Scaling Laws

Data filtering has become a powerful tool for improving model performance while reducing computational cost. However, as large language model compute budgets continue to grow, the limited data volume provided by heavily filtered and deduplicated datasets will become a practical constraint. In efforts to better understand how to proceed, we study model performance at various compute budgets and across multiple pre-training datasets created through data filtering and deduplication. We find that, given appropriate modifications to the training recipe, repeating existing aggressively filtered datasets for up to ten epochs can outperform training on the ten times larger superset for a single epoch across multiple compute budget orders of magnitude. While this finding relies on repeating the dataset for many epochs, we also investigate repeats within these datasets at the document level. We find that not all documents within a dataset are equal, and we can create better datasets relative to a token budget by explicitly manipulating the counts of individual documents. We conclude by arguing that even as large language models scale, data filtering remains an important direction of research.

Generative protein language models (PLMs) are powerful tools for designing proteins purpose-built to solve problems in medicine, agriculture, and industrial processes. Recent work has trained ever larger language models, but there has been little systematic study of the optimal training distributions and the influence of model scale on the sequences generated by PLMs. We introduce the ProGen3 family of sparse generative PLMs, and we develop compute-optimal scaling laws to scale up to a 46B-parameter model pre-trained on 1.5T amino acid tokens. ProGen3's pre-training data is sampled from an optimized data distribution over the PPA v1, a carefully curated dataset of 3.4B full-length proteins. We evaluate for the first time in the wet lab the influence of model scale on the sequences generated by PLMs, and we find that larger models generate viable proteins for a much wider diversity of protein families. Finally, we find both computationally and experimentally that larger models are more responsive to alignment with laboratory data, resulting in improved protein fitness prediction and sequence generation capabilities. These results indicate that larger PLMs like ProGen3-46B trained on larger, well-curated datasets are powerful foundation models that push the frontier of protein design.

Foundation models trained at scale exhibit remarkable emergent behaviors, learning new capabilities beyond their initial training objectives. We find such emergent behaviors in biological vision models via large-scale contrastive vision-language training. To achieve this, we first curate TreeOfLife-200M, comprising 214 million images of living organisms, the largest and most diverse biological organism image dataset to date. We then train BioCLIP 2 on TreeOfLife-200M to distinguish different species. Despite the narrow training objective, BioCLIP 2 yields extraordinary accuracy when applied to various biological visual tasks such as habitat classification and trait prediction. We identify emergent properties in the learned embedding space of BioCLIP 2. At the inter-species level, the embedding distribution of different species aligns closely with functional and ecological meanings (e.g., beak sizes and habitats). At the intra-species level, instead of being diminished, the intra-species variations (e.g., life stages and sexes) are preserved and better separated in subspaces orthogonal to inter-species distinctions. We provide formal proof and analyses to explain why hierarchical supervision and contrastive objectives encourage these emergent properties. Crucially, our results reveal that these properties become increasingly significant with larger-scale training data, leading to a biologically meaningful embedding space.

Large foundation models are typically trained on data from multiple domains, with the data mixture--the proportion of each domain used--playing a critical role in model performance. The standard approach to selecting this mixture relies on trial and error, which becomes impractical for large-scale pretraining. We propose a systematic method to determine the optimal data mixture for any target domain using scaling laws. Our approach accurately predicts the loss of a model of size $N$ trained with $D$ tokens and a specific domain weight vector $h$. We validate the universality of these scaling laws by demonstrating their predictive power in three distinct and large-scale settings: large language model (LLM), native multimodal model (NMM), and large vision models (LVM) pretraining. We further show that these scaling laws can extrapolate to new data mixtures and across scales: their parameters can be accurately estimated using a few small-scale training runs, and used to estimate the performance at larger scales and unseen domain weights. The scaling laws allow to derive the optimal domain weights for any target domain under a given training budget ($N$,$D$), providing a principled alternative to costly trial-and-error methods.

Diffusion and flow models have emerged as powerful generative approaches capable of modeling diverse and multimodal behavior. However, applying these models to offline RL remains challenging due to the iterative nature of their noise sampling processes, making policy optimization difficult. In this paper, we introduce Scalable Offline Reinforcement Learning (SORL), a new offline RL algorithm that leverages shortcut models – a novel class of generative models – to scale both training and inference. SORL's policy can capture complex data distributions and can be trained simply and efficiently in a one-stage training procedure. At test time, SORL supports both sequential and parallel inference scaling by using the learned Q-function as a verifier. We demonstrate that SORL achieves strong performance across a range of offline RL tasks and exhibits positive scaling behavior with increased test-time compute.

