Daily Papers

Guided by the belief of the scaling law, large language models (LLMs) have achieved impressive performance in recent years. However, scaling law only gives a qualitative estimation of loss, which is influenced by various factors such as model architectures, data distributions, tokenizers, and computation precision. Thus, estimating the real performance of LLMs with different training settings rather than loss may be quite useful in practical development. In this article, we present an empirical equation named "Performance Law" to directly predict the MMLU score of an LLM, which is a widely used metric to indicate the general capability of LLMs in real-world conversations and applications. Based on only a few key hyperparameters of the LLM architecture and the size of training data, we obtain a quite accurate MMLU prediction of various LLMs with diverse sizes and architectures developed by different organizations in different years. Performance law can be used to guide the choice of LLM architecture and the effective allocation of computational resources without extensive experiments.

While large language models (LLMs) have showcased impressive capabilities, they struggle with addressing legal queries due to the intricate complexities and specialized expertise required in the legal field. In this paper, we introduce InternLM-Law, a specialized LLM tailored for addressing diverse legal queries related to Chinese laws, spanning from responding to standard legal questions (e.g., legal exercises in textbooks) to analyzing complex real-world legal situations. We meticulously construct a dataset in the Chinese legal domain, encompassing over 1 million queries, and implement a data filtering and processing pipeline to ensure its diversity and quality. Our training approach involves a novel two-stage process: initially fine-tuning LLMs on both legal-specific and general-purpose content to equip the models with broad knowledge, followed by exclusive fine-tuning on high-quality legal data to enhance structured output generation. InternLM-Law achieves the highest average performance on LawBench, outperforming state-of-the-art models, including GPT-4, on 13 out of 20 subtasks. We make InternLM-Law and our dataset publicly available to facilitate future research in applying LLMs within the legal domain.

The recent application of LLMs to the legal field has spurred the creation of benchmarks across various jurisdictions and languages. However, no benchmark has yet been specifically designed for the Portuguese legal system. In this work, we present LegalBench.PT, the first comprehensive legal benchmark covering key areas of Portuguese law. To develop LegalBench.PT, we first collect long-form questions and answers from real law exams, and then use GPT-4o to convert them into multiple-choice, true/false, and matching formats. Once generated, the questions are filtered and processed to improve the quality of the dataset. To ensure accuracy and relevance, we validate our approach by having a legal professional review a sample of the generated questions. Although the questions are synthetically generated, we show that their basis in human-created exams and our rigorous filtering and processing methods applied result in a reliable benchmark for assessing LLMs' legal knowledge and reasoning abilities. Finally, we evaluate the performance of leading LLMs on LegalBench.PT and investigate potential biases in GPT-4o's responses. We also assess the performance of Portuguese lawyers on a sample of questions to establish a baseline for model comparison and validate the benchmark.

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.

Large language models with a huge number of parameters, when trained on near internet-sized number of tokens, have been empirically shown to obey neural scaling laws: specifically, their performance behaves predictably as a power law in either parameters or dataset size until bottlenecked by the other resource. To understand this better, we first identify the necessary properties allowing such scaling laws to arise and then propose a statistical model -- a joint generative data model and random feature model -- that captures this neural scaling phenomenology. By solving this model in the dual limit of large training set size and large number of parameters, we gain insight into (i) the statistical structure of datasets and tasks that lead to scaling laws, (ii) the way nonlinear feature maps, such as those provided by neural networks, enable scaling laws when trained on these datasets, (iii) the optimality of the equiparameterization scaling of training sets and parameters, and (iv) whether such scaling laws can break down and how they behave when they do. Key findings are the manner in which the power laws that occur in the statistics of natural datasets are extended by nonlinear random feature maps and then translated into power-law scalings of the test loss and how the finite extent of the data's spectral power law causes the model's performance to plateau.

Large language models (LLMs) are increasingly being applied to programming tasks, ranging from single-turn code completion to autonomous agents. Current code agent designs frequently depend on complex, hand-crafted workflows and tool sets. However, this reliance on elaborate scaffolding presents several challenges: agent performance becomes overly dependent on prompt tuning and custom design choices, heavy human intervention obscures a model's true underlying capabilities, and intricate pipelines are costly to build and maintain. Furthermore, optimizing complex task prompts increases the risk of data leakage. Currently, when introducing new models, LLM providers like OpenAI and Anthropic often publish benchmark scores to demonstrate their models' coding proficiency, but keep their proprietary evaluation frameworks confidential. To address these limitations, we introduce Lita (Lite Agent), which operationalizes liteness, a principle of minimizing manual design while retaining the essential elements of a fully autonomous agent. Lita enables a more faithful and unified evaluation without elaborate scaffolding. Experiments on the Aider Polyglot and SWE-Bench with frontier models demonstrate that Lita achieves competitive or superior performance compared to workflow-based and agentic baselines. Crucially, Lita also consumes fewer tokens and requires significantly less design effort. Our results suggest that Lita is sufficient to reveal the underlying coding competence of modern LLMs. Finally, we propose the Agent Complexity Law: the performance gap between agents of varying complexity, from simple to sophisticated designs, will shrink as the core model improves, ultimately converging to a negligible difference.

We present Liquid, an auto-regressive generation paradigm that seamlessly integrates visual comprehension and generation by tokenizing images into discrete codes and learning these code embeddings alongside text tokens within a shared feature space for both vision and language. Unlike previous multimodal large language model (MLLM), Liquid achieves this integration using a single large language model (LLM), eliminating the need for external pretrained visual embeddings such as CLIP. For the first time, Liquid uncovers a scaling law that performance drop unavoidably brought by the unified training of visual and language tasks diminishes as the model size increases. Furthermore, the unified token space enables visual generation and comprehension tasks to mutually enhance each other, effectively removing the typical interference seen in earlier models. We show that existing LLMs can serve as strong foundations for Liquid, saving 100x in training costs while outperforming Chameleon in multimodal capabilities and maintaining language performance comparable to mainstream LLMs like LLAMA2. Liquid also outperforms models like SD v2.1 and SD-XL (FID of 5.47 on MJHQ-30K), excelling in both vision-language and text-only tasks. This work demonstrates that LLMs such as LLAMA3.2 and GEMMA2 are powerful multimodal generators, offering a scalable solution for enhancing both vision-language understanding and generation. The code and models will be released.

Scaling laws have transformed our understanding of large language models by linking upstream metrics like cross-entropy loss to design factors such as model size, training data, and compute. However, these conventional laws fail to capture downstream task performance, where context plays a critical role. In this work, we propose a straightforward, interpretable framework that jointly models downstream performance as a function of the training compute and the provided context. We empirically validate our framework by fitting it on the observed downstream performance of extended-context variants of Llama-2-7B and Llama-2-13B across 65,500 unique instances spanning three tasks: arithmetic reasoning, common sense reasoning, and machine translation. Our results demonstrate that our framework accurately models in-distribution downstream performance, generalizes across three orders of magnitude in training compute, and reliably extrapolates performance as the amount of context increases. These findings offer valuable insights into the interplay between training compute and context utilization, providing guidance for designing more efficient long-context LLMs for diverse downstream tasks. Our code is available at https://github.com/wang-research-lab/context-scaling.

Scaling laws provide important insights that can guide the design of large language models (LLMs). Existing work has primarily focused on studying scaling laws for pretraining (upstream) loss. However, in transfer learning settings, in which LLMs are pretrained on an unsupervised dataset and then finetuned on a downstream task, we often also care about the downstream performance. In this work, we study the scaling behavior in a transfer learning setting, where LLMs are finetuned for machine translation tasks. Specifically, we investigate how the choice of the pretraining data and its size affect downstream performance (translation quality) as judged by two metrics: downstream cross-entropy and BLEU score. Our experiments indicate that the size of the finetuning dataset and the distribution alignment between the pretraining and downstream data significantly influence the scaling behavior. With sufficient alignment, both downstream cross-entropy and BLEU score improve monotonically with more pretraining data. In such cases, we show that it is possible to predict the downstream BLEU score with good accuracy using a log-law. However, there are also cases where moderate misalignment causes the BLEU score to fluctuate or get worse with more pretraining, whereas downstream cross-entropy monotonically improves. By analyzing these observations, we provide new practical insights for choosing appropriate pretraining data.

Understanding how language model performance varies with scale is critical to benchmark and algorithm development. Scaling laws are one approach to building this understanding, but the requirement of training models across many different scales has limited their use. We propose an alternative, observational approach that bypasses model training and instead builds scaling laws from ~80 publically available models. Building a single scaling law from multiple model families is challenging due to large variations in their training compute efficiencies and capabilities. However, we show that these variations are consistent with a simple, generalized scaling law where language model performance is a function of a low-dimensional capability space, and model families only vary in their efficiency in converting training compute to capabilities. Using this approach, we show the surprising predictability of complex scaling phenomena: we show that several emergent phenomena follow a smooth, sigmoidal behavior and are predictable from small models; we show that the agent performance of models such as GPT-4 can be precisely predicted from simpler non-agentic benchmarks; and we show how to predict the impact of post-training interventions like Chain-of-Thought and Self-Consistency as language model capabilities continue to improve.

Neural scaling laws define a predictable relationship between a model's parameter count and its performance after training in the form of a power law. However, most research to date has not explicitly investigated whether scaling laws can be used to accelerate model development. In this work, we perform such an empirical investigation across a wide range of language understanding tasks, starting from models with as few as 10K parameters, and evaluate downstream performance across 9 language understanding tasks. We find that scaling laws emerge at finetuning time in some NLP tasks, and that they can also be exploited for debugging convergence when training large models. Moreover, for tasks where scaling laws exist, they can be used to predict the performance of larger models, which enables effective model selection. However, revealing scaling laws requires careful hyperparameter tuning and multiple runs for the purpose of uncertainty estimation, which incurs additional overhead, partially offsetting the computational benefits.

Increasing the size of a Transformer model does not always lead to enhanced performance. This phenomenon cannot be explained by the empirical scaling laws. Furthermore, improved generalization ability occurs as the model memorizes the training samples. We present a theoretical framework that sheds light on the memorization process and performance dynamics of transformer-based language models. We model the behavior of Transformers with associative memories using Hopfield networks, such that each transformer block effectively conducts an approximate nearest-neighbor search. Based on this, we design an energy function analogous to that in the modern continuous Hopfield network which provides an insightful explanation for the attention mechanism. Using the majorization-minimization technique, we construct a global energy function that captures the layered architecture of the Transformer. Under specific conditions, we show that the minimum achievable cross-entropy loss is bounded from below by a constant approximately equal to 1. We substantiate our theoretical results by conducting experiments with GPT-2 on various data sizes, as well as training vanilla Transformers on a dataset of 2M tokens.

Scaling laws for large language models (LLMs) predict model performance based on parameters like size and training data. However, differences in training configurations and data processing across model families lead to significant variations in benchmark performance, making it difficult for a single scaling law to generalize across all LLMs. On the other hand, training family-specific scaling laws requires training models of varying sizes for every family. In this work, we propose Skills Scaling Laws (SSLaws, pronounced as Sloth), a novel scaling law that leverages publicly available benchmark data and assumes LLM performance is driven by low-dimensional latent skills, such as reasoning and instruction following. These latent skills are influenced by computational resources like model size and training tokens but with varying efficiencies across model families. Sloth exploits correlations across benchmarks to provide more accurate and interpretable predictions while alleviating the need to train multiple LLMs per family. We present both theoretical results on parameter identification and empirical evaluations on 12 prominent benchmarks, from Open LLM Leaderboard v1/v2, demonstrating that Sloth predicts LLM performance efficiently and offers insights into scaling behaviors for complex downstream tasks and increased test-time compute.

