brain field potentials are scale-free: their power spectra follow a $1/f^β$ law whose aperiodic exponent $β$ tracks cortical state, and sleep depth in particular is a shift in $β$. We ask whether a transformer endowed with an explicit renormalization-group (RG) inductive bias the RG-Flow Transformer, which couples ordinary self-attention to a scale-aware stream with a learnable anomalous dimension $γ$, block-spin coarse-graining, and an entropy-gated synchronization bridge has an advantage over a parameter-matched vanilla transformer on \emph{real, scarce} EEG. Using the PhysioNet Sleep-EDF corpus with a strict leakage-free by-subject hold-out, we (i) benchmark RG-Flow against a param-matched vanilla transformer and a hierarchy-only ablation on 5-class AASM sleep staging, (ii) sweep the per-subject data budget to look for the inductive-bias crossover predicted when data are scarce, and (iii) test whether RG-Flow's learned $γ$ tracks the measured spectral exponent $β$ out-of-sample a quantity the vanilla model does not possess. Across $5$ subjects and $5$ seeds under leave-one-subject-out cross-validation, RG-Flow and the vanilla transformer are statistically indistinguishable on 5-class staging (77.3% vs 77.0% accuracy; paired $p=0.294$), and the predicted scarce-data crossover does not appear: vanilla is numerically ahead at every data-limited budget. What does separate the models is interpretability RG-Flow recovers the continuous spectral exponent out-of-sample ($β$-recovery $R^2 = 0.416$), a capability the vanilla architecture has no analogue for.

We study how unsupervised autoencoders trained on microscopic spin configurations from the Ising model learn macroscopic, theory-relevant variables underlying the data-generating process. Without embedding domain knowledge, we mimic a typical discovery setting: We quantify learning across multiple spatial (coarse-graining) scales and reveal two distinct dynamical regimes controlled by main hyperparameters (model depth, width, and learning rate) -- a magnetization-dominated regime and an energy-dominated regime characterized by trade-offs in their representation quality. The first regime is a transitory state exhibiting dynamical scaling and fluctuations that follow an ordering-to-scale; the second gradually shifts resolution towards smaller scales relevant for the energy representation. Deep models trained at moderate and fast rates become arrested before reaching these regimes. With a novel analysis of recursive-dynamic trajectories, we demonstrate that prediction errors induce flow fields that produce a common trajectory topology across all representation spaces. A dynamical viewpoint of learning is established in which intrinsic properties expose the effects of forced changes in representation during training. We utilize the intuition that learning operates as a process driven far from equilibrium by fluctuations from the training data and optimizer to provide an interpretive basis grounded in both the physical world and the machine models that represent it.

In this short note we consider the gradient descent dynamics of deep scalar linear networks, $f(x) = \prod_{l=1}^L w_l x$, which enjoy exact time-course solutions for any integer depth. We show that even in this minimal model, the optimal depth-wise learning rate scaling depends on data, whereas data-agnostic scaling rules fail to transfer across depths. Under the data-dependent optimal scaling, the learning dynamics is independent of data and weakly dependent on depth, resulting in a constant linear convergence rate across all depths including infinity. We further show similar data-dependent effects in deep scalar linear networks with residual connections.

Pretraining scaling laws reveal that model capability improves predictably with data and compute. But learning from real world environments after deployment remains far less understood. Analyzing roughly 38,000 hours of agent interaction with the environment across 134 real world tasks, we find, to the best of our knowledge, the first evidence that overall performance during environment learning follows a log-sigmoid scaling law with remarkably high precision, reaching R^2 = 0.998. Across model generations, we also find that agent learning speed roughly doubles every three months. This discovery stems from EdgeBench, a suite of 134 real world tasks with ultra-long horizons, spanning scientific discovery, software engineering, combinatorial optimization, professional knowledge work, formal mathematics, and interactive games. Each task sustains at least 12 hours of continuous agent operation under rich, multilevel feedback, and is built through substantial expert effort. We publicly release 51 tasks and our full evaluation framework to accelerate the study of how agents learn from real world experience.

