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.

Training Large Language Models (LLMs) is prohibitively expensive, creating a critical scaling gap where insights from small-scale experiments often fail to transfer to resource-intensive production systems, thereby hindering efficient innovation. To bridge this, we introduce Farseer, a novel and refined scaling law offering enhanced predictive accuracy across scales. By systematically constructing a model loss surface L(N,D), Farseer achieves a significantly better fit to empirical data than prior laws (e.g., Chinchilla's law). Our methodology yields accurate, robust, and highly generalizable predictions, demonstrating excellent extrapolation capabilities, outperforming Chinchilla's law, whose extrapolation error is 433% higher. This allows for the reliable evaluation of competing training strategies across all (N,D) settings, enabling conclusions from small-scale ablation studies to be confidently extrapolated to predict large-scale performance. Furthermore, Farseer provides new insights into optimal compute allocation, better reflecting the nuanced demands of modern LLM training. To validate our approach, we trained an extensive suite of approximately 1,000 LLMs across diverse scales and configurations, consuming roughly 3 million NVIDIAH100 GPU hours. To foster further research, we are comprehensively open-sourcing all code, data, results 3, all training logs4, all models used in scaling law fitting 5.

It is commonly believed that scaling language models should commit a significant space or time cost, by increasing the parameters (parameter scaling) or output tokens (inference-time scaling). We introduce another and more inference-efficient scaling paradigm: increasing the model's parallel computation during both training and inference time. We apply P diverse and learnable transformations to the input, execute forward passes of the model in parallel, and dynamically aggregate the P outputs. This method, namely parallel scaling (PARSCALE), scales parallel computation by reusing existing parameters and can be applied to any model structure, optimization procedure, data, or task. We theoretically propose a new scaling law and validate it through large-scale pre-training, which shows that a model with P parallel streams is similar to scaling the parameters by O(logP) while showing superior inference efficiency. For example, PARSCALE can use up to 22 less memory increase and 6 less latency increase compared to parameter scaling that achieves the same performance improvement. It can also recycle an off-the-shelf pre-trained model into a parallelly scaled one by post-training on a small amount of tokens, further reducing the training budget. The new scaling law we discovered potentially facilitates the deployment of more powerful models in low-resource scenarios, and provides an alternative perspective for the role of computation in machine learning. Our code and 67 trained model checkpoints are publicly available at https://github.com/QwenLM/ParScale

Scaling laws have emerged as a unifying lens for understanding and guiding the training of large language models (LLMs). However, existing studies predominantly focus on the final-step loss, leaving open whether the entire loss dynamics obey similar laws and, crucially, how the learning rate schedule (LRS) shapes them. We address these gaps in a controlled theoretical setting by analyzing stochastic gradient descent (SGD) on a power-law kernel regression model. The key insight is a novel intrinsic-time viewpoint, which captures the training progress more faithfully than iteration count. We then establish a Functional Scaling Law (FSL) that captures the full loss trajectory under arbitrary LRSs, with the schedule's influence entering through a simple convolutional functional. We further instantiate the theory for three representative LRSs--constant, exponential decay, and warmup-stable-decay (WSD)--and derive explicit scaling relations in both data-and compute-limited regimes. These comparisons explain key empirical phenomena: (i) higher-capacity models are more data-and compute-efficient; (ii) learning-rate decay improves training efficiency; and (iii) WSD-type schedules outperform pure decay. Finally, experiments on LLMs ranging from 0.1B to 1B parameters demonstrate the practical relevance of FSL as a surrogate model for fitting and predicting loss trajectories in large-scale pre-training.

Scaling the effective context length is essential for advancing large language models (LLMs) toward artificial general intelligence (AGI). However, the quadratic increase in computational complexity inherent in traditional attention mechanisms presents a prohibitive overhead. Existing approaches either impose strongly biased structures, such as sink or window attention which are task-specific, or radically modify the attention mechanism into linear approximations, whose performance in complex reasoning tasks remains inadequately explored. In this work, we propose a solution that adheres to the "less structure" principle, allowing the model to determine where to attend autonomously, rather than introducing predefined biases. We introduce Mixture of Block Attention (MoBA), an innovative approach that applies the principles of Mixture of Experts (MoE) to the attention mechanism. This novel architecture demonstrates superior performance on long-context tasks while offering a key advantage: the ability to seamlessly transition between full and sparse attention, enhancing efficiency without the risk of compromising performance. MoBA has already been deployed to handle actual production workloads with long-context requirements, demonstrating significant advancements in efficient attention computation for LLMs. Our code is available at https://github.com/MoonshotAI/MoBA.

With the growing size of data and models in Large Recommendation Models, the time required for debugging has become increasingly prohibitive, underscoring the urgent need for effective guidance in parameter configuration. The Scaling Law (SL) offers analogous guidance in the Sequential Language domain, having achieved significant success by predicting model loss when scaling model size. However, the existing guidance from SL for Sequential Recommendation (SR) remains qualitative, which is because quantitative analysis of SL on SR encounters challenges with quality measurement on redundant sequences along with loss-performance discrepancy. In response, we introduce the Performance Law (P-Law) for SR models, which predicts model performance across various settings, intending to provide a quantitative framework for guiding the parameter optimization of future models. Initially, Performance Law utilizes Real Entropy to measure data quality, aiming to remove the low-quality influence of low-entropy redundant sequences. Subsequently, Performance Law investigates a fitting decay term, which facilitated the prediction of the major loss-performance discrepancy phenomena of overfitting, ultimately achieving quantitative performance prediction. Extensive experiment on various datasets demonstrates the effectiveness of Performance Law by displaying exceptional quantitative prediction ability against the original and modified qualitative SL. Additional application experiments on optimal parameter prediction and model expansion potential prediction also demonstrated the broad applicability of the Performance Law.

