AI Scaling Laws

We investigate the scaling laws and compute-optimal training of physiological waveform foundation models (FMs). We train Aether, a family of over one hundred FMs ranging from 20M to 2.1B parameters, on up to 36.3M hours of physiological waveforms. We construct eight clinical prediction tasks from MIMIC-III and evaluate the FMs through linear probing. The 720M FM outperforms all existing baseline FMs across all eight tasks. A scaling law of model size, pretraining hours, and labeled patients predicts downstream ranking error, i.e. $1-\mathrm{AUROC}$, effectively with $0.5\%$ prediction MAE at held-out resource scales and $0.9\%$ MAE when extrapolating to 2.1B parameters. We present three findings: (1) Compute-optimal training scales both FM size and pretraining hours. Under the fitted law, a $10.0\times$ increase in compute FLOPs scales model size by $1.2\times$ and pretraining hours by $8.2\times$. (2) Larger FMs use waveform data more efficiently, and greater pretraining exposure increases the benefit of model scaling. Starting from 25M parameters and 4.8M pretraining hours, doubling FM size reduces the predicted hours needed for the same performance by $51.8\%$. (3) Pretraining and clinical supervision reinforce each other: more labeled patients increase the return to pretraining, while larger FMs and longer pretraining reduce labeling requirements. For the example of the 720M FM, extending pretraining from 120K to 36.3M hours reduces the predicted patient requirement by $61\%$ at a target ranking error. These findings provide a quantitative training recipe and a promising and durable scaling path for physiological waveform modeling and downstream clinical prediction.

Looped transformers and Mixture-of-Experts (MoE) offer complementary routes to efficient scaling: recurrence increases computational depth at fixed parameters, while MoE sparsity expands total capacity at fixed active compute. Yet existing scaling laws model recurrence or sparsity in isolation. In this work, we introduce Loop Scaling Laws, the first scaling law to jointly model recurrence and sparsity alongside model size and data. At its core is a bounded, sparsity-conditional recurrence mapping that characterizes the effective-parameter gain from looping and how sparsity raises this gain. The laws predict the held-out loss of looped models more accurately than prior alternatives, and recover the standard dense and MoE scaling laws as special cases. Beyond prediction, the fitted laws provide a principled foundation for designing looped MoE models under compute and memory constraints. Downstream evaluations further demonstrate the complementary benefits of the two axes: sparsity delivers ~3x active-parameter efficiency, recurrence yields ~2x total-parameter efficiency on reasoning, and joint scaling further advances the performance frontier. As a practical extension, we show these gains hold at trillion-token scale: at matched training compute, a looped MoE with law-derived recurrence matches a ~2x larger non-looped MoE on the reasoning benchmarks, while enabling test-time scaling through recurrence.

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.

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.

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.

The deep learning revolution, kicked off by AlexNet, taught us that end-to-end training beats decomposing a problem into hand-designed stages. Generative modeling, however, has remained the exception-despite generative models being remarkably capable, they are still not trained end-to-end. This is because, at its core, generative modeling is about handling distributions with many modes, and existing scalable approaches handle this the same way, by factoring the generation procedure, which prevents end-to-end generation. In this work, we introduce Explorative Modeling, a new paradigm that instead factors the training loop, exploring K candidate matches between model generations and data, and training on the best, so predictions commit to modes rather than blurring them. We find Explorative Models (XMs) useful in two settings. First, increasing exploration adds a third pretraining axis beyond parameters and data for existing generative models-where scaling exploration monotonically improves performance across both continuous and discrete domains (images, video, and language). Notably, gains from exploration increase with scale, climbing from 7% to 36% as data scales and from 13% to 23% as models grow, with efficiency gains more than doubling at 3x the compute. Concretely, exploration improves FLOP efficiency by 4.1x, sample efficiency by 6.2x, parameter efficiency by 47%, lifts the strongest of image-generation recipes to a near-state-of-the-art 1.43 FID on ImageNet without guidance, enables scaling how end-to-end existing models are, and unlocks scaling generalization. Second, XMs enable end-to-end reconstructive generative modeling, matching diffusion on control tasks with 16-256x fewer inference steps. Together, these results establish XMs as both a new pretraining axis for existing generative models and a standalone end-to-end generative modeling paradigm.

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.

Pixel-wise Earth-observation (EO) foundation models are now achieving state-of-the-art performance via generated spatial embeddings. However, how these models scale and how best to spend a pretraining budget remain poorly understood. We present the largest controlled scaling study for EO to date: 395 training runs on 1,024 GH200 superchips within a fixed pixel-wise Barlow Twins family, each evaluated on 15 downstream tasks. We find that pretraining loss barely predicts downstream performance (|Pearson r| < 0.2), so selecting models by loss wastes a large share of the compute. We also find that, as the training budget grows, the encoder and the data should grow together while the projector stays fixed, which gives a simple rule for allocating compute. Using this rule, we train a family of pixel-wise models (0.5B and 1B, with a 2B model in training) and distill them into compact students for embeddings-as-data deployment. The 21-million-parameter distilled TESSERA v2-1B-M in aggregate outperforms all open and proprietary models tested, some of which are orders of magnitude larger. These students produce Matryoshka representations that are inexpensive to serve: a 16-dimensional prefix keeps 92% of the full 128-dimensional performance at 1/8 of the storage. Upon completion of training we plan to release v2 global embeddings covering 2017-2025. Together, these results give a concrete, empirically grounded recipe for scaling pixel-wise EO foundation models: train large encoders, select by downstream performance, and distil into flexible student models. All code will be released at https://github.com/ucam-eo/tessera.