Pretraining data of large language models composes multiple domains (e.g., web texts, academic papers, codes), whose mixture proportions crucially impact the competence of outcome models. While existing endeavors rely on heuristics or qualitative strategies to tune the proportions, we discover the quantitative predictability of model performance regarding the mixture proportions in function forms, which we refer to as the data mixing laws. Fitting such functions on sample mixtures unveils model performance on unseen mixtures before actual runs, thus guiding the selection of an ideal data mixture. Furthermore, we propose nested use of the scaling laws of training steps, model sizes, and our data mixing law to enable predicting the performance of large models trained on massive data under various mixtures with only small-scale training. Moreover, experimental results verify that our method effectively optimizes the training mixture of a 1B model trained for 100B tokens in RedPajama, reaching a performance comparable to the one trained for 48% more steps on the default mixture. Extending the application of data mixing laws to continual training accurately predicts the critical mixture proportion that avoids catastrophic forgetting and outlooks the potential for dynamic data schedules

CTR models play a vital role in improving user experience and boosting business revenue in many online personalized services. However, current CTR models generally encounter bottlenecks in performance improvement. Inspired by the scaling law phenomenon of LLMs, we propose a new paradigm for improving CTR predictions: first, constructing a CTR model with accuracy scalable to the model grade and data size, and then distilling the knowledge implied in this model into its lightweight model that can serve online users. To put it into practice, we construct a CTR model named SUAN (Stacked Unified Attention Network). In SUAN, we propose the UAB as a behavior sequence encoder. A single UAB unifies the modeling of the sequential and non-sequential features and also measures the importance of each user behavior feature from multiple perspectives. Stacked UABs elevate the configuration to a high grade, paving the way for performance improvement. In order to benefit from the high performance of the high-grade SUAN and avoid the disadvantage of its long inference time, we modify the SUAN with sparse self-attention and parallel inference strategies to form LightSUAN, and then adopt online distillation to train the low-grade LightSUAN, taking a high-grade SUAN as a teacher. The distilled LightSUAN has superior performance but the same inference time as the LightSUAN, making it well-suited for online deployment. Experimental results show that SUAN performs exceptionally well and holds the scaling laws spanning three orders of magnitude in model grade and data size, and the distilled LightSUAN outperforms the SUAN configured with one grade higher. More importantly, the distilled LightSUAN has been integrated into an online service, increasing the CTR by 2.81% and CPM by 1.69% while keeping the average inference time acceptable. Our source code is available at https://github.com/laiweijiang/SUAN.

When using Large Language Models (LLMs) to support Knowledge Graph Engineering (KGE), one of the first indications when searching for an appropriate model is its size. According to the scaling laws, larger models typically show higher capabilities. However, in practice, resource costs are also an important factor and thus it makes sense to consider the ratio between model performance and costs. The LLM-KG-Bench framework enables the comparison of LLMs in the context of KGE tasks and assesses their capabilities of understanding and producing KGs and KG queries. Based on a dataset created in an LLM-KG-Bench run covering 26 open state-of-the-art LLMs, we explore the model size scaling laws specific to KGE tasks. In our analyses, we assess how benchmark scores evolve between different model size categories. Additionally, we inspect how the general score development of single models and families of models correlates to their size. Our analyses revealed that, with a few exceptions, the model size scaling laws generally also apply to the selected KGE tasks. However, in some cases, plateau or ceiling effects occurred, i.e., the task performance did not change much between a model and the next larger model. In these cases, smaller models could be considered to achieve high cost-effectiveness. Regarding models of the same family, sometimes larger models performed worse than smaller models of the same family. These effects occurred only locally. Hence it is advisable to additionally test the next smallest and largest model of the same family.

Large language models (LLMs) present significant deployment challenges due to their immense computational and memory requirements. While semi-structured pruning, particularly 2:4 sparsity, offers a path to practical hardware acceleration, existing methods often incur substantial performance degradation. To bridge this gap, we introduce ARMOR: (Adaptive Representation with Matrix-factORization), a novel one-shot post-training pruning algorithm. Instead of directly pruning weights, ARMOR factorizes each weight matrix into a 2:4 sparse core wrapped by two low-overhead, block diagonal matrices. These wrappers act as efficient pre and post-transformation error correctors, offering greater flexibility to preserve model quality compared to conventional 2:4 pruning techniques. The sparse core and block diagonal wrappers are chosen through a block coordinate descent algorithm that minimizes a layer-wise proxy loss. We theoretically prove this optimization is guaranteed to converge to a solution with a proxy loss less than or equal to state-of-the-art pruning algorithms. Experiments on Llama (Touvron et al., 2023; Dubey et al., 2024) and Qwen (Yang et al., 2025) model families demonstrate that ARMOR consistently and significantly outperforms state-of-the-art 2:4 pruning methods across a wide range of downstream tasks and perplexity evaluations. ARMOR achieves this superior performance while retaining the inference speedups and substantial memory usage reductions of 2:4 pruning, establishing a more effective trade-off between model compression and task accuracy

Many deep learning models, developed in recent years, reach higher ImageNet accuracy than ResNet50, with fewer or comparable FLOPS count. While FLOPs are often seen as a proxy for network efficiency, when measuring actual GPU training and inference throughput, vanilla ResNet50 is usually significantly faster than its recent competitors, offering better throughput-accuracy trade-off. In this work, we introduce a series of architecture modifications that aim to boost neural networks' accuracy, while retaining their GPU training and inference efficiency. We first demonstrate and discuss the bottlenecks induced by FLOPs-optimizations. We then suggest alternative designs that better utilize GPU structure and assets. Finally, we introduce a new family of GPU-dedicated models, called TResNet, which achieve better accuracy and efficiency than previous ConvNets. Using a TResNet model, with similar GPU throughput to ResNet50, we reach 80.8 top-1 accuracy on ImageNet. Our TResNet models also transfer well and achieve state-of-the-art accuracy on competitive single-label classification datasets such as Stanford cars (96.0%), CIFAR-10 (99.0%), CIFAR-100 (91.5%) and Oxford-Flowers (99.1%). They also perform well on multi-label classification and object detection tasks. Implementation is available at: https://github.com/mrT23/TResNet.

As pretrained transformer language models continue to achieve state-of-the-art performance, the Natural Language Processing community has pushed for advances in model compression and efficient attention mechanisms to address high computational requirements and limited input sequence length. Despite these separate efforts, no investigation has been done into the intersection of these two fields. In this work, we provide an evaluation of model compression via knowledge distillation on efficient attention transformers. We provide cost-performance trade-offs for the compression of state-of-the-art efficient attention architectures and the gains made in performance in comparison to their full attention counterparts. Furthermore, we introduce a new long-context Named Entity Recognition dataset, GONERD, to train and test the performance of NER models on long sequences. We find that distilled efficient attention transformers can preserve a significant amount of original model performance, preserving up to 98.6% across short-context tasks (GLUE, SQUAD, CoNLL-2003), up to 94.6% across long-context Question-and-Answering tasks (HotpotQA, TriviaQA), and up to 98.8% on long-context Named Entity Recognition (GONERD), while decreasing inference times by up to 57.8%. We find that, for most models on most tasks, performing knowledge distillation is an effective method to yield high-performing efficient attention models with low costs.

Large language models (LLMs) have advanced many domains and applications but face high fine-tuning costs, inference latency, limited edge deployability, and reliability concerns. Small language models (SLMs), compact, efficient, and adaptable, offer complementary remedies. Recent work explores collaborative frameworks that fuse SLMs' specialization and efficiency with LLMs' generalization and reasoning to meet diverse objectives across tasks and deployment scenarios. Motivated by these developments, this paper presents a systematic survey of SLM-LLM collaboration organized by collaboration objectives. We propose a taxonomy with four goals: performance enhancement, cost-effectiveness, cloud-edge privacy, and trustworthiness. Within this framework, we review representative methods, summarize design paradigms, and outline open challenges and future directions toward efficient, secure, and scalable SLM-LLM collaboration.

The performance of embodied agents has been shown to improve by increasing model parameters, dataset size, and compute. This has been demonstrated in domains from robotics to video games, when generative learning objectives on offline datasets (pre-training) are used to model an agent's behavior (imitation learning) or their environment (world modeling). This paper characterizes the role of scale in these tasks more precisely. Going beyond the simple intuition that `bigger is better', we show that the same types of power laws found in language modeling (e.g. between loss and optimal model size), also arise in world modeling and imitation learning. However, the coefficients of these laws are heavily influenced by the tokenizer, task \& architecture -- this has important implications on the optimal sizing of models and data.

The performance of a language model has been shown to be effectively modeled as a power-law in its parameter count. Here we study the scaling behaviors of Routing Networks: architectures that conditionally use only a subset of their parameters while processing an input. For these models, parameter count and computational requirement form two independent axes along which an increase leads to better performance. In this work we derive and justify scaling laws defined on these two variables which generalize those known for standard language models and describe the performance of a wide range of routing architectures trained via three different techniques. Afterwards we provide two applications of these laws: first deriving an Effective Parameter Count along which all models scale at the same rate, and then using the scaling coefficients to give a quantitative comparison of the three routing techniques considered. Our analysis derives from an extensive evaluation of Routing Networks across five orders of magnitude of size, including models with hundreds of experts and hundreds of billions of parameters.

Pruning has become a widely adopted technique for reducing the hardware requirements of large language models (LLMs). To recover model performance after pruning, post-training is commonly employed to mitigate the resulting performance degradation. While post-training benefits from larger datasets, once the dataset size is already substantial, increasing the training data provides only limited performance gains. To balance post-training cost and model performance, it is necessary to explore the optimal amount of post-training data.Through extensive experiments on the Llama-3 and Qwen-2.5 series models, pruned using various common pruning methods, we uncover the scaling Law for Post-training after model Pruning, referred to as the P^2 Law.This law identifies four key factors for predicting the pruned model's post-training loss: the model size before pruning, the number of post-training tokens, the pruning rate, and the model's loss before pruning. Moreover, P^2 Law can generalize to larger dataset sizes, larger model sizes, and higher pruning rates, offering valuable insights for the post-training of pruned LLMs.

Building general-purpose models that can effectively perceive the world through multimodal signals has been a long-standing goal. Current approaches involve integrating separately pre-trained components, such as connecting vision encoders to LLMs and continuing multimodal training. While such approaches exhibit remarkable sample efficiency, it remains an open question whether such late-fusion architectures are inherently superior. In this work, we revisit the architectural design of native multimodal models (NMMs)--those trained from the ground up on all modalities--and conduct an extensive scaling laws study, spanning 457 trained models with different architectures and training mixtures. Our investigation reveals no inherent advantage to late-fusion architectures over early-fusion ones, which do not rely on image encoders. On the contrary, early-fusion exhibits stronger performance at lower parameter counts, is more efficient to train, and is easier to deploy. Motivated by the strong performance of the early-fusion architectures, we show that incorporating Mixture of Experts (MoEs) allows for models that learn modality-specific weights, significantly enhancing performance.

The largest experiments in machine learning now require resources far beyond the budget of all but a few institutions. Fortunately, it has recently been shown that the results of these huge experiments can often be extrapolated from the results of a sequence of far smaller, cheaper experiments. In this work, we show that not only can the extrapolation be done based on the size of the model, but on the size of the problem as well. By conducting a sequence of experiments using AlphaZero and Hex, we show that the performance achievable with a fixed amount of compute degrades predictably as the game gets larger and harder. Along with our main result, we further show that the test-time and train-time compute available to an agent can be traded off while maintaining performance.

The development and evaluation of Large Language Models (LLMs) have largely focused on individual capabilities. However, this overlooks the intersection of multiple abilities across different types of expertise that are often required for real-world tasks, which we term cross capabilities. To systematically explore this concept, we first define seven core individual capabilities and then pair them to form seven common cross capabilities, each supported by a manually constructed taxonomy. Building on these definitions, we introduce CrossEval, a benchmark comprising 1,400 human-annotated prompts, with 100 prompts for each individual and cross capability. To ensure reliable evaluation, we involve expert annotators to assess 4,200 model responses, gathering 8,400 human ratings with detailed explanations to serve as reference examples. Our findings reveal that, in both static evaluations and attempts to enhance specific abilities, current LLMs consistently exhibit the "Law of the Weakest Link," where cross-capability performance is significantly constrained by the weakest component. Specifically, across 58 cross-capability scores from 17 models, 38 scores are lower than all individual capabilities, while 20 fall between strong and weak, but closer to the weaker ability. These results highlight the under-performance of LLMs in cross-capability tasks, making the identification and improvement of the weakest capabilities a critical priority for future research to optimize performance in complex, multi-dimensional scenarios.