We present a bidirectional framework for estimating the energy consumption of text-to-video (T2V) and text-to-video-audio (T2VA) models from architectural first principles and observable generation parameters such as resolution and duration, requiring no access to weights, model size, or implementation details. Forward, it predicts energy from generation parameters and architectural principles; backward, it recovers architectural scaling behavior from observed inference times, with accuracy serving as a criterion for architectural validity. Building on the established compute-bound nature of video diffusion models, we demonstrate that each model's energy profile obeys theoretically derived scaling laws, decomposing into quadratic and linear terms whose coefficients directly reflect the underlying architectural complexity. Validated across six open-source models spanning 8.3B-27B parameters and three GPU configurations, this decomposition achieves below 3% MAPE across all architectures. This approach offers a standardized, empirically and theoretically grounded framework for sustainability benchmarking across T2V models and architectures.

Synthetic data can be scaled along two routes: Source Expansion (SE), which enlarges the source by adding seed materials or generators, and Fixed-Source Synthesis (FSS), which holds the source fixed and scales the generation budget. Existing scaling studies typically expand the source as the data grows, conflating SE with FSS and leaving FSS underexplored. We isolate FSS by holding the seed-question pool and teacher model fixed, varying only the per-question response budget under Rejection Sampling (RS). We adapt the rectified scaling law to FSS, deriving it from how repeated sampling covers a fixed source. Empirically, the derived form, fit on low budgets, predicts performance at the held-out highest budget for every evaluated teacher--student pair. At matched total-sample budgets, SE and FSS are comparable at small budgets; at large budgets, adding seed questions outperforms spending the same budget on more responses. Within FSS, however, neither synthesizing additional questions from the existing seeds nor varying the synthesis protocol outperforms plain RS at matched budgets. FSS is thus a bounded scaling axis and a controlled setting for comparing synthesis protocols. We will release our code and data to facilitate further research.

We propose a scaling law that takes into account model size and training data while explicitly splitting the latter into training steps and batch size (called three-term law). Fitting the proposed law on a large set of training runs, we find that it correctly recovers the scaling of the optimal batch size. Moreover, because it makes use of training runs with suboptimal batch size, our proposed law can be robustly fit with a significantly smaller amount of training runs. We further show that the three-term law can be used to derive scaling laws for suboptimal batch sizes, and that it matches previous empirical findings related to the critical batch size.

High-throughput RLHF systems often decouple rollout generation from policy optimization, leading to the use of stale rollouts during learner updates. In this work, we study the effect of such staleness in asynchronous GRPO. We make the behavior policy explicit in the GRPO surrogate objective and distinguish between the surrogate-gradient mapping used by the learner and the true total derivative of a distribution-dependent population objective. Under assumptions of local boundedness, distributional smoothness, and behavior-policy smoothness, we show that stale rollouts introduce a per-step surrogate-gradient bias of order O(S * eta), where S denotes the maximum rollout lag and eta denotes the learning rate. We further derive a conditional collapse-time scaling law: when within-cycle drift remains below a batch-level clipping radius, collapse is governed primarily by cumulative learner drift T * eta; when the stale-rollout constraint is active, stability instead depends explicitly on S * eta. This yields a two-constraint stability condition eta << min{R_batch / (S * G_upd), R_crit / (T * G_upd)}, explaining why the maximum stable learning rate may appear weakly dependent on staleness in the horizon-limited regime.

Grid-based approaches to approximate nearest neighbor (ANN) search have been absent from modern scaling analyses. We present a systematic characterization of a multiprobe grid algorithm with respect to dataset size $N$ and dimensionality $d$. Our experiments reveal a previously unreported $d$-scaling crossover on the GloVe embedding family, in which multiprobe grid search maintains an approximately constant dimensional scaling exponent while other graph-, tree-, and partitioning-based methods exhibit degrading throughput. The advantage comes with near-linear query scaling in $N$, but also with lower indexing cost than competing ANN methods. Our results suggest that grid-based methods such as multiprobe grid may be competitive in rebuild-heavy or high-dimensional settings where indexing cost and dimensional robustness dictate performance. More broadly, recent work has formalized self-attention as an ANN operation. Thus, the $N$- and $d$-scaling properties of ANN algorithms may guide cost analysis of efficient transformer architectures. Code is available at: https://github.com/weiz345/MultiProbeANN.