As exponential compute scaling continues, will the capabilities of frontier AI models outstrip what is accessible to developers on a small fixed budget? Or will capabilities converge, with "meek models inheriting the earth"? Building on Gundlach et al. (2025b), we show that the answer depends on how we value and measure AI capabilities. We discuss conventional performance measures and show that, while validation loss shows a shrinking gap, on other metrics frontier models grow their lead forever. Classifying performance metrics by their functional forms in relation to training (and inference) compute, we provide tight mathematical conditions for determining which metrics favor meek models, and show that bounded performance metrics always do. But careful interpretation of performance metrics is essential: we show that many common bounded metrics have closely-related counterpart metrics that are unbounded (and vice versa). Determining the apt metric in a domain is a prerequisite for policy, since bounded and unbounded metrics may suggest opposing policy responses. If a particular capability -- like software engineering, synthetic biology, or rhetorical persuasiveness -- is unbounded when measured in the terms we care about, frontier-level capability will likely be concentrated in the hands of a few wealthy actors. Conversely, if that capability is instead bounded, frontier-level capabilities proliferate through meek models into the hands of the many.

Modern deep neural networks usually have large parameter scales and nonlinear hierarchical structures, and they have achieved strong performance in computer vision. However, the source of their generalization performance remains difficult to explain using traditional statistical learning theory. Among the factors that may affect visual generalization, data scale, model complexity, and input modalities are fundamental and controllable variables. This study empirically analyzes how these three factors influence model generalization performance. Specifically, in a preliminary experiment, we construct a one-dimensional nonlinear function and vary the number of training samples and the polynomial degree to observe the effects of data scale and model complexity on model performance. In the main experiments, we compare model performance on CIFAR-10 and CIFAR-100 under different training data scales, model architectures, and input modalities. The experimental results show that increasing the training data scale consistently improves generalization performance, whereas changes in model complexity do not provide stable gains. In addition, removing color information degrades model performance, while explicit prior features such as gradients, edges, and wavelets have inconsistent effects across different model architectures. Overall, this study provides an empirical analysis of the relationships among data scale, model complexity, input modalities, and visual generalization performance. Code and experimental logs are available at: https://github.com/YidiZhouluo/DeepLearning-Empirical-Studies/tree/main/Exp_01.

Mixture-of-Experts (MoE) has become the de facto architecture for hundred-billion-parameter language models, yet its advantages at sub-billion scales for on-device deployment remain largely unexplored. To close this gap, we present MobileMoE, a family of on-device MoE language models with sub-billion active parameters (0.3-0.9B active and 1.3-5.3B total) that establish a new Pareto frontier for on-device LLMs. We first formulate an on-device MoE scaling law that jointly optimizes MoE architecture under mobile memory and compute constraints, identifying an on-device sweet spot - moderate sparsity with fine-grained and shared experts - that is simultaneously memory and compute-optimal. Building on the derived architectures, we train MobileMoE with a four-stage recipe covering pre-training, mid-training, instruction fine-tuning, and quantization-aware training, all on open-source datasets. Across 14 benchmarks, MobileMoE matches or exceeds leading on-device dense LLMs with 2-4$\times$ fewer inference FLOPs, and matches or surpasses the state-of-the-art MoE OLMoE-1B-7B with up to 60% fewer parameters. To bridge the last mile to mobile deployment, we provide the first efficient MoE inference on commodity smartphones with comprehensive on-device profiling. At comparable INT4 weight memory, MobileMoE-S delivers $1.8$-$3.8\times$ faster prefill and $2.2$-$3.4\times$ faster decode than the dense baseline MobileLLM-Pro.

Specialized foundation models are beginning to emerge in various medical subdomains, but pretraining methodologies and parametric scaling with the size of the pretraining dataset are rarely assessed systematically and in a like-for-like manner. This work focuses on foundation models for electrocardiography (ECG) data, one of the most widely captured physiological time series world-wide. We present a comprehensive assessment of pretraining methodologies, covering five different contrastive and non-contrastive self-supervised learning objectives for ECG foundation models, and investigate their scaling behavior with pretraining dataset sizes up to 11M input samples, exclusively from publicly available sources. Pretraining strategy has a meaningful and consistent impact on downstream performance, with contrastive predictive coding (slightly ahead of JEPA) yielding the most transferable representations across diverse clinical tasks. Scaling pretraining data continues to yield meaningful improvements up to 11M samples for most objectives. We also compare model architectures across all pretraining methodologies and find evidence for a clear superiority of structured state space models compared to transformers and CNN models. We hypothesize that the strong inductive biases of structured state space models, rather than pretraining scale alone, are the primary driver of effective ECG representation learning, with important implications for future foundation model development in this and potentially other physiological signal domains.

Mixture-of-Experts (MoE) architectures have become standard in large language models, yet many of their core design choices - expert count, granularity, shared experts, load balancing, token dropping - have only been studied one or two at a time over narrow configuration ranges. It remains an open question whether these choices can be optimized independently, without considering interactions. We present the first systematic study of over 2,000 pretraining runs spanning models up to 6.6B total parameters, in which we exhaustively vary total experts, expert dimension, heterogeneous expert sizing within a single layer, shared expert size and load-balancing mechanisms. We find that at every active-parameter scale that we study, performance consistently improves with total MoE parameters even at extreme active expert parameter ratios like 128.Further, the optimal expert size is nearly invariant to total parameter count and depends only on active parameter count. Third, we see that other choices like shared experts, heterogeneous experts and load-balancing settings have small effects relative to expert count and granularity, although dropless routing yields a consistent gain. Overall, our results suggest a simpler recipe: focus on expert count and granularity, other choices have minimal effect on final quality.

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.