Comprehensively understanding and accurately predicting the performance of large language models across diverse downstream tasks has emerged as a pivotal challenge in NLP research. The pioneering scaling law on downstream works demonstrated intrinsic similarities within model families and utilized such similarities for performance prediction. However, they tend to overlook the similarities between model families and only consider design factors listed in the original scaling law. To overcome these limitations, we introduce a novel framework, Collaborative Performance Prediction (CPP), which significantly enhances prediction accuracy by leveraging the historical performance of various models on downstream tasks and other design factors for both model and task. We also collect a collaborative data sourced from online platforms containing both historical performance and additional design factors. With the support of the collaborative data, CPP not only surpasses traditional scaling laws in predicting the performance of scaled LLMs but also facilitates a detailed analysis of factor importance, an area previously overlooked.

The waning of Moore's Law has shifted the focus of the tech industry towards alternative methods for continued performance gains. While optimizing compilers are a standard tool to help increase program efficiency, programmers continue to shoulder much responsibility in crafting and refactoring code with better performance characteristics. In this paper, we investigate the ability of large language models (LLMs) to suggest functionally correct, performance improving code edits. We hypothesize that language models can suggest such edits in ways that would be impractical for static analysis alone. We investigate these questions by curating a large-scale dataset of Performance-Improving Edits, PIE. PIE contains trajectories of programs, where a programmer begins with an initial, slower version and iteratively makes changes to improve the program's performance. We use PIE to evaluate and improve the capacity of large language models. Specifically, use examples from PIE to fine-tune multiple variants of CODEGEN, a billion-scale Transformer-decoder model. Additionally, we use examples from PIE to prompt OpenAI's CODEX using a few-shot prompting. By leveraging PIE, we find that both CODEX and CODEGEN can generate performance-improving edits, with speedups of more than 2.5x for over 25% of the programs, for C++ and Python, even after the C++ programs were compiled using the O3 optimization level. Crucially, we show that PIE allows CODEGEN, an open-sourced and 10x smaller model than CODEX, to match the performance of CODEX on this challenging task. Overall, this work opens new doors for creating systems and methods that can help programmers write efficient code.

Recent large language models have been trained on vast datasets, but also often on repeated data, either intentionally for the purpose of upweighting higher quality data, or unintentionally because data deduplication is not perfect and the model is exposed to repeated data at the sentence, paragraph, or document level. Some works have reported substantial negative performance effects of this repeated data. In this paper we attempt to study repeated data systematically and to understand its effects mechanistically. To do this, we train a family of models where most of the data is unique but a small fraction of it is repeated many times. We find a strong double descent phenomenon, in which repeated data can lead test loss to increase midway through training. A predictable range of repetition frequency leads to surprisingly severe degradation in performance. For instance, performance of an 800M parameter model can be degraded to that of a 2x smaller model (400M params) by repeating 0.1% of the data 100 times, despite the other 90% of the training tokens remaining unique. We suspect there is a range in the middle where the data can be memorized and doing so consumes a large fraction of the model's capacity, and this may be where the peak of degradation occurs. Finally, we connect these observations to recent mechanistic interpretability work - attempting to reverse engineer the detailed computations performed by the model - by showing that data repetition disproportionately damages copying and internal structures associated with generalization, such as induction heads, providing a possible mechanism for the shift from generalization to memorization. Taken together, these results provide a hypothesis for why repeating a relatively small fraction of data in large language models could lead to disproportionately large harms to performance.

Large language models (LLMs) achieve strong performance across diverse tasks, largely driven by high-quality web data used in pre-training. However, recent studies indicate this data source is rapidly depleting. Synthetic data emerges as a promising alternative, but it remains unclear whether synthetic datasets exhibit predictable scalability comparable to raw pre-training data. In this work, we systematically investigate the scaling laws of synthetic data by introducing SynthLLM, a scalable framework that transforms pre-training corpora into diverse, high-quality synthetic datasets. Our approach achieves this by automatically extracting and recombining high-level concepts across multiple documents using a graph algorithm. Key findings from our extensive mathematical experiments on SynthLLM include: (1) SynthLLM generates synthetic data that reliably adheres to the rectified scaling law across various model sizes; (2) Performance improvements plateau near 300B tokens; and (3) Larger models approach optimal performance with fewer training tokens. For instance, an 8B model peaks at 1T tokens, while a 3B model requires 4T. Moreover, comparisons with existing synthetic data generation and augmentation methods demonstrate that SynthLLM achieves superior performance and scalability. Our findings highlight synthetic data as a scalable and reliable alternative to organic pre-training corpora, offering a viable path toward continued improvement in model performance.

We present the first systematic investigation of supervised scaling laws outside of an ImageNet-like context - on images of galaxies. We use 840k galaxy images and over 100M annotations by Galaxy Zoo volunteers, comparable in scale to Imagenet-1K. We find that adding annotated galaxy images provides a power law improvement in performance across all architectures and all tasks, while adding trainable parameters is effective only for some (typically more subjectively challenging) tasks. We then compare the downstream performance of finetuned models pretrained on either ImageNet-12k alone vs. additionally pretrained on our galaxy images. We achieve an average relative error rate reduction of 31% across 5 downstream tasks of scientific interest. Our finetuned models are more label-efficient and, unlike their ImageNet-12k-pretrained equivalents, often achieve linear transfer performance equal to that of end-to-end finetuning. We find relatively modest additional downstream benefits from scaling model size, implying that scaling alone is not sufficient to address our domain gap, and suggest that practitioners with qualitatively different images might benefit more from in-domain adaption followed by targeted downstream labelling.

Traditionally, data selection has been studied in settings where all samples from prospective sources are fully revealed to a machine learning developer. However, in practical data exchange scenarios, data providers often reveal only a limited subset of samples before an acquisition decision is made. Recently, there have been efforts to fit scaling laws that predict model performance at any size and data source composition using the limited available samples. However, these scaling functions are black-box, computationally expensive to fit, highly susceptible to overfitting, or/and difficult to optimize for data selection. This paper proposes a framework called <projektor>, which predicts model performance and supports data selection decisions based on partial samples of prospective data sources. Our approach distinguishes itself from existing work by introducing a novel *two-stage* performance inference process. In the first stage, we leverage the Optimal Transport distance to predict the model's performance for any data mixture ratio within the range of disclosed data sizes. In the second stage, we extrapolate the performance to larger undisclosed data sizes based on a novel parameter-free mapping technique inspired by neural scaling laws. We further derive an efficient gradient-based method to select data sources based on the projected model performance. Evaluation over a diverse range of applications demonstrates that <projektor> significantly improves existing performance scaling approaches in terms of both the accuracy of performance inference and the computation costs associated with constructing the performance predictor. Also, <projektor> outperforms by a wide margin in data selection effectiveness compared to a range of other off-the-shelf solutions.

Scaling language models improves performance but comes with significant computational costs. This paper proposes UL2R, a method that substantially improves existing language models and their scaling curves with a relatively tiny amount of extra compute. The key idea is to continue training a state-of-the-art large language model (e.g., PaLM) on a few more steps with UL2's mixture-of-denoiser objective. We show that, with almost negligible extra computational costs and no new sources of data, we are able to substantially improve the scaling properties of large language models on downstream metrics. In this paper, we continue training PaLM with UL2R, introducing a new set of models at 8B, 62B, and 540B scale which we call U-PaLM. Impressively, at 540B scale, we show an approximately 2x computational savings rate where U-PaLM achieves the same performance as the final PaLM 540B model at around half its computational budget (i.e., saving sim4.4 million TPUv4 hours). We further show that this improved scaling curve leads to 'emergent abilities' on challenging BIG-Bench tasks -- for instance, U-PaLM does much better than PaLM on some tasks or demonstrates better quality at much smaller scale (62B as opposed to 540B). Overall, we show that U-PaLM outperforms PaLM on many few-shot setups, i.e., English NLP tasks (e.g., commonsense reasoning, question answering), reasoning tasks with chain-of-thought (e.g., GSM8K), multilingual tasks (MGSM, TydiQA), MMLU and challenging BIG-Bench tasks. Finally, we provide qualitative examples showing the new capabilities of U-PaLM for single and multi-span infilling.

We study empirical scaling laws for language model performance on the cross-entropy loss. The loss scales as a power-law with model size, dataset size, and the amount of compute used for training, with some trends spanning more than seven orders of magnitude. Other architectural details such as network width or depth have minimal effects within a wide range. Simple equations govern the dependence of overfitting on model/dataset size and the dependence of training speed on model size. These relationships allow us to determine the optimal allocation of a fixed compute budget. Larger models are significantly more sample-efficient, such that optimally compute-efficient training involves training very large models on a relatively modest amount of data and stopping significantly before convergence.

Language model (LM) distillation is a trending area that aims to distil the knowledge resided in a large teacher LM to a small student one. While various methods have been proposed to push the distillation to its limits, it is still a pain distilling LMs when a large capacity gap is exhibited between the teacher and the student LMs. The pain is mainly resulted by the curse of capacity gap, which describes that a larger teacher LM cannot always lead to a better student LM than one distilled from a smaller teacher LM due to the affect of capacity gap increment. That is, there is likely an optimal point yielding the best student LM along the scaling course of the teacher LM. Even worse, the curse of capacity gap can be only partly yet not fully lifted as indicated in previous studies. However, the tale is not ever one-sided. Although a larger teacher LM has better performance than a smaller teacher LM, it is much more resource-demanding especially in the context of recent large LMs (LLMs). Consequently, instead of sticking to lifting the curse, leaving the curse as is should be arguably fine. Even better, in this paper, we reveal that the optimal capacity gap is almost consistent across different student scales and architectures, fortunately turning the curse into the law of capacity gap. The law later guides us to distil a 3B student LM (termed MiniMA) from a 7B teacher LM (adapted LLaMA2-7B). MiniMA is demonstrated to yield a new compute-performance pareto frontier among existing 3B LMs on commonly used benchmarks, and its instruction-tuned version (termed MiniChat) outperforms a wide range of 3B competitors in GPT4 evaluation and could even compete with several 7B chat models.

Neural scaling laws (NSL) refer to the phenomenon where model performance improves with scale. Sharma & Kaplan analyzed NSL using approximation theory and predict that MSE losses decay as N^{-alpha}, alpha=4/d, where N is the number of model parameters, and d is the intrinsic input dimension. Although their theory works well for some cases (e.g., ReLU networks), we surprisingly find that a simple 1D problem y=x^2 manifests a different scaling law (alpha=1) from their predictions (alpha=4). We opened the neural networks and found that the new scaling law originates from lottery ticket ensembling: a wider network on average has more "lottery tickets", which are ensembled to reduce the variance of outputs. We support the ensembling mechanism by mechanistically interpreting single neural networks, as well as studying them statistically. We attribute the N^{-1} scaling law to the "central limit theorem" of lottery tickets. Finally, we discuss its potential implications for large language models and statistical physics-type theories of learning.

We identify empirical scaling laws for the cross-entropy loss in four domains: generative image modeling, video modeling, multimodal imageleftrightarrowtext models, and mathematical problem solving. In all cases autoregressive Transformers smoothly improve in performance as model size and compute budgets increase, following a power-law plus constant scaling law. The optimal model size also depends on the compute budget through a power-law, with exponents that are nearly universal across all data domains. The cross-entropy loss has an information theoretic interpretation as S(True) + D_{KL}(True||Model), and the empirical scaling laws suggest a prediction for both the true data distribution's entropy and the KL divergence between the true and model distributions. With this interpretation, billion-parameter Transformers are nearly perfect models of the YFCC100M image distribution downsampled to an 8times 8 resolution, and we can forecast the model size needed to achieve any given reducible loss (ie D_{KL}) in nats/image for other resolutions. We find a number of additional scaling laws in specific domains: (a) we identify a scaling relation for the mutual information between captions and images in multimodal models, and show how to answer the question "Is a picture worth a thousand words?"; (b) in the case of mathematical problem solving, we identify scaling laws for model performance when extrapolating beyond the training distribution; (c) we finetune generative image models for ImageNet classification and find smooth scaling of the classification loss and error rate, even as the generative loss levels off. Taken together, these results strengthen the case that scaling laws have important implications for neural network performance, including on downstream tasks.

High-precision modeling of systems is one of the main areas of industrial data analysis. Models of systems, their digital twins, are used to predict their behavior under various conditions. We have developed several models of a storage system using machine learning-based generative models. The system consists of several components: hard disk drive (HDD) and solid-state drive (SSD) storage pools with different RAID schemes and cache. Each storage component is represented by a probabilistic model that describes the probability distribution of the component performance in terms of IOPS and latency, depending on their configuration and external data load parameters. The results of the experiments demonstrate the errors of 4-10 % for IOPS and 3-16 % for latency predictions depending on the components and models of the system. The predictions show up to 0.99 Pearson correlation with Little's law, which can be used for unsupervised reliability checks of the models. In addition, we present novel data sets that can be used for benchmarking regression algorithms, conditional generative models, and uncertainty estimation methods in machine learning.