Low-rank adaptation (LoRA) is commonly viewed as an update-space approximation to full fine-tuning, yet this view is incomplete for self-gated Transformer feed-forward networks. In gated FFNs, a low-rank residual can change not only projected features but also the nonlinear selection weights that determine which channels contribute to the output. We formalize this effect as selection misalignment and connect it to the local effective homogeneity of self-gated activations. This motivates a nonlinearity-aware principle for parameter-efficient fine-tuning: low-rank updates should allocate capacity to gate channels whose nonlinear states remain responsive and should shape the temporal evolution of selection. We propose NA-LoRA, a training-only method with two lightweight mechanisms: a derivative-based temporal-importance mask for gate-related LoRA updates and an activation-specific step-scaling rule when a meaningful coarse effective-homogeneity partition is available. NA-LoRA adds no auxiliary loss and incurs no inference-time overhead. Experiments on language-model fine-tuning and vision-language transfer benchmarks show that NA-LoRA consistently improves over vanilla LoRA and is competitive with or better than strong PEFT variants.

The Bitter Lesson, which posits that general-purpose methods that scale with computation and data ultimately outperform those with built-in human knowledge, has become a dominant paradigm in the era of Large Language Models. We revisit this principle by observing a new and critical scaling dimension: the duration of the Feedback Information Loop (FIL), the time required for a system to receive a verification signal after generating a prediction. Most historic successes in Artificial Intelligence (AI) have benefited from near instantaneous feedback (e.g., games or classification tasks), but we argue that future AI applications in science and the physical world will inherently involve FILs ranging from hours to weeks. This trend poses a fundamental scaling limit, as obtaining enough verification steps required by purely data-driven methods becomes practically impossible. Additionally, we propose a method that is orthogonal to purely data-driven approaches, based on human-inspired expert knowledge. The method relies on inductive biases and constraining the solution space. We provide an initial validation of the hypothesis and the method, by studying the real-world GPU programming task, a domain with non-trivial FIL, and demonstrate that incorporating inductive biases yields superior performance over data-driven approaches. The code is released under: https://github.com/ai-nikolai/robust_kernelbench

Delayed generalization (\ie~grokking) refers to the phenomenon in which a neural network fits its training data early in training but only begins to generalize after a prolonged delay, often through an abrupt transition. Despite extensive empirical study, its underlying mechanism remains poorly understood. In this work, we first theoretically characterize a shell--core topological configuration of the reachable solution space induced by Adam's optimization dynamics with weight-shrinkage regularization, supported by empirical evidence. This optimization-induced topological configuration gives rise to grokking. In model's parameter space, random initialization solutions concentrate on a thin outer spherical shell, enclosing another spherical shell of memorization solutions, which in turn contains a core corresponding to the generalization solutions. Leveraging stopping-time theory, we then analyze the geometry of this topological configuration and the solution transition time at which optimization trajectories escape the memorization manifold and first reach the boundary of the generalization manifold. Our theoretical analysis derives grokking scaling laws for the learning rate, batch size, and $\ell_2$ regularization coefficient, which are further validated through experiments and shown to recover results from prior literature.

Language model loss follows remarkably regular scaling laws over model and data size, yet it remains unclear why the aggregate loss should exhibit a power-law form. Existing explanations often attribute this regularity to a heavy-tailed spectrum of pattern difficulty in natural language, but this view has not been directly validated at token-level granularity in large-scale real-data training. We present a token-level framework that decomposes scaling laws into localized learning events of individual contextualized tokens. By fitting token loss trajectories with sigmoids, we show that token learning is concentrated in localized transitions, giving rise to a learning-time spectrum that dominates the scaling-law shape. Across more than one hundred pre-training runs on large and diverse real-language corpora with modern LLM architectures, scaling up to 6B parameters and 300B training tokens, the measured learning-time spectrum quantitatively reconstructs the validation loss derivative along the training-step $T$, data-scale $D$, and model-scale $M$ axes. We further show that the same signal is actionable: by reshaping the training distribution according to when tokens become learnable, we alter the optimization trajectory and achieve 11% faster validation-loss reduction. These results provide direct empirical evidence that scaling laws are governed primarily by the distribution of token-level learning times, and that this distribution can be used not only to explain scaling behavior but also to improve training performance.