With the growing size of data and models in Large Recommendation Models, the time required for debugging has become increasingly prohibitive, underscoring the urgent need for effective guidance in parameter configuration. The Scaling Law (SL) offers analogous guidance in the Sequential Language domain, having achieved significant success by predicting model loss when scaling model size. However, the existing guidance from SL for Sequential Recommendation (SR) remains qualitative, which is because quantitative analysis of SL on SR encounters challenges with quality measurement on redundant sequences along with loss-performance discrepancy. In response, we introduce the Performance Law (P-Law) for SR models, which predicts model performance across various settings, intending to provide a quantitative framework for guiding the parameter optimization of future models. Initially, Performance Law utilizes Real Entropy to measure data quality, aiming to remove the low-quality influence of low-entropy redundant sequences. Subsequently, Performance Law investigates a fitting decay term, which facilitated the prediction of the major loss-performance discrepancy phenomena of overfitting, ultimately achieving quantitative performance prediction. Extensive experiment on various datasets demonstrates the effectiveness of Performance Law by displaying exceptional quantitative prediction ability against the original and modified qualitative SL. Additional application experiments on optimal parameter prediction and model expansion potential prediction also demonstrated the broad applicability of the Performance Law.

We introduce a model for neural scaling laws under sparse activations. In the model, test loss is often dominated by rare coordinates that are never observed in the training input. This mechanism induces a novel bottleneck absent from dense models. We derive the asymptotic population loss in both the underparameterized and overparameterized regimes, and show that the loss exhibits a double-descent peak near the interpolation threshold -- where the number of parameters is just sufficient to fit the training data -- resulting in a loss curve governed by two distinct scaling exponents -- one for the overparameterized regime and one for the underparameterized regime -- with a gap determined by the degree of sparsity. Additionally, we derive a compute-optimal frontier that favors increasing dataset size over model capacity under fixed compute budgets. We also analyze gradient-descent dynamics and identify a scaling law for the probability that fixed-step gradient descent becomes unstable. We further show that the sparsity-induced effect persists under nonlinear activations.

Chinchilla Approach 2 is among the most widely used methods for fitting neural scaling laws. Its parabolic approximation introduces systematic biases in compute-optimal allocation estimates, even on noise-free synthetic data. Applied to published Llama 3 IsoFLOP data at open frontier compute scales, these biases imply a parameter underallocation corresponding to 6.5% of the $3.8\times10^{25}$ FLOP training budget and \$1.4M (90% CI: \$412K-\$2.9M) in unnecessary compute at 50% H100 MFU. Simulated multimodal model misallocations show even greater opportunity costs due to higher loss surface asymmetry. Three sources of this error are examined: IsoFLOP sampling grid width (Taylor approximation accuracy), uncentered IsoFLOP sampling, and loss surface asymmetry ($α\neq β$). Chinchilla Approach 3 largely eliminates these biases but is often regarded as less data-efficient, numerically unstable, prone to local minima, and harder to implement. Each concern is shown to be unfounded or addressable, especially when the partially linear structure of the objective is exploited via Variable Projection, enabling unbiased inference on all five loss surface parameters through a two-dimensional optimization that is well-conditioned, analytically differentiable, and amenable to dense, or even exhaustive, grid search. It may serve as a more convenient replacement for Approach 2 or a more scalable alternative for adaptations of Approach 3 to richer scaling law formulations. See https://github.com/Open-Athena/vpnls for details and https://openathena.ai/scaling-law-analysis for other results from this study.

Continual Pre-Training (CPT) on Large Language Models (LLMs) has been widely used to expand the model's fundamental understanding of specific downstream domains (e.g., math and code). For the CPT on domain-specific LLMs, one important question is how to choose the optimal mixture ratio between the general-corpus (e.g., Dolma, Slim-pajama) and the downstream domain-corpus. Existing methods usually adopt laborious human efforts by grid-searching on a set of mixture ratios, which require high GPU training consumption costs. Besides, we cannot guarantee the selected ratio is optimal for the specific domain. To address the limitations of existing methods, inspired by the Scaling Law for performance prediction, we propose to investigate the Scaling Law of the Domain-specific Continual Pre-Training (D-CPT Law) to decide the optimal mixture ratio with acceptable training costs for LLMs of different sizes. Specifically, by fitting the D-CPT Law, we can easily predict the general and downstream performance of arbitrary mixture ratios, model sizes, and dataset sizes using small-scale training costs on limited experiments. Moreover, we also extend our standard D-CPT Law on cross-domain settings and propose the Cross-Domain D-CPT Law to predict the D-CPT law of target domains, where very small training costs (about 1\% of the normal training costs) are needed for the target domains. Comprehensive experimental results on six downstream domains demonstrate the effectiveness and generalizability of our proposed D-CPT Law and Cross-Domain D-CPT Law.