Research on scaling large language models (LLMs) has primarily focused on model parameters and training data size, overlooking the role of vocabulary size. % Intuitively, larger vocabularies enable more efficient tokenization by representing sentences with fewer tokens, but they also increase the risk of under-fitting representations for rare tokens. We investigate how vocabulary size impacts LLM scaling laws by training models ranging from 33M to 3B parameters on up to 500B characters with various vocabulary configurations. We propose three complementary approaches for predicting the compute-optimal vocabulary size: IsoFLOPs analysis, derivative estimation, and parametric fit of the loss function. Our approaches converge on the same result that the optimal vocabulary size depends on the available compute budget and that larger models deserve larger vocabularies. However, most LLMs use too small vocabulary sizes. For example, we predict that the optimal vocabulary size of Llama2-70B should have been at least 216K, 7 times larger than its vocabulary of 32K. We validate our predictions empirically by training models with 3B parameters across different FLOPs budgets. Adopting our predicted optimal vocabulary size consistently improves downstream performance over commonly used vocabulary sizes. By increasing the vocabulary size from the conventional 32K to 43K, we improve performance on ARC-Challenge from 29.1 to 32.0 with the same 2.3e21 FLOPs. Our work emphasizes the necessity of jointly considering model parameters and vocabulary size for efficient scaling.

In this paper, we investigate the underlying factors that potentially enhance the mathematical reasoning capabilities of large language models (LLMs). We argue that the data scaling law for math reasoning capabilities in modern LLMs is far from being saturated, highlighting how the model's quality improves with increases in data quantity. To support this claim, we introduce the Skywork-Math model series, supervised fine-tuned (SFT) on common 7B LLMs using our proposed 2.5M-instance Skywork-MathQA dataset. Skywork-Math 7B has achieved impressive accuracies of 51.2% on the competition-level MATH benchmark and 83.9% on the GSM8K benchmark using only SFT data, outperforming an early version of GPT-4 on MATH. The superior performance of Skywork-Math models contributes to our novel two-stage data synthesis and model SFT pipelines, which include three different augmentation methods and a diverse seed problem set, ensuring both the quantity and quality of Skywork-MathQA dataset across varying difficulty levels. Most importantly, we provide several practical takeaways to enhance math reasoning abilities in LLMs for both research and industry applications.

Low-precision training is considered an effective strategy for reducing both training and downstream inference costs. Previous scaling laws for precision mainly focus on integer quantization, which pay less attention to the constituents in floating-point quantization and thus cannot well fit the LLM losses in this scenario. In contrast, while floating-point quantization training is more commonly implemented in production, the research on it has been relatively superficial. In this paper, we thoroughly explore the effects of floating-point quantization targets, exponent bits, mantissa bits, and the calculation granularity of the scaling factor in floating-point quantization training performance of LLM models. While presenting an accurate floating-point quantization unified scaling law, we also provide valuable suggestions for the community: (1) Exponent bits contribute slightly more to the model performance than mantissa bits. We provide the optimal exponent-mantissa bit ratio for different bit numbers, which is available for future reference by hardware manufacturers; (2) We discover the formation of the critical data size in low-precision LLM training. Too much training data exceeding the critical data size will inversely bring in degradation of LLM performance; (3) The optimal floating-point quantization precision is directly proportional to the computational power, but within a wide computational power range, we estimate that the best cost-performance precision lies between 4-8 bits.

The interest in linear complexity models for large language models is on the rise, although their scaling capacity remains uncertain. In this study, we present the scaling laws for linear complexity language models to establish a foundation for their scalability. Specifically, we examine the scaling behaviors of three efficient linear architectures. These include TNL, a linear attention model with data-independent decay; HGRN2, a linear RNN with data-dependent decay; and cosFormer2, a linear attention model without decay. We also include LLaMA as a baseline architecture for softmax attention for comparison. These models were trained with six variants, ranging from 70M to 7B parameters on a 300B-token corpus, and evaluated with a total of 1,376 intermediate checkpoints on various downstream tasks. These tasks include validation loss, commonsense reasoning, and information retrieval and generation. The study reveals that existing linear complexity language models exhibit similar scaling capabilities as conventional transformer-based models while also demonstrating superior linguistic proficiency and knowledge retention.

Since the introduction of Vision Transformer (ViT), patchification has long been regarded as a de facto image tokenization approach for plain visual architectures. By compressing the spatial size of images, this approach can effectively shorten the token sequence and reduce the computational cost of ViT-like plain architectures. In this work, we aim to thoroughly examine the information loss caused by this patchification-based compressive encoding paradigm and how it affects visual understanding. We conduct extensive patch size scaling experiments and excitedly observe an intriguing scaling law in patchification: the models can consistently benefit from decreased patch sizes and attain improved predictive performance, until it reaches the minimum patch size of 1x1, i.e., pixel tokenization. This conclusion is broadly applicable across different vision tasks, various input scales, and diverse architectures such as ViT and the recent Mamba models. Moreover, as a by-product, we discover that with smaller patches, task-specific decoder heads become less critical for dense prediction. In the experiments, we successfully scale up the visual sequence to an exceptional length of 50,176 tokens, achieving a competitive test accuracy of 84.6% with a base-sized model on the ImageNet-1k benchmark. We hope this study can provide insights and theoretical foundations for future works of building non-compressive vision models. Code is available at https://github.com/wangf3014/Patch_Scaling.

Cross-lingual transfer enables vision-language models (VLMs) to perform vision tasks in various languages with training data only in one language. Current approaches rely on large pre-trained multilingual language models. However, they face the curse of multilinguality, sacrificing downstream task performance for multilingual capabilities, struggling with lexical ambiguities, and falling behind recent advances. In this work, we study the scaling laws of systematic generalization with monolingual VLMs for multilingual tasks, focusing on the impact of model size and seen training samples. We propose Florenz, a monolingual encoder-decoder VLM with 0.4B to 11.2B parameters combining the pre-trained VLM Florence-2 and the large language model Gemma-2. Florenz is trained with varying compute budgets on a synthetic dataset that features intentionally incomplete language coverage for image captioning, thus, testing generalization from the fully covered translation task. We show that not only does indirectly learning unseen task-language pairs adhere to a scaling law, but also that with our data generation pipeline and the proposed Florenz model family, image captioning abilities can emerge in a specific language even when only data for the translation task is available. Fine-tuning on a mix of downstream datasets yields competitive performance and demonstrates promising scaling trends in multimodal machine translation (Multi30K, CoMMuTE), lexical disambiguation (CoMMuTE), and image captioning (Multi30K, XM3600, COCO Karpathy).

Large language models (LLMs), including both proprietary and open-source models, have showcased remarkable capabilities in addressing a wide range of downstream tasks. Nonetheless, when it comes to practical Chinese legal tasks, these models fail to meet the actual requirements. Proprietary models do not ensure data privacy for sensitive legal cases, while open-source models demonstrate unsatisfactory performance due to their lack of legal knowledge. To address this problem, we introduce LawGPT, the first open-source model specifically designed for Chinese legal applications. LawGPT comprises two key components: legal-oriented pre-training and legal supervised fine-tuning. Specifically, we employ large-scale Chinese legal documents for legal-oriented pre-training to incorporate legal domain knowledge. To further improve the model's performance on downstream legal tasks, we create a knowledge-driven instruction dataset for legal supervised fine-tuning. Our experimental results demonstrate that LawGPT outperforms the open-source LLaMA 7B model. Our code and resources are publicly available at https://github.com/pengxiao-song/LaWGPT and have received 5.7K stars on GitHub.

Discovering scaling laws for predicting model performance at scale is a fundamental and open-ended challenge, mostly reliant on slow, case specific human experimentation. To investigate the potential for LLMs to automate this process, we collect over 5,000 experiments from existing literature and curate seven diverse scaling law discovery tasks. While existing agents struggle to produce accurate law formulas, this paper introduces SLDAgent, an evolution-based agent that co-optimize the scaling law model and the parameters, enabling it to autonomously explore complex relationships between variables. For the first time, we demonstrates that SLDAgent can automatically discover laws that exhibit consistently more accurate extrapolation than their established, human-derived counterparts across all tasks. Through comprehensive analysis, we elucidate why these discovered laws are superior and verify their practical utility in both pretraining and finetuning applications. This work establishes a new paradigm for agentic scientific discovery, showing that AI systems can understand their own scaling behavior, and can contribute novel and practical knowledge back to the research community.

Achieving optimal performance of video diffusion transformers within given data and compute budget is crucial due to their high training costs. This necessitates precisely determining the optimal model size and training hyperparameters before large-scale training. While scaling laws are employed in language models to predict performance, their existence and accurate derivation in visual generation models remain underexplored. In this paper, we systematically analyze scaling laws for video diffusion transformers and confirm their presence. Moreover, we discover that, unlike language models, video diffusion models are more sensitive to learning rate and batch size, two hyperparameters often not precisely modeled. To address this, we propose a new scaling law that predicts optimal hyperparameters for any model size and compute budget. Under these optimal settings, we achieve comparable performance and reduce inference costs by 40.1% compared to conventional scaling methods, within a compute budget of 1e10 TFlops. Furthermore, we establish a more generalized and precise relationship among validation loss, any model size, and compute budget. This enables performance prediction for non-optimal model sizes, which may also be appealed under practical inference cost constraints, achieving a better trade-off.

Post-training quantization of Large Language Models (LLMs) has proven effective in reducing the computational requirements for running inference on these models. In this study, we focus on a straightforward question: When aiming for a specific accuracy or perplexity target for low-precision quantization, how many high-precision numbers or calculations are required to preserve as we scale LLMs to larger sizes? We first introduce a critical metric named the quantization ratio, which compares the number of parameters quantized to low-precision arithmetic against the total parameter count. Through extensive and carefully controlled experiments across different model families, arithmetic types, and quantization granularities (e.g. layer-wise, matmul-wise), we identify two central phenomenons. 1) The larger the models, the better they can preserve performance with an increased quantization ratio, as measured by perplexity in pre-training tasks or accuracy in downstream tasks. 2) The finer the granularity of mixed-precision quantization (e.g., matmul-wise), the more the model can increase the quantization ratio. We believe these observed phenomena offer valuable insights for future AI hardware design and the development of advanced Efficient AI algorithms.

In this work, we provide a large-scale empirical study of the scaling properties of multilingual neural machine translation models. We examine how increases in the model size affect the model performance and investigate the role of the training mixture composition on the scaling behavior. We find that changing the weightings of the individual language pairs in the training mixture only affect the multiplicative factor of the scaling law. In particular, we observe that multilingual models trained using different mixing rates all exhibit the same scaling exponent. Through a novel joint scaling law formulation, we compute the effective number of parameters allocated to each language pair and examine the role of language similarity in the scaling behavior of our models. We find little evidence that language similarity has any impact. In contrast, the direction of the multilinguality plays a significant role, with models translating from multiple languages into English having a larger number of effective parameters per task than their reversed counterparts. Finally, we leverage our observations to predict the performance of multilingual models trained with any language weighting at any scale, significantly reducing efforts required for language balancing in large multilingual models. Our findings apply to both in-domain and out-of-domain test sets and to multiple evaluation metrics, such as ChrF and BLEURT.

Driven by new types of wireless devices and the proliferation of bandwidth-intensive applications, data traffic and the corresponding network load are increasing dramatically. Network densification has been recognized as a promising and efficient way to provide higher network capacity and enhanced coverage. Most prior work on performance analysis of ultra-dense networks (UDNs) has focused on random spatial deployment with idealized singular path loss models and Rayleigh fading. In this paper, we consider a more precise and general model, which incorporates multi-slope path loss and general fading distributions. We derive the tail behavior and scaling laws for the coverage probability and the capacity considering strongest base station association in a Poisson field network. Our analytical results identify the regimes in which the signal-to-interference-plus-noise ratio (SINR) either asymptotically grows, saturates, or decreases with increasing network density. We establish general results on when UDNs lead to worse or even zero SINR coverage and capacity, and we provide crisp insights on the fundamental limits of wireless network densification.