Scalable oversight, the process by which weaker AI systems supervise stronger ones, has been proposed as a key strategy to control future superintelligent systems. However, it is still unclear how scalable oversight itself scales. To address this gap, we propose a framework that quantifies the probability of successful oversight as a function of the capabilities of the overseer and the system being overseen. Specifically, our framework models oversight as a game between capability-mismatched players; the players have oversight-specific Elo scores that are a piecewise-linear function of their general intelligence, with two plateaus corresponding to task incompetence and task saturation. We validate our framework with a modified version of the game Nim and then apply it to four oversight games: Mafia, Debate, Backdoor Code and Wargames. For each game, we find scaling laws that approximate how domain performance depends on general AI system capability. We then build on our findings in a theoretical study of Nested Scalable Oversight (NSO), a process in which trusted models oversee untrusted stronger models, which then become the trusted models in the next step. We identify conditions under which NSO succeeds and derive numerically (and in some cases analytically) the optimal number of oversight levels to maximize the probability of oversight success. We also apply our theory to our four oversight games, where we find that NSO success rates at a general Elo gap of 400 are 13.5\% for Mafia, 51.7\% for Debate, 10.0\% for Backdoor Code, and 9.4\% for Wargames; these rates decline further when overseeing stronger systems.

Large Language Models (LLMs) are typically trained on data mixtures: most data come from web scrapes, while a small portion is curated from high-quality sources with dense domain-specific knowledge. In this paper, we show that when training LLMs on such data mixtures, knowledge acquisition from knowledge-dense datasets—unlike training exclusively on knowledge-dense data—does not always follow a smooth scaling law but can exhibit phase transitions with respect to the mixing ratio and model size. Through controlled experiments on a synthetic biography dataset mixed with web-scraped data, we demonstrate that: (1) as we increase the model size to a critical value, the model suddenly transitions from memorizing very few to most of the biographies; (2) below a critical mixing ratio, the model memorizes almost nothing even with extensive training, but beyond this threshold, it rapidly memorizes more biographies. We attribute these phase transitions to a capacity allocation phenomenon: a model with bounded capacity must act like a knapsack problem solver to minimize the overall test loss, and the optimal allocation across datasets can change discontinuously as the model size or mixing ratio varies. We formalize this intuition in an information-theoretic framework and reveal that these phase transitions are predictable, with the critical mixing ratio following a power-law relationship with the model size. Our findings highlight a concrete case where a good mixing recipe for large models may not be optimal for small models, and vice versa.

Recent trends in test-time scaling for reasoning models (e.g., OpenAI o1, DeepSeek R1) have led to a popular belief that extending thinking traces using prompts like “Wait” or “Let me rethink” can improve performance. This raises a natural question: Does thinking more at test-time truly lead to better reasoning? To answer this question, we perform a detailed empirical study across models and benchmarks, which reveals a consistent pattern of initial performance improvements from additional thinking followed by a decline, due to "overthinking". To understand this non-monotonic trend, we consider a simple probabilistic model, which reveals that additional thinking increases output variance—creating an illusion of improved reasoning while ultimately undermining precision. Thus, observed gains from "more thinking" are not true indicators of improved reasoning, but artifacts stemming from the connection between model uncertainty and evaluation metric. This suggests that test-time scaling through extended thinking is not an effective way to utilize the inference thinking budget. Recognizing these limitations, we introduce an alternative test-time scaling approach, parallel thinking, inspired by Best-of-N sampling. Our method generates multiple independent reasoning paths within the same inference budget and selects the most consistent response via majority vote, achieving up to 20% higher accuracy compared to extended thinking. This provides a simple yet effective mechanism for test-time scaling of reasoning models.