Learning-rate transfer can reduce the cost of training large language models: instead of sweeping learning rates at target scale, practitioners extrapolate from smaller runs. Existing approaches often assume that the optimal learning rate follows a log-linear scaling law in data scale and model size. We carefully examine and evaluate this scaling law. In our empirical study of GPT-2--style models from 22M to 707M parameters trained on 5B to 100B tokens, the optimal learning rate develops upward curvature at larger scales, leading to inaccurate extrapolation. We find that this curvature largely disappears when learning rates are replaced by effective learning rate (the step size in normalized weight space), and when data $D$ extrapolation is used instead of model size $N$ extrapolation. Next, we explain nonlinearity in scaling: weight-norm converges to equilibrium slower when optimal learning is small, requiring a larger step size to reduce the transient phase. Experiments with AdamH, which directly controls the effective learning rate, further support this explanation.

Understanding how performance scales jointly with model size and data is a central problem in modern machine learning. Existing theoretical works on scaling laws typically describe generalization as a function of data or compute, often in fixed-feature or infinite-width regimes and for online SGD. Here, we instead study how generalization scales with the number of trainable parameters and the number of samples in a feature-learning model. We analyze $\ell_2$-regularized empirical test error minimization in a quadratic two-layer network in a finite-sample setting with structured data. This setting allows for an explicit characterization of the generalization error as a function of the number of samples, model width, and regularization. Our results reveal a phase diagram with distinct scaling regimes as the number of parameters varies. In particular, the generalization error follows data-dependent power laws controlled by the spectral structure of the target. We further characterize the transitions between regimes, including the onset of interpolation, and their impact on generalization.

Scaling laws describe how learning performance varies with model size, data size, and compute. While recent theoretical work has established scaling laws for sketched linear regression, much less is understood for contrastive representation learning. In this paper, we study a sketched linear model for contrastive learning under a paired Gaussian latent-variable setup. The learner observes only sketched views of two correlated variables and trains a bilinear contrastive score by full-batch empirical gradient descent. We analyze a Gaussian-negative quadratic contrastive surrogate under aligned power-law spectra and a contrastive source condition, where we derive a risk decomposition into irreducible risk, approximation error, GD bias, GD variance, and a cross term. The cross term is controlled by the bias and variance and therefore does not affect the upper-bound scaling. Our main theorem gives an explicit scaling law with respect to sketch dimension $M$, sample size $N$, and effective optimization horizon $L_{\mathrm{eff}}γ$. Compared with standard linear-regression scaling laws, the contrastive setting must learn interactions between two views, and this changes how optimization and finite-sample noise scale with model size, data, and training time. This provides a first theoretical step toward understanding scaling behavior in contrastive learning and gives guidance for balancing model size, data, and optimization compute.

Sphere neural networks have achieved symbolic level syllogistic reasoning without training data, raising the question of where the limit of the scaling law for logical reasoning lies, i.e., whether data-driven machine learning systems can achieve the same level by increasing training data and training time. We show two methodological limitations that prevent supervised deep learning from reaching the symbolic-level syllogistic reasoning: (1) training data can not distinguish all 24 types of valid syllogistic reasoning; (2) end-to-end mapping from premises to conclusion introduces contradictory training targets between neural components for pattern recognition and logical reasoning. Beside theoretical analysis, we experimentally illustrate that Euler Net cannot achieve rigorous syllogistic reasoning. We further challenge the most recent ChatGPTs (GPT-5-nano and GPT-5) to determine the satisfiability of syllogistic statements in four surface forms (patterns): words, double words, simple symbols, and long random symbols, showing that surface forms affect the reasoning performance and that ChatGPT GPT-5 may reach 100% accuracy but still provide incorrect explanations. As empirical training processes are stopped after achieving 100% accuracy, we conclude that supervised machine learning systems will not attain the rigour of symbolic logical reasoning.