We present the "Law of Vision Representation" in multimodal large language models (MLLMs). It reveals a strong correlation between the combination of cross-modal alignment, correspondence in vision representation, and MLLM performance. We quantify the two factors using the cross-modal Alignment and Correspondence score (AC score). Through extensive experiments involving thirteen different vision representation settings and evaluations across eight benchmarks, we find that the AC score is linearly correlated to model performance. By leveraging this relationship, we are able to identify and train the optimal vision representation only, which does not require finetuning the language model every time, resulting in a 99.7% reduction in computational cost.

We provide a distillation scaling law that estimates distilled model performance based on a compute budget and its allocation between the student and teacher. Our findings reduce the risks associated with using distillation at scale; compute allocation for both the teacher and student models can now be done to maximize student performance. We provide compute optimal distillation recipes for when 1) a teacher exists, or 2) a teacher needs training. If many students are to be distilled, or a teacher already exists, distillation outperforms supervised pretraining until a compute level which grows predictably with student size. If one student is to be distilled and a teacher also needs training, supervised learning should be done instead. Additionally, we provide insights across our large scale study of distillation, which increase our understanding of distillation and inform experimental design.

Activation sparsity denotes the existence of substantial weakly-contributed elements within activation outputs that can be eliminated, benefiting many important applications concerned with large language models (LLMs). Although promoting greater activation sparsity within LLMs deserves deep studies, existing works lack comprehensive and quantitative research on the correlation between activation sparsity and potentially influential factors. In this paper, we present a comprehensive study on the quantitative scaling properties and influential factors of the activation sparsity within decoder-only Transformer-based LLMs. Specifically, we propose PPL-p% sparsity, a precise and performance-aware activation sparsity metric that is applicable to any activation function. Through extensive experiments, we find several important phenomena. Firstly, different activation functions exhibit comparable performance but opposite training-time sparsity trends. The activation ratio (i.e., 1-sparsity ratio) evolves as a convergent increasing power-law and decreasing logspace power-law with the amount of training data for SiLU-activated and ReLU-activated LLMs, respectively. These demonstrate that ReLU is more efficient as the activation function than SiLU and can leverage more training data to improve activation sparsity. Secondly, the activation ratio linearly increases with the width-depth ratio below a certain bottleneck point, indicating the potential advantage of a deeper architecture at a fixed parameter scale. Finally, at similar width-depth ratios, we surprisingly find that the limit value of activation sparsity varies weakly with the parameter scale, i.e., the activation patterns within LLMs are insensitive to the parameter scale. These empirical laws towards LLMs with greater activation sparsity have important implications for making LLMs more efficient and interpretable.

Data scaling has revolutionized fields like natural language processing and computer vision, providing models with remarkable generalization capabilities. In this paper, we investigate whether similar data scaling laws exist in robotics, particularly in robotic manipulation, and whether appropriate data scaling can yield single-task robot policies that can be deployed zero-shot for any object within the same category in any environment. To this end, we conduct a comprehensive empirical study on data scaling in imitation learning. By collecting data across numerous environments and objects, we study how a policy's generalization performance changes with the number of training environments, objects, and demonstrations. Throughout our research, we collect over 40,000 demonstrations and execute more than 15,000 real-world robot rollouts under a rigorous evaluation protocol. Our findings reveal several intriguing results: the generalization performance of the policy follows a roughly power-law relationship with the number of environments and objects. The diversity of environments and objects is far more important than the absolute number of demonstrations; once the number of demonstrations per environment or object reaches a certain threshold, additional demonstrations have minimal effect. Based on these insights, we propose an efficient data collection strategy. With four data collectors working for one afternoon, we collect sufficient data to enable the policies for two tasks to achieve approximately 90% success rates in novel environments with unseen objects.

Statutory article retrieval (SAR), the task of retrieving statute law articles relevant to a legal question, is a promising application of legal text processing. In particular, high-quality SAR systems can improve the work efficiency of legal professionals and provide basic legal assistance to citizens in need at no cost. Unlike traditional ad-hoc information retrieval, where each document is considered a complete source of information, SAR deals with texts whose full sense depends on complementary information from the topological organization of statute law. While existing works ignore these domain-specific dependencies, we propose a novel graph-augmented dense statute retriever (G-DSR) model that incorporates the structure of legislation via a graph neural network to improve dense retrieval performance. Experimental results show that our approach outperforms strong retrieval baselines on a real-world expert-annotated SAR dataset.

Large Language Models (LLMs) excel in diverse tasks but often underperform in specialized fields due to limited domain-specific or proprietary corpus. Continual pre-training (CPT) enhances LLM capabilities by imbuing new domain-specific or proprietary knowledge while replaying general corpus to prevent catastrophic forgetting. The data mixture ratio of general corpus and domain-specific corpus, however, has been chosen heuristically, leading to sub-optimal training efficiency in practice. In this context, we attempt to re-visit the scaling behavior of LLMs under the hood of CPT, and discover a power-law relationship between loss, mixture ratio, and training tokens scale. We formalize the trade-off between general and domain-specific capabilities, leading to a well-defined Critical Mixture Ratio (CMR) of general and domain data. By striking the balance, CMR maintains the model's general ability and achieves the desired domain transfer, ensuring the highest utilization of available resources. Considering the balance between efficiency and effectiveness, CMR can be regarded as the optimal mixture ratio. Through extensive experiments, we ascertain the predictability of CMR, propose CMR scaling law and have substantiated its generalization. These findings offer practical guidelines for optimizing LLM training in specialized domains, ensuring both general and domain-specific performance while efficiently managing training resources.

We propose an approach to estimate the number of samples required for a model to reach a target performance. We find that the power law, the de facto principle to estimate model performance, leads to large error when using a small dataset (e.g., 5 samples per class) for extrapolation. This is because the log-performance error against the log-dataset size follows a nonlinear progression in the few-shot regime followed by a linear progression in the high-shot regime. We introduce a novel piecewise power law (PPL) that handles the two data regimes differently. To estimate the parameters of the PPL, we introduce a random forest regressor trained via meta learning that generalizes across classification/detection tasks, ResNet/ViT based architectures, and random/pre-trained initializations. The PPL improves the performance estimation on average by 37% across 16 classification and 33% across 10 detection datasets, compared to the power law. We further extend the PPL to provide a confidence bound and use it to limit the prediction horizon that reduces over-estimation of data by 76% on classification and 91% on detection datasets.

BERT has achieved impressive performance in several NLP tasks. However, there has been limited investigation on its adaptation guidelines in specialised domains. Here we focus on the legal domain, where we explore several approaches for applying BERT models to downstream legal tasks, evaluating on multiple datasets. Our findings indicate that the previous guidelines for pre-training and fine-tuning, often blindly followed, do not always generalize well in the legal domain. Thus we propose a systematic investigation of the available strategies when applying BERT in specialised domains. These are: (a) use the original BERT out of the box, (b) adapt BERT by additional pre-training on domain-specific corpora, and (c) pre-train BERT from scratch on domain-specific corpora. We also propose a broader hyper-parameter search space when fine-tuning for downstream tasks and we release LEGAL-BERT, a family of BERT models intended to assist legal NLP research, computational law, and legal technology applications.

Legislation can be viewed as a body of prescriptive rules expressed in natural language. The application of legislation to facts of a case we refer to as statutory reasoning, where those facts are also expressed in natural language. Computational statutory reasoning is distinct from most existing work in machine reading, in that much of the information needed for deciding a case is declared exactly once (a law), while the information needed in much of machine reading tends to be learned through distributional language statistics. To investigate the performance of natural language understanding approaches on statutory reasoning, we introduce a dataset, together with a legal-domain text corpus. Straightforward application of machine reading models exhibits low out-of-the-box performance on our questions, whether or not they have been fine-tuned to the legal domain. We contrast this with a hand-constructed Prolog-based system, designed to fully solve the task. These experiments support a discussion of the challenges facing statutory reasoning moving forward, which we argue is an interesting real-world task that can motivate the development of models able to utilize prescriptive rules specified in natural language.

Scaling laws have shaped recent advances in machine learning by enabling predictable scaling of model performance based on model size, computation, and data volume. Concurrently, the rise in computational cost for AI has motivated model compression techniques, notably quantization and sparsification, which have emerged to mitigate the steep computational demands associated with large-scale training and inference. This paper investigates the interplay between scaling laws and compression formats, exploring whether a unified scaling framework can accurately predict model performance when training occurs over various compressed representations, such as sparse, scalar-quantized, sparse-quantized or even vector-quantized formats. Our key contributions include validating a general scaling law formulation and showing that it is applicable both individually but also composably across compression types. Based on this, our main finding is demonstrating both theoretically and empirically that there exists a simple "capacity" metric -- based on the representation's ability to fit random Gaussian data -- which can robustly predict parameter efficiency across multiple compressed representations. On the practical side, we extend our formulation to directly compare the accuracy potential of different compressed formats, and to derive better algorithms for training over sparse-quantized formats.

This paper presents a systematic study of scaling laws for the deepfake detection task. Specifically, we analyze the model performance against the number of real image domains, deepfake generation methods, and training images. Since no existing dataset meets the scale requirements for this research, we construct ScaleDF, the largest dataset to date in this field, which contains over 5.8 million real images from 51 different datasets (domains) and more than 8.8 million fake images generated by 102 deepfake methods. Using ScaleDF, we observe power-law scaling similar to that shown in large language models (LLMs). Specifically, the average detection error follows a predictable power-law decay as either the number of real domains or the number of deepfake methods increases. This key observation not only allows us to forecast the number of additional real domains or deepfake methods required to reach a target performance, but also inspires us to counter the evolving deepfake technology in a data-centric manner. Beyond this, we examine the role of pre-training and data augmentations in deepfake detection under scaling, as well as the limitations of scaling itself.

Scaling up neural networks has led to remarkable performance across a wide range of tasks. Moreover, performance often follows reliable scaling laws as a function of training set size, model size, and compute, which offers valuable guidance as large-scale experiments are becoming increasingly expensive. However, previous work on scaling laws has primarily used private data \& models or focused on uni-modal language or vision learning. To address these limitations, we investigate scaling laws for contrastive language-image pre-training (CLIP) with the public LAION dataset and the open-source OpenCLIP repository. Our large-scale experiments involve models trained on up to two billion image-text pairs and identify power law scaling for multiple downstream tasks including zero-shot classification, retrieval, linear probing, and end-to-end fine-tuning. We find that the training distribution plays a key role in scaling laws as the OpenAI and OpenCLIP models exhibit different scaling behavior despite identical model architectures and similar training recipes. We open-source our evaluation workflow and all models, including the largest public CLIP models, to ensure reproducibility and make scaling laws research more accessible. Source code and instructions to reproduce this study will be available at https://github.com/LAION-AI/scaling-laws-openclip

Neural scaling laws offer valuable insights for designing robust sequence processing architectures. While these laws have been extensively characterized in other modalities, their behavior in speech remains comparatively underexplored. In this work, we introduce OWLS, an open-access, reproducible suite of multilingual speech recognition and translation models spanning 0.25B to 18B parameters, with the 18B version being the largest speech model, to the best of our knowledge. OWLS leverages up to 360K hours of public speech data across 150 languages, enabling a systematic investigation into how data, model, and compute scaling each influence performance in multilingual speech tasks. We use OWLS to derive neural scaling laws, showing how final performance can be reliably predicted when scaling. One of our key findings is that scaling enhances performance on low-resource languages/dialects, helping to mitigate bias and improve the accessibility of speech technologies. Finally, we show how OWLS can be used to power new research directions by discovering emergent abilities in large-scale speech models. Model checkpoints will be released on https://huggingface.co/collections/espnet/owls-scaling-laws-for-speech-recognition-and-translation-67ab7f991c194065f057ce8d for future studies.