Determining the precise geographic location of an image at a global scale remains an unsolved challenge. Standard image retrieval techniques are inefficient due to the sheer volume of images (>100M) and fail when coverage is insufficient. Scalable solutions, however, involve a trade-off: global classification typically yields coarse results (10+ kilometers), while cross-view retrieval between ground and aerial imagery suffers from a domain gap and has been primarily studied on smaller regions. This paper introduces a hybrid approach that achieves fine-grained geo-localization across a large geographic expanse the size of a continent. We leverage a proxy classification task during training to learn rich feature representations that implicitly encode precise location information. We combine these learned prototypes with embeddings of aerial imagery to increase robustness to the sparsity of ground-level data. This enables direct, fine-grained retrieval over areas spanning multiple countries. Our extensive evaluation demonstrates that our approach can localize within 200m more than 68\% of queries of a dataset covering a large part of Europe. The code is publicly available at [scaling-geoloc.github.io](https://scaling-geoloc.github.io).

We investigate scaling laws for stochastic momentum algorithms on the power law random features model, parameterized by data complexity, target complexity, and model size. When trained with a stochastic momentum algorithm, our analysis reveals four distinct loss curve shapes determined by varying data-target complexities. While traditional stochastic gradient descent with momentum (SGD-M) yields identical scaling law exponents to SGD, dimension-adapted Nesterov acceleration (DANA) improves these exponents by scaling momentum hyperparameters based on model size and data complexity. This outscaling phenomenon, which also improves compute-optimal scaling behavior, is achieved by DANA across a broad range of data and target complexities, while traditional methods fall short. Extensive experiments on high-dimensional synthetic quadratics validate our theoretical predictions and large-scale text experiments with LSTMs show DANA's improved loss exponents over SGD hold in a practical setting.

We present a universal theoretical framework for understanding *long-context language modeling* based on a *bipartite* mutual information scaling law that we rigorously verify in natural language. We demonstrate that bipartite mutual information captures multi-token interactions distinct from and scaling independently of conventional two-point mutual information, and show that this provides a more complete characterization of the dependencies needed for accurately modeling long sequences. Leveraging this scaling law, we formulate the **L**ong-context **L**anguage **M**odeling (**L**$^2$**M**) condition, which lower bounds the necessary scaling of a model's history state—the latent variables responsible for storing past information—for effective long-context modeling. We validate the framework and its predictions on transformer and state-space models. Our work provides a principled foundation to understand long-context modeling and to design more efficient architectures with stronger long-context capabilities, with potential applications beyond natural language.

Modern Large Language Models, such as the LLaMA, Qwen and DeepSeek series, predominantly adopt the Pre-LayerNorm (Pre-LN) Transformer architecture. While being stable during pretraining and scalable to large model sizes, Pre-LN suffers from an exponential growth in activation variance across layers, causing the shortcut to dominate over sub-layer outputs in the residual connection and limiting the learning capacity of deeper layers. To mitigate this issue, we propose Gradient-Preserving Activation Scaling (GPAS), a simple technique that can be used in combination with existing approaches. GPAS works by scaling down the intermediate activations while keeping their gradients unchanged. This leaves information in the activations intact, and avoids the gradient vanishing problem associated with gradient downscaling. Extensive experiments across various model sizes from 71M to 1B show that GPAS achieves consistent performance gains. Beyond enhancing Pre-LN Transformers, GPAS also shows promise in improving alternative architectures such as Sandwich-LN and DeepNorm, demonstrating its versatility and potential for improving training dynamics in a wide range of settings. Our code is available at https://github.com/dandingsky/GPAS.

Computational epidemiology is crucial in understanding and controlling infectious diseases, as highlighted by large-scale outbreaks such as COVID-19. Given the inherent uncertainty and variability of disease spread, Monte Carlo (MC) simulations are widely used to predict infection peaks, estimate reproduction numbers, and evaluate the impact of non-pharmaceutical interventions (NPIs). While effective, MC-based methods require numerous runs to achieve statistically reliable estimates and variance, which suffer from high computational costs. In this work, we present a unified theoretical framework for analyzing disease spread dynamics on both directed and undirected contact networks, and propose an algorithm, **RAPID**, that significantly improves computational efficiency. Our contributions are threefold. First, we derive an asymptotic variance lower bound for MC estimates and identify the key factors influencing estimation variance. Second, we provide a theoretical analysis of the probabilistic disease spread process using linear approximations and derive the convergence conditions under non-reinfection epidemic models. Finally, we conduct extensive experiments on six real-world datasets, demonstrating our method's effectiveness and robustness in estimating the nodes' final state distribution. Specifically, our proposed method consistently produces accurate estimates aligned with results from a large number of MC simulations, while maintaining a runtime comparable to a single MC simulation. Our code and datasets are available at https://github.com/GuanghuiMin/RAPID.