We study whether a scaling-law-style inference-compute frontier appears in limit order book prediction. Using FI-2010 and a suite of models ranging from small decision trees to neural LOB architectures, we find that the realized empirical frontier of predictive loss versus structural forward work is well summarized by a power law. In particular, with MLPLOB held out as an architecture family, a power-law fit to the low- and mid-compute non-MLPLOB frontier extrapolates across multiple orders of magnitude and attains $R^2=0.941$ on the excluded high-compute MLPLOB target frontier. A similar exercise in latency space gives substantially weaker results, showing that latency is not merely noisy compute. We use this gap to motivate FastBiNLOB, a dense axis-separable LOB mixer built from hardware-friendly temporal and feature mixing operations. In a five-seed experiment, FastBiNLOB exceeds the published $y_{10}$ and $y_{100}$ macro-F1 targets at notably lower latency than existing published SOTA architectures.

Neural scaling laws describe how pre-training loss decays as power laws with training time, model size, and compute. This position paper argues that the exponents of these power laws are fixed by generic mechanisms: a one-third time scaling due to the strong nonlinearity of Softmax, an inverse width scaling due to representational superposition, and an inverse depth scaling due to ensemble averaging of Transformer layers. These mechanisms are robust to a wide range of data structures and architectural details, placing current large language models in a universality class with fixed exponents. The coefficients, however, are expected to be sensitive to data and architecture details, and directly determine practical quantities such as the optimal model shape and the compute-optimal frontier. We therefore argue that understanding the coefficients is the key to near-term performance improvements, and that a closer examination of the current universality class may reveal pathways to better universality classes.

Large Language Models (LLMs) achieve strong performance across a growing range of domains, yet their scale poses deployment challenges in applications where latency and cost constraints are critical. This paper derives empirical scaling laws for domain-specific LLM compression, quantifying how in-domain and general knowledge performance scale with dataset size, compression ratio, supervision format, and iterative pruning schedule. Using quantitative finance as our application domain, we compare logit-based and LoRA-based distillation under iterative structural pruning, introducing a blended chain-of-thought supervision loss that stabilizes KL-divergence distillation over reasoning traces. In-domain task quality degrades predictably under compression while general-knowledge benchmarks collapse well before the same point; supervision format is the key driver of this tradeoff, with chain-of-thought supervision actively recovering general knowledge that pruning erases. We release the headline dataset FinHeadlineMix, scaling law results, and practical recommendations to provide a reusable framework for domain-specific compression decisions.

Language models are running out of high-quality training data, and even aggressively deduplicated corpora retain some amount of repetition. Earlier controlled studies predated Chinchilla-style scaling laws and could only measure the cost of repetition indirectly. We revisit repetition in the Chinchilla era, using a fitted no-repetition scaling law to report Compute-Equivalent Gain and Compute-Equivalent Loss. We show that under this modernized paradigm, repetition damage is systematic in three ways. First, holding compute allocated to repeated data constant, eval loss peaks at an intermediate repeat count $\Rep$; repeating a moderately sized subset a moderate number of times damages performance more than repeating a large subset a few times or a small subset many times. Second, the location of this peak is well-fit by a power law in model size; this scaling law reveals that the most damaging number of repeated data grows more quickly than compute. Finally, when repeated documents consume 10% of the FLOPs budget in a controlled exact-document repetition setting, the compute-equivalent loss can be large: on FineWeb-Edu-Dedup, the most damaging repeat count for a Qwen3-style 344M-parameter model at $\OT=1$ matches the loss of a no-repetition run using 67% of the FLOPs. We demonstrate that these phenomena are not language-model-specific, and can be analytically understood in a simple statistical model: a misspecified linear regression with verbatim duplicates reproduces the same qualitative loss peak, quantifying how such peaks can arise from a statistical tradeoff between memorization and generalization. Our findings add precision to the study of duplication in language models, allowing practitioners to quantify the wasted compute incurred by the presence and repeat structure of duplicates in pretraining corpora.

We discuss reasons why the scaling exponents of current Large Language Models (LLMs) applications are indicating an unsustainable regime in terms of energy resources. We further show that attributing the smallness of such exponents to a numerical bias due to the neglect of a non-zero value of the loss function in the limit of infinite data (``pedestal effect") does not remove the unsustainability issue. Finally, the effects of the smoothness (roughness) of the data on the scaling exponents is commented upon based on an analogy with phenomenological models of fluid turbulence.