Large language models (LLMs), both proprietary and open-source, have demonstrated remarkable capabilities across various natural language processing tasks. However, they face significant limitations in legal reasoning tasks. Proprietary models introduce data privacy risks and high inference costs, while open-source models underperform due to insufficient legal domain training data. To address these limitations, we study data generation for legal reasoning to improve the legal reasoning performance of open-source LLMs with the help of proprietary LLMs. This is challenging due to the lack of legal knowledge in proprietary LLMs and the difficulty in verifying the generated data. We propose KgDG, a knowledge-guided data generation framework for legal reasoning. Our framework enables leveraging legal knowledge to enhance generation diversity and introduces a refinement and verification process to ensure the quality of generated data. Moreover, we expand the generated dataset to further enhance the LLM reasoning capabilities. Using KgDG, we create a synthetic legal reasoning dataset containing 50K high-quality examples. Our trained model LawGPT outperforms existing legal-specific LLMs and achieves performance comparable to proprietary LLMs, demonstrating the effectiveness of KgDG and LawGPT. Our code and resources is publicly available at https://anonymous.4open.science/r/KgDG-45F5 .

Vision-language models (VLMs) are trained for thousands of GPU hours on carefully curated web datasets. In recent times, data curation has gained prominence with several works developing strategies to retain 'high-quality' subsets of 'raw' scraped data. For instance, the LAION public dataset retained only 10% of the total crawled data. However, these strategies are typically developed agnostic of the available compute for training. In this paper, we first demonstrate that making filtering decisions independent of training compute is often suboptimal: the limited high-quality data rapidly loses its utility when repeated, eventually requiring the inclusion of 'unseen' but 'lower-quality' data. To address this quality-quantity tradeoff (QQT), we introduce neural scaling laws that account for the non-homogeneous nature of web data, an angle ignored in existing literature. Our scaling laws (i) characterize the differing 'utility' of various quality subsets of web data; (ii) account for how utility diminishes for a data point at its 'nth' repetition; and (iii) formulate the mutual interaction of various data pools when combined, enabling the estimation of model performance on a combination of multiple data pools without ever jointly training on them. Our key message is that data curation cannot be agnostic of the total compute that a model will be trained for. Our scaling laws allow us to curate the best possible pool for achieving top performance on Datacomp at various compute budgets, carving out a pareto-frontier for data curation. Code is available at https://github.com/locuslab/scaling_laws_data_filtering.

Representations from transformer-based unidirectional language models are known to be effective at predicting brain responses to natural language. However, most studies comparing language models to brains have used GPT-2 or similarly sized language models. Here we tested whether larger open-source models such as those from the OPT and LLaMA families are better at predicting brain responses recorded using fMRI. Mirroring scaling results from other contexts, we found that brain prediction performance scales log-linearly with model size from 125M to 30B parameter models, with ~15% increased encoding performance as measured by correlation with a held-out test set across 3 subjects. Similar log-linear behavior was observed when scaling the size of the fMRI training set. We also characterized scaling for acoustic encoding models that use HuBERT, WavLM, and Whisper, and we found comparable improvements with model size. A noise ceiling analysis of these large, high-performance encoding models showed that performance is nearing the theoretical maximum for brain areas such as the precuneus and higher auditory cortex. These results suggest that increasing scale in both models and data will yield incredibly effective models of language processing in the brain, enabling better scientific understanding as well as applications such as decoding.

Although transformer architectures have achieved state-of-the-art performance across diverse domains, their quadratic computational complexity with respect to sequence length remains a significant bottleneck, particularly for latency-sensitive long-context applications. While recent linear-complexity alternatives are increasingly powerful, effectively training them from scratch is still resource-intensive. To overcome these limitations, we propose LAWCAT (Linear Attention with Convolution Across Time), a novel linearization framework designed to efficiently transfer the capabilities of pre-trained transformers into a performant linear attention architecture. LAWCAT integrates causal Conv1D layers to enhance local dependency modeling and employs normalized gated linear attention to improve generalization across varying context lengths. Our comprehensive evaluations demonstrate that, distilling Mistral-7B with only 1K-length sequences yields over 90\% passkey retrieval accuracy up to 22K tokens, significantly extending its effective context window. Similarly, Llama3.2-1B LAWCAT variant achieves competitive performance on S-NIAH 1\&2\&3 tasks (1K-8K context length) and BABILong benchmark (QA2\&QA3, 0K-16K context length), requiring less than 0.1\% pre-training tokens compared with pre-training models. Furthermore, LAWCAT exhibits faster prefill speeds than FlashAttention-2 for sequences exceeding 8K tokens. LAWCAT thus provides an efficient pathway to high-performance, long-context linear models suitable for edge deployment, reducing reliance on extensive long-sequence training data and computational resources.

In recent years, the state-of-the-art in deep learning has been dominated by very large models that have been pre-trained on vast amounts of data. The paradigm is very simple: Investing more computational resources (optimally) leads to better performance, and even predictably so; neural scaling laws have been derived that accurately forecast the performance of a network for a desired level of compute. This leads to the notion of a "compute-optimal" model, i.e. a model that allocates a given level of compute during training optimally to maximise performance. In this work, we extend the concept of optimality by allowing for an "adaptive" model, i.e. a model that can change its shape during the course of training. By allowing the shape to adapt, we can optimally traverse between the underlying scaling laws, leading to a significant reduction in the required compute to reach a given target performance. We focus on vision tasks and the family of Vision Transformers, where the patch size as well as the width naturally serve as adaptive shape parameters. We demonstrate that, guided by scaling laws, we can design compute-optimal adaptive models that beat their "static" counterparts.

Scaling laws have been recently employed to derive compute-optimal model size (number of parameters) for a given compute duration. We advance and refine such methods to infer compute-optimal model shapes, such as width and depth, and successfully implement this in vision transformers. Our shape-optimized vision transformer, SoViT, achieves results competitive with models that exceed twice its size, despite being pre-trained with an equivalent amount of compute. For example, SoViT-400m/14 achieves 90.3% fine-tuning accuracy on ILSRCV2012, surpassing the much larger ViT-g/14 and approaching ViT-G/14 under identical settings, with also less than half the inference cost. We conduct a thorough evaluation across multiple tasks, such as image classification, captioning, VQA and zero-shot transfer, demonstrating the effectiveness of our model across a broad range of domains and identifying limitations. Overall, our findings challenge the prevailing approach of blindly scaling up vision models and pave a path for a more informed scaling.

In reinforcement learning from human feedback, it is common to optimize against a reward model trained to predict human preferences. Because the reward model is an imperfect proxy, optimizing its value too much can hinder ground truth performance, in accordance with Goodhart's law. This effect has been frequently observed, but not carefully measured due to the expense of collecting human preference data. In this work, we use a synthetic setup in which a fixed "gold-standard" reward model plays the role of humans, providing labels used to train a proxy reward model. We study how the gold reward model score changes as we optimize against the proxy reward model using either reinforcement learning or best-of-n sampling. We find that this relationship follows a different functional form depending on the method of optimization, and that in both cases its coefficients scale smoothly with the number of reward model parameters. We also study the effect on this relationship of the size of the reward model dataset, the number of reward model and policy parameters, and the coefficient of the KL penalty added to the reward in the reinforcement learning setup. We explore the implications of these empirical results for theoretical considerations in AI alignment.

Large Language Models (LLMs) have emerged as a milestone in artificial intelligence, and their performance can improve as the model size increases. However, this scaling brings great challenges to training and inference efficiency, particularly for deploying LLMs in resource-constrained environments, and the scaling trend is becoming increasingly unsustainable. This paper introduces the concept of ``capacity density'' as a new metric to evaluate the quality of the LLMs across different scales and describes the trend of LLMs in terms of both effectiveness and efficiency. To calculate the capacity density of a given target LLM, we first introduce a set of reference models and develop a scaling law to predict the downstream performance of these reference models based on their parameter sizes. We then define the effective parameter size of the target LLM as the parameter size required by a reference model to achieve equivalent performance, and formalize the capacity density as the ratio of the effective parameter size to the actual parameter size of the target LLM. Capacity density provides a unified framework for assessing both model effectiveness and efficiency. Our further analysis of recent open-source base LLMs reveals an empirical law (the densing law)that the capacity density of LLMs grows exponentially over time. More specifically, using some widely used benchmarks for evaluation, the capacity density of LLMs doubles approximately every three months. The law provides new perspectives to guide future LLM development, emphasizing the importance of improving capacity density to achieve optimal results with minimal computational overhead.

Recent advancements in Large Language Models (LLMs) have revolutionized decision-making by breaking down complex problems into more manageable language sequences referred to as ``thoughts''. An effective thought design should consider three key perspectives: performance, efficiency, and flexibility. However, existing thought can at most exhibit two of these attributes. To address these limitations, we introduce a novel thought prompting approach called ``Everything of Thoughts'' (XoT) to defy the law of ``Penrose triangle of existing thought paradigms. XoT leverages pretrained reinforcement learning and Monte Carlo Tree Search (MCTS) to incorporate external domain knowledge into thoughts, thereby enhancing LLMs' capabilities and enabling them to generalize to unseen problems efficiently. Through the utilization of the MCTS-LLM collaborative thought revision framework, this approach autonomously produces high-quality comprehensive cognitive mappings with minimal LLM interactions. Additionally, XoT empowers LLMs to engage in unconstrained thinking, allowing for flexible cognitive mappings for problems with multiple solutions.

While scaling laws provide a reliable methodology for predicting train loss across compute scales for a single data distribution, less is known about how these predictions should change as we change the distribution. In this paper, we derive a strategy for predicting one loss from another and apply it to predict across different pre-training datasets and from pre-training data to downstream task data. Our predictions extrapolate well even at 20x the largest FLOP budget used to fit the curves. More precisely, we find that there are simple shifted power law relationships between (1) the train losses of two models trained on two separate datasets when the models are paired by training compute (train-to-train), (2) the train loss and the test loss on any downstream distribution for a single model (train-to-test), and (3) the test losses of two models trained on two separate train datasets (test-to-test). The results hold up for pre-training datasets that differ substantially (some are entirely code and others have no code at all) and across a variety of downstream tasks. Finally, we find that in some settings these shifted power law relationships can yield more accurate predictions than extrapolating single-dataset scaling laws.

Recent advancements in Reinforcement Post-Training (RPT) have significantly enhanced the capabilities of Large Reasoning Models (LRMs), sparking increased interest in the generalization of RL-based reasoning. While existing work has primarily focused on investigating its generalization across tasks or modalities, this study proposes a novel cross-linguistic perspective to investigate reasoning generalization. This raises a crucial question: Does the reasoning capability achieved from English RPT effectively transfer to other languages? We address this by systematically evaluating English-centric LRMs on multilingual reasoning benchmarks and introducing a metric to quantify cross-lingual transferability. Our findings reveal that cross-lingual transferability varies significantly across initial model, target language, and training paradigm. Through interventional studies, we find that models with stronger initial English capabilities tend to over-rely on English-specific patterns, leading to diminished cross-lingual generalization. To address this, we conduct a thorough parallel training study. Experimental results yield three key findings: First-Parallel Leap, a substantial leap in performance when transitioning from monolingual to just a single parallel language, and a predictable Parallel Scaling Law, revealing that cross-lingual reasoning transfer follows a power-law with the number of training parallel languages. Moreover, we identify the discrepancy between actual monolingual performance and the power-law prediction as Monolingual Generalization Gap, indicating that English-centric LRMs fail to fully generalize across languages. Our study challenges the assumption that LRM reasoning mirrors human cognition, providing critical insights for the development of more language-agnostic LRMs.

We present the Power Law Graph Transformer, a transformer model with well defined deductive and inductive tasks for prediction and representation learning. The deductive task learns the dataset level (global) and instance level (local) graph structures in terms of learnable power law distribution parameters. The inductive task outputs the prediction probabilities using the deductive task output, similar to a transductive model. We trained our model with Turkish-English and Portuguese-English datasets from TED talk transcripts for machine translation and compared the model performance and characteristics to a transformer model with scaled dot product attention trained on the same experimental setup. We report BLEU scores of 17.79 and 28.33 on the Turkish-English and Portuguese-English translation tasks with our model, respectively. We also show how a duality between a quantization set and N-dimensional manifold representation can be leveraged to transform between local and global deductive-inductive outputs using successive application of linear and non-linear transformations end-to-end.