Diffusion- and flow-based models have emerged as state-of-the-art generative modeling approaches, but they require many sampling steps. Consistency models can distill these models into efficient one-step generators; however, unlike flow- and diffusion-based methods, their performance inevitably degrades when increasing the number of steps, which we show both analytically and empirically. Flow maps generalize these approaches by connecting any two noise levels in a single step and remain effective across all step counts. In this paper, we introduce two new continuous-time objectives for training flow maps, along with additional novel training techniques, generalizing existing consistency and flow matching objectives. We further demonstrate that autoguidance can improve performance, using a low-quality model for guidance during distillation, and an additional boost can be achieved by adversarial finetuning, with minimal loss in sample diversity. We extensively validate our flow map models, called *Align Your Flow*, on challenging image generation benchmarks and achieve state-of-the-art few-step generation performance on both ImageNet 64x64 and 512x512, using small and efficient neural networks. Finally, we show text-to-image flow map models that outperform all existing non-adversarially trained few-step samplers in text-conditioned synthesis.

We rethink test-time scaling laws from a practical efficiency perspective, revealing that the effectiveness of smaller models is significantly overestimated. Prior work, grounded in compute-optimality, overlooks critical memory access bottlenecks introduced by inference-time strategies (e.g., Best-of-N, long CoTs). Our holistic analysis, spanning models from 0.6B to 32B parameters, reveals a new Kinetics Scaling Law that better guides resource allocation by incorporating both computation and memory access costs. The Kinetics Scaling Law suggests that test-time compute is more effective when used on models above a threshold (14B) than on smaller ones. A key reason is that in test-time scaling, attention—rather than parameter count—emerges as the dominant cost factor. Motivated by this, we propose a new scaling paradigm centered on sparse attention, which lowers per-token cost and enables longer generations and more parallel samples within the same resource budget. Empirically, we show that sparse attention models consistently outperform dense counterparts, achieving over 60-point gains in low-cost regimes and over 5-point gains in high-cost regimes for problem-solving accuracy on AIME and LiveCodeBench. These results suggest that sparse attention is essential for realizing the full potential of test-time scaling because, unlike training where parameter scaling saturates, test-time accuracy continues to improve through increased generation.

The prediction of learning curves for Natural Language Processing (NLP) models enables informed decision-making to meet specific performance objectives, while reducing computational overhead and lowering the costs associated with dataset acquisition and curation. In this work, we formulate the prediction task as a multitask learning problem, where each task’s data is modelled as being organized within a two-layer hierarchy. To model the shared information and dependencies across tasks and hierarchical levels, we employ latent variable multi-output Gaussian Processes, enabling to account for task correlations and supporting zero-shot prediction of learning curves (LCs). We demonstrate that this approach facilitates the development of probabilistic scaling laws at lower costs. Applying an active learning strategy, LCs can be queried to reduce predictive uncertainty and provide predictions close to ground truth scaling laws. We validate our framework on three small-scale NLP datasets with up to $30$ LCs. These are obtained from nanoGPT models, from bilingual translation using mBART and Transformer models, and from multilingual translation using M2M100 models of varying sizes.

Tabular data is one of the most ubiquitous sources of information worldwide, spanning a wide variety of domains. This inherent heterogeneity has slowed the development of Tabular Foundation Models (TFMs) capable of fast generalization to unseen datasets. In-Context Learning (ICL) has recently emerged as a promising solution for TFMs, enabling dynamic adaptation to new tasks without additional tuning. While many studies have attempted to re-purpose large language models for tabular ICL, they have had limited success, so recent works have focused on developing tabular-specific foundation models. In this work, we propose an approach to combine ICL-based retrieval with self supervised learning to train tabular foundation models. We also investigate the utility of real vs. synthetic data for model pre-training, and show that real data can contain useful signal not easily captured in synthetic training. Specifically, we show that incorporating real data during the pre-training phase can lead to significantly faster training and better downstream generalization to unseen data. Our resulting model, **TabDPT**, achieves strong performance on both regression (CTR23) and classification (CC18) benchmarks. Importantly, we also demonstrate that with our pre-training procedure, scaling both model and data size leads to consistent performance improvements that follow power laws. This echoes scaling laws in LLMs and other foundation models, and suggests that large-scale TFMs can be achievable. We open-source our full pipeline: inference code including trained model weights can be found [here](https://github.com/layer6ai-labs/TabDPT-inference), and the training code to reproduce experiments can be found [here](https://github.com/layer6ai-labs/TabDPT-training).