LoRA (Low-Rank Adaptation) is a widely used model fine-tuning method. In fine-tuning, the law among model performance, model parameters, and data complexity has been a focal issue in the field. Existing methods often leverage external metrics (such as cross-entropy or perplexity) to evaluate model performance. In the fine-tuning process for large models, two types of knowledge are typically involved: the frozen, general knowledge acquired by the model during pre-training and the new knowledge learned through the LoRA module from the current data. Generally, the less LoRA's learned knowledge relies on the large model, the more it captures the specific knowledge of new data, thereby enhancing its adaptability to new tasks. However, external metrics do not readily capture the dependency relationship between these two types of knowledge. Therefore, we designed an internal metric based on the Mutual Information Upper Bound (MIUB) theory to investigate the scaling law of large-model LoRA fine-tuning. In our experiments, we validated this approach on benchmark datasets, using the Llama3-8B and Phi3-3B models. The results show that the proposed MIUB metric aligns more accurately and stably with the scaling law of LoRA fine-tuning compared to cross-entropy and perplexity.

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.

Reinforcement Learning from Human Feedback (RLHF) has been crucial to the recent success of Large Language Models (LLMs), however, it is often a complex and brittle process. In the classical RLHF framework, a reward model is first trained to represent human preferences, which is in turn used by an online reinforcement learning (RL) algorithm to optimize the LLM. A prominent issue with such methods is reward over-optimization or reward hacking, where performance as measured by the learned proxy reward model increases, but true quality plateaus or even deteriorates. Direct Alignment Algorithms (DDAs) like Direct Preference Optimization have emerged as alternatives to the classical RLHF pipeline by circumventing the reward modeling phase. However, although DAAs do not use a separate proxy reward model, they still commonly deteriorate from over-optimization. While the so-called reward hacking phenomenon is not well-defined for DAAs, we still uncover similar trends: at higher KL budgets, DAA algorithms exhibit similar degradation patterns to their classic RLHF counterparts. In particular, we find that DAA methods deteriorate not only across a wide range of KL budgets but also often before even a single epoch of the dataset is completed. Through extensive empirical experimentation, this work formulates and formalizes the reward over-optimization or hacking problem for DAAs and explores its consequences across objectives, training regimes, and model scales.

We explore the impact of parameter sparsity on the scaling behavior of Transformers trained on massive datasets (i.e., "foundation models"), in both vision and language domains. In this setting, we identify the first scaling law describing the relationship between weight sparsity, number of non-zero parameters, and amount of training data, which we validate empirically across model and data scales; on ViT/JFT-4B and T5/C4. These results allow us to characterize the "optimal sparsity", the sparsity level which yields the best performance for a given effective model size and training budget. For a fixed number of non-zero parameters, we identify that the optimal sparsity increases with the amount of data used for training. We also extend our study to different sparsity structures (such as the hardware-friendly n:m pattern) and strategies (such as starting from a pretrained dense model). Our findings shed light on the power and limitations of weight sparsity across various parameter and computational settings, offering both theoretical understanding and practical implications for leveraging sparsity towards computational efficiency improvements.

Scale has become a main ingredient in obtaining strong machine learning models. As a result, understanding a model's scaling properties is key to effectively designing both the right training setup as well as future generations of architectures. In this work, we argue that scale and training research has been needlessly complex due to reliance on the cosine schedule, which prevents training across different lengths for the same model size. We investigate the training behavior of a direct alternative - constant learning rate and cooldowns - and find that it scales predictably and reliably similar to cosine. Additionally, we show that stochastic weight averaging yields improved performance along the training trajectory, without additional training costs, across different scales. Importantly, with these findings we demonstrate that scaling experiments can be performed with significantly reduced compute and GPU hours by utilizing fewer but reusable training runs.

The extrapolation capability of Large Language Models (LLMs) based on Rotary Position Embedding is currently a topic of considerable interest. The mainstream approach to addressing extrapolation with LLMs involves modifying RoPE by replacing 10000, the rotary base of theta_n={10000}^{-2n/d} in the original RoPE, with a larger value and providing longer fine-tuning text. In this work, we first observe that fine-tuning a RoPE-based LLM with either a smaller or larger base in pre-training context length could significantly enhance its extrapolation performance. After that, we propose \textit{Scaling Laws of RoPE-based Extrapolation}, a unified framework from the periodic perspective, to describe the relationship between the extrapolation performance and base value as well as tuning context length. In this process, we also explain the origin of the RoPE-based extrapolation issue by \textit{critical dimension for extrapolation}. Besides these observations and analyses, we achieve extrapolation up to 1 million context length within only 16K training length on LLaMA2 7B and 13B.

Pretraining large language models (LLMs) is resource-intensive, often requiring months of training time even with high-end GPU clusters. There are two approaches of mitigating such computational demands: reusing smaller models to train larger ones (upcycling), and training computationally efficient models like mixture-of-experts (MoE). In this paper, we study the upcycling of LLMs to MoE models, of which the scaling behavior remains underexplored. Through extensive experiments, we identify empirical scaling laws that describe how performance depends on dataset size and model configuration. Particularly, we show that, while scaling these factors improves performance, there is a novel interaction term between the dense and upcycled training dataset that limits the efficiency of upcycling at large computational budgets. Based on these findings, we provide guidance to scale upcycling, and establish conditions under which upcycling outperforms from-scratch trainings within budget constraints.

Scaling laws have emerged as important components of large language model (LLM) training as they can predict performance gains through scale, and provide guidance on important hyper-parameter choices that would otherwise be expensive. LLMs also rely on large, high-quality training datasets, like those sourced from (sometimes sensitive) user data. Training models on this sensitive user data requires careful privacy protections like differential privacy (DP). However, the dynamics of DP training are significantly different, and consequently their scaling laws are not yet fully understood. In this work, we establish scaling laws that accurately model the intricacies of DP LLM training, providing a complete picture of the compute-privacy-utility tradeoffs and the optimal training configurations in many settings.

Vision generative models have recently made significant advancements along two primary paradigms: diffusion-style and language-style, both of which have demonstrated excellent scaling laws. Quantization is crucial for efficiently deploying these models, as it reduces memory and computation costs. In this work, we systematically investigate the impact of quantization on these two paradigms. Surprisingly, despite achieving comparable performance in full precision, language-style models consistently outperform diffusion-style models across various quantization settings. This observation suggests that language-style models have superior bit-level scaling laws, offering a better tradeoff between model quality and total bits. To dissect this phenomenon, we conduct extensive experiments and find that the primary reason is the discrete representation space of language-style models, which is more tolerant of information loss during quantization. Furthermore, our analysis indicates that improving the bit-level scaling law of quantized vision generative models is challenging, with model distillation identified as a highly effective approach. Specifically, we propose TopKLD to optimize the transfer of distilled knowledge by balancing ``implicit knowledge'' and ``explicit knowledge'' during the distillation process. This approach elevates the bit-level scaling laws by one level across both integer and floating-point quantization settings.

Scaling laws offer valuable insights into the design of time series foundation models (TSFMs). However, previous research has largely focused on the scaling laws of TSFMs for in-distribution (ID) data, leaving their out-of-distribution (OOD) scaling behavior and the influence of model architectures less explored. In this work, we examine two common TSFM architectures, encoder-only and decoder-only Transformers, and investigate their scaling behavior on both ID and OOD data. These models are trained and evaluated across varying parameter counts, compute budgets, and dataset sizes. Our experiments reveal that the log-likelihood loss of TSFMs exhibits similar scaling behavior in both OOD and ID settings. We further compare the scaling properties across different architectures, incorporating two state-of-the-art TSFMs as case studies, showing that model architecture plays a significant role in scaling. The encoder-only Transformers demonstrate better scalability than the decoder-only Transformers, while the architectural enhancements in the two advanced TSFMs primarily improve ID performance but reduce OOD scalability. While scaling up TSFMs is expected to drive performance breakthroughs, the lack of a comprehensive understanding of TSFM scaling laws has hindered the development of a robust framework to guide model scaling. We fill this gap in this work by synthesizing our findings and providing practical guidelines for designing and scaling larger TSFMs with enhanced model capabilities.

We study and quantify the problem of forgetting when fine-tuning pre-trained large language models (LLMs) on a downstream task. We find that parameter-efficient fine-tuning (PEFT) strategies, such as Low-Rank Adapters (LoRA), still suffer from catastrophic forgetting. In particular, we identify a strong inverse linear relationship between the fine-tuning performance and the amount of forgetting when fine-tuning LLMs with LoRA. We further obtain precise scaling laws that show forgetting increases as a shifted power law in the number of parameters fine-tuned and the number of update steps. We also examine the impact of forgetting on knowledge, reasoning, and the safety guardrails trained into Llama 2 7B chat. Our study suggests that forgetting cannot be avoided through early stopping or by varying the number of parameters fine-tuned. We believe this opens up an important safety-critical direction for future research to evaluate and develop fine-tuning schemes which mitigate forgetting

Alternatives to backpropagation have long been studied to better understand how biological brains may learn. Recently, they have also garnered interest as a way to train neural networks more efficiently. By relaxing constraints inherent to backpropagation (e.g., symmetric feedforward and feedback weights, sequential updates), these methods enable promising prospects, such as local learning. However, the tradeoffs between different methods in terms of final task performance, convergence speed, and ultimately compute and data requirements are rarely outlined. In this work, we use scaling laws to study the ability of Direct Feedback Alignment~(DFA) to train causal decoder-only Transformers efficiently. Scaling laws provide an overview of the tradeoffs implied by a modeling decision, up to extrapolating how it might transfer to increasingly large models. We find that DFA fails to offer more efficient scaling than backpropagation: there is never a regime for which the degradation in loss incurred by using DFA is worth the potential reduction in compute budget. Our finding comes at variance with previous beliefs in the alternative training methods community, and highlights the need for holistic empirical approaches to better understand modeling decisions.

Multimodal Large Languages models have been progressing from uni-modal understanding toward unifying visual, audio and language modalities, collectively termed omni models. However, the correlation between uni-modal and omni-modal remains unclear, which requires comprehensive evaluation to drive omni model's intelligence evolution. In this work, we propose a novel, high quality and UNified Omni model benchmark, UNO-Bench, which effectively assesses both UNi-modal and Omni-modal capabilities. The benchmark consists of 3730 human curated samples, with 98% cross-modality solvability, across 44 task types, and an innovative multi-step open-ended question type for assessing complex reasoning. Besides, a general scoring model supporting 6 question types is proposed for automated evaluation with 95% accuracy. Experimental result shows the Compositional Law between omni-modal and uni-modal performance and the omni-modal capability manifests as a bottleneck effect on weak models, while exhibiting synergistic promotion on strong models. The code and data are available at https://github.com/meituan-longcat/UNO-Bench

Neural networks often suffer from catastrophic interference (CI): performance on previously learned tasks drops off significantly when learning a new task. This contrasts strongly with humans, who can sequentially learn new tasks without appreciably forgetting previous tasks. Prior work has explored various techniques for mitigating CI such as regularization, rehearsal, generative replay, and distillation methods. The current work takes a different approach, one guided by cognitive science research showing that in naturalistic environments, the probability of encountering a task decreases as a power-law of the time since it was last performed. We argue that a realistic evaluation of techniques for the mitigation of CI should be performed in simulated naturalistic learning environments. Thus, we evaluate the extent of mitigation of CI when training simple rehearsal-based methods in power-law environments similar to the ones humans face. Our work explores this novel rehearsal-based approach for a domain-incremental task: learning permutations in the MNIST task. We compare our rehearsal environment with other baselines to show its efficacy in promoting continual learning. Additionally, we investigate whether this environment shows forward facilitation, i.e., faster learning of later tasks. Next, we explore the robustness of our learning environment to the number of tasks, model size, and amount of data rehearsed after each task. Notably, our results show that the performance is comparable or superior to that of models trained using popular regularization methods and also to rehearsals in non-power-law environments. The benefits of this training paradigm include simplicity and the lack of a need for extra neural circuitry. In addition, because our method is orthogonal to other methods, future research can combine training in power-law environments with other continual learning mechanisms.

Implementing a reward function that perfectly captures a complex task in the real world is impractical. As a result, it is often appropriate to think of the reward function as a proxy for the true objective rather than as its definition. We study this phenomenon through the lens of Goodhart's law, which predicts that increasing optimisation of an imperfect proxy beyond some critical point decreases performance on the true objective. First, we propose a way to quantify the magnitude of this effect and show empirically that optimising an imperfect proxy reward often leads to the behaviour predicted by Goodhart's law for a wide range of environments and reward functions. We then provide a geometric explanation for why Goodhart's law occurs in Markov decision processes. We use these theoretical insights to propose an optimal early stopping method that provably avoids the aforementioned pitfall and derive theoretical regret bounds for this method. Moreover, we derive a training method that maximises worst-case reward, for the setting where there is uncertainty about the true reward function. Finally, we evaluate our early stopping method experimentally. Our results support a foundation for a theoretically-principled study of reinforcement learning under reward misspecification.