Large‑language‑model inference with long contexts often produces key–value (KV) caches whose footprint exceeds the capacity of high‑bandwidth memory on a GPU. Prior LLM inference frameworks such as vLLM mitigate this pressure by swapping KV cache pages to host DRAM. However, the high cost of large DRAM pools makes this solution economically unattractive. Although offloading to SSDs can be a cost-effective way to expand memory capacity relative to DRAM, conventional frameworks such as FlexGen experience a substantial throughput drop since the data path that routes SSD traffic through CPU to GPU is severely bandwidth-constrained. To overcome these limitations, we introduce HiFC, a novel DRAM‑free swapping scheme that enables direct access to SSD-resident memory with low latency and high effective bandwidth. HiFC stores KV pages in pseudo-SLC (pSLC) regions of commodity NVMe SSDs, sustaining high throughput under sequential I/O and improving write endurance by up to 8$\times$. Leveraging GPU Direct Storage, HiFC enables direct transfers between SSD and GPU, bypassing host DRAM and alleviating PCIe bottlenecks. HiFC employs fine-grained block mapping to confine writes to high-performance pSLC zones, stabilizing latency and throughput under load. HiFC achieves inference throughput comparable to DRAM-based swapping under diverse long-context workloads, such as NarrativeQA, while significantly lowering the memory expansion cost of a GPU server system by 4.5$\times$ over three years.

Current image generation methods are based on a two-stage training approach. In stage 1, an auto-encoder is trained to compress an image into a latent space; in stage 2, a generative model is trained to learn a distribution over that latent space. This reveals a fundamental trade-off, do we compress more aggressively to make the latent distribution easier for the stage 2 model to learn even if it makes reconstruction worse? We study this problem in the context of discrete, auto-regressive image generation. Through the lens of scaling laws, we show that smaller stage 2 models can benefit from more compressed stage 1 latents even if reconstruction performance worsens, demonstrating that generation modeling capacity plays a role in this trade-off. Diving deeper, we rigorously study the connection between compute scaling and the stage 1 rate-distortion trade-off. Next, we introduce Causally Regularized Tokenization (CRT), which uses knowledge of the stage 2 generation modeling procedure to embed useful inductive biases in stage 1 latents. This regularization improves stage 2 generation performance better by making the tokens easier to model without affecting the stage 1 compression rate and marginally affecting distortion: we are able to improve compute efficiency 2-3$\times$ over baseline. Finally, we use CRT with further optimizations to the visual tokenizer setup to result in a generative pipeline that matches LlamaGen-3B generation performance (2.18 FID) with half the tokens per image (256 vs. 576) and a fourth the total model parameters (775M vs. 3.1B) while using the same architecture and inference procedure.

Test-time reasoning algorithms such as chain-of-thought, self-consistency, and MCTS enhance LLM problem-solving but can wastefully generate many tokens without improving accuracy. At the same time, we observe that these algorithms exhibit answer stabilization: their intermediate solutions often cease to change after a certain point, and further investment of compute does not change their final answer. To quantify this phenomenon, we introduce Certaindex, an algorithm-agnostic metric measuring this evolving stability, signaling when further computation is unlikely to alter the final result. Certaindex is lightweight, can accelerate reasoning program inference via early exit, and further enables dynamic token allocation, gang scheduling, and many opportunities when integrated with real-world LLM serving systems. To quantify real-world benefits, we built Certaindex as a scheduler into Dynasor, our reasoning-aware LLM serving system, and demonstrate up to 50\% compute savings and 3.3$\times$ higher throughput in real workloads with no accuracy drop. Our code is available at https://github.com/hao-ai-lab/Dynasor.git