This research aims to build generative language models specialized for the legal domain. The manuscript presents the development of LexGPT models based on GPT-J models and pre-trained with Pile of Law. The foundation model built in this manuscript is the initial step for the development of future applications in the legal domain, such as further training with reinforcement learning from human feedback. Another objective of this manuscript is to assist legal professionals in utilizing language models through the ``No Code'' approach. By fine-tuning models with specialized data and without modifying any source code, legal professionals can create custom language models for downstream tasks with minimum effort and technical knowledge. The downstream task in this manuscript is to turn a LexGPT model into a classifier, although the performance is notably lower than the state-of-the-art result. How to enhance downstream task performance without modifying the model or its source code is a research topic for future exploration.

We study the robust interpolation problem of arbitrary data distributions supported on a bounded space and propose a two-fold law of robustness. Robust interpolation refers to the problem of interpolating n noisy training data points in R^d by a Lipschitz function. Although this problem has been well understood when the samples are drawn from an isoperimetry distribution, much remains unknown concerning its performance under generic or even the worst-case distributions. We prove a Lipschitzness lower bound Omega(n/p) of the interpolating neural network with p parameters on arbitrary data distributions. With this result, we validate the law of robustness conjecture in prior work by Bubeck, Li, and Nagaraj on two-layer neural networks with polynomial weights. We then extend our result to arbitrary interpolating approximators and prove a Lipschitzness lower bound Omega(n^{1/d}) for robust interpolation. Our results demonstrate a two-fold law of robustness: i) we show the potential benefit of overparametrization for smooth data interpolation when n=poly(d), and ii) we disprove the potential existence of an O(1)-Lipschitz robust interpolating function when n=exp(omega(d)).

In a legal system, judgment consistency is regarded as one of the most important manifestations of fairness. However, due to the complexity of factual elements that impact sentencing in real-world scenarios, few works have been done on quantitatively measuring judgment consistency towards real-world data. In this paper, we propose an evaluation metric for judgment inconsistency, Legal Inconsistency Coefficient (LInCo), which aims to evaluate inconsistency between data groups divided by specific features (e.g., gender, region, race). We propose to simulate judges from different groups with legal judgment prediction (LJP) models and measure the judicial inconsistency with the disagreement of the judgment results given by LJP models trained on different groups. Experimental results on the synthetic data verify the effectiveness of LInCo. We further employ LInCo to explore the inconsistency in real cases and come to the following observations: (1) Both regional and gender inconsistency exist in the legal system, but gender inconsistency is much less than regional inconsistency; (2) The level of regional inconsistency varies little across different time periods; (3) In general, judicial inconsistency is negatively correlated with the severity of the criminal charges. Besides, we use LInCo to evaluate the performance of several de-bias methods, such as adversarial learning, and find that these mechanisms can effectively help LJP models to avoid suffering from data bias.

Software engineering (SWE) has recently emerged as a crucial testbed for next-generation LLM agents, demanding inherent capabilities in two critical dimensions: sustained iterative problem-solving (e.g., >50 interaction rounds) and long-context dependency resolution (e.g., >32k tokens). However, the data curation process in SWE remains notoriously time-consuming, as it heavily relies on manual annotation for code file filtering and the setup of dedicated runtime environments to execute and validate unit tests. Consequently, most existing datasets are limited to only a few thousand GitHub-sourced instances. To this end, we propose an incremental, automated data-curation pipeline that systematically scales both the volume and diversity of SWE datasets. Our dataset comprises 10,169 real-world Python task instances from 2,531 distinct GitHub repositories, each accompanied by a task specified in natural language and a dedicated runtime-environment image for automated unit-test validation. We have carefully curated over 8,000 successfully runtime-validated training trajectories from our proposed SWE dataset. When fine-tuning the Skywork-SWE model on these trajectories, we uncover a striking data scaling phenomenon: the trained model's performance for software engineering capabilities in LLMs continues to improve as the data size increases, showing no signs of saturation. Notably, our Skywork-SWE model achieves 38.0% pass@1 accuracy on the SWE-bench Verified benchmark without using verifiers or multiple rollouts, establishing a new state-of-the-art (SOTA) among the Qwen2.5-Coder-32B-based LLMs built on the OpenHands agent framework. Furthermore, with the incorporation of test-time scaling techniques, the performance further improves to 47.0% accuracy, surpassing the previous SOTA results for sub-32B parameter models. We release the Skywork-SWE-32B model checkpoint to accelerate future research.

We reveal that low-bit quantization favors undertrained large language models (LLMs) by observing that models with larger sizes or fewer training tokens experience less quantization-induced degradation (QiD) when applying low-bit quantization, whereas smaller models with extensive training tokens suffer significant QiD. To gain deeper insights into this trend, we study over 1500 quantized LLM checkpoints of various sizes and at different training levels (undertrained or fully trained) in a controlled setting, deriving scaling laws for understanding the relationship between QiD and factors such as the number of training tokens, model size and bit width. With the derived scaling laws, we propose a novel perspective that we can use QiD to measure an LLM's training levels and determine the number of training tokens required for fully training LLMs of various sizes. Moreover, we use the scaling laws to predict the quantization performance of different-sized LLMs trained with 100 trillion tokens. Our projection shows that the low-bit quantization performance of future models, which are expected to be trained with over 100 trillion tokens, may NOT be desirable. This poses a potential challenge for low-bit quantization in the future and highlights the need for awareness of a model's training level when evaluating low-bit quantization research. To facilitate future research on this problem, we release all the 1500+ quantized checkpoints used in this work at https://huggingface.co/Xu-Ouyang.

In studies of transferable learning, scaling laws are obtained for various important foundation models to predict their properties and performance at larger scales. We show here how scaling law derivation can also be used for model and dataset comparison, allowing to decide which procedure is to be preferred for pre-training. For the first time, full scaling laws based on dense measurements across a wide span of model and samples seen scales are derived for two important language-vision learning procedures, CLIP and MaMMUT, that use either contrastive only or contrastive and captioning text generative loss. Ensuring sufficient prediction accuracy for held out points, we use derived scaling laws to compare both models, obtaining evidence for MaMMUT's stronger improvement with scale and better sample efficiency than standard CLIP. To strengthen validity of the comparison, we show scaling laws for various downstream tasks, classification, retrieval, and segmentation, and for different open datasets, DataComp, DFN and Re-LAION, observing consistently the same trends. We show that comparison can also be performed when deriving scaling laws with a constant learning rate schedule, reducing compute cost. Accurate derivation of scaling laws provides thus means to perform model and dataset comparison across scale spans, avoiding misleading conclusions based on measurements from single reference scales only, paving the road for systematic comparison and improvement of open foundation models and datasets for their creation. We release all the pre-trained models with their intermediate checkpoints, including openMaMMUT-L/14, which achieves 80.3% zero-shot ImageNet-1k accuracy, trained on 12.8B samples from DataComp-1.4B. Code for reproducing experiments in the paper and raw experiments data can be found at https://github.com/LAION-AI/scaling-laws-for-comparison.

We develop task scaling laws and model ladders to predict the individual task performance of pretrained language models (LMs) in the overtrained setting. Standard power laws for language modeling loss cannot accurately model task performance. Therefore, we leverage a two-step prediction approach: first use model and data size to predict a task-specific loss, and then use this task loss to predict task performance. We train a set of small-scale "ladder" models, collect data points to fit the parameterized functions of the two prediction steps, and make predictions for two target models: a 7B model trained to 4T tokens and a 13B model trained to 5T tokens. Training the ladder models only costs 1% of the compute used for the target models. On four multiple-choice tasks written in ranked classification format, we can predict the accuracy of both target models within 2 points of absolute error. We have higher prediction error on four other tasks (average absolute error 6.9) and find that these are often tasks with higher variance in task metrics. We also find that using less compute to train fewer ladder models tends to deteriorate predictions. Finally, we empirically show that our design choices and the two-step approach lead to superior performance in establishing scaling laws.

Recent studies have showcased remarkable capabilities of decoder-only models in many NLP tasks, including translation. Yet, the machine translation field has been largely dominated by encoder-decoder models based on the Transformer architecture. As a consequence, scaling laws of encoder-decoder models for neural machine translation have already been well studied, but decoder-only models have received less attention. This work explores the scaling laws of decoder-only models on the multilingual and multidomain translation task. We trained a collection of six decoder-only models, ranging from 70M to 7B parameters, on a sentence-level, multilingual and multidomain dataset. We conducted a series of experiments showing that the loss of decoder-only models can be estimated using a scaling law similar to the one discovered for large language models, but we also show that this scaling law has difficulties to generalize to too large models or to a different data distribution. We also study different scaling methods and show that scaling the depth and the width of a model lead to similar test loss improvements, but with different impact on the model's efficiency.

The deployment of ever-larger machine learning models reflects a growing consensus that the more expressive the modelx2013and the more data one has access tox2013the more one can improve performance. As models get deployed in a variety of real world scenarios, they inevitably face strategic environments. In this work, we consider the natural question of how the interplay of models and strategic interactions affects scaling laws. We find that strategic interactions can break the conventional view of scaling lawsx2013meaning that performance does not necessarily monotonically improve as models get larger and/ or more expressive (even with infinite data). We show the implications of this phenomenon in several contexts including strategic regression, strategic classification, and multi-agent reinforcement learning through examples of strategic environments in whichx2013by simply restricting the expressivity of one's model or policy classx2013one can achieve strictly better equilibrium outcomes. Motivated by these examples, we then propose a new paradigm for model-selection in games wherein an agent seeks to choose amongst different model classes to use as their action set in a game.

The escalating demand for efficient decoding in large language models (LLMs) is particularly critical for reasoning-intensive architectures like OpenAI-o3 and DeepSeek-R1, which depend on extended chain-of-thought reasoning. This study investigates speculative decoding techniques through dense LLM architectures to establish foundational insights for accelerating reasoning tasks. While speculative decoding methods leveraging parallel draft-verification cycles have emerged as promising acceleration techniques, the scaling laws governing decoding efficiency remain under-explored compared to conventional backbone LLMs developed through Pretraining->SFT->RLHF training paradigms. In this work, we discover Log-linear Scaling Laws (Theorem 1.1, 1.2 and 1.3) governing draft model acceptance rate (or decoding speed) across three dimensions: pretraining token volume, draft model capacity, and decoding batch size. Building on these laws, we achieve Scylla, which coordinates multi-dimensional scaling for popular LLMs (Llama2/3, Qwen2.5). Empirical validation shows Scylla achieves 1.5-2.2 higher acceptance rate than EAGLE2 and 0.3 higher than EAGLE3 at temperature T = 0, with peak performance gains on summarization and QA tasks (Figure 2). Industrial inference engine deployments demonstrate 2X decoding throughput improvements over EAGLE2 (Table 5), validating the transformative potential of systematic scaling for efficient LLM inference. Code will be released later.

Diffusion transformers (DiT) have already achieved appealing synthesis and scaling properties in content recreation, e.g., image and video generation. However, scaling laws of DiT are less explored, which usually offer precise predictions regarding optimal model size and data requirements given a specific compute budget. Therefore, experiments across a broad range of compute budgets, from 1e17 to 6e18 FLOPs are conducted to confirm the existence of scaling laws in DiT for the first time. Concretely, the loss of pretraining DiT also follows a power-law relationship with the involved compute. Based on the scaling law, we can not only determine the optimal model size and required data but also accurately predict the text-to-image generation loss given a model with 1B parameters and a compute budget of 1e21 FLOPs. Additionally, we also demonstrate that the trend of pre-training loss matches the generation performances (e.g., FID), even across various datasets, which complements the mapping from compute to synthesis quality and thus provides a predictable benchmark that assesses model performance and data quality at a reduced cost.

Scaling law principles indicate a power-law correlation between loss and variables such as model size, dataset size, and computational resources utilized during training. These principles play a vital role in optimizing various aspects of model pre-training, ultimately contributing to the success of large language models such as GPT-4, Llama and Gemini. However, the original scaling law paper by OpenAI did not disclose the complete details necessary to derive the precise scaling law formulas, and their conclusions are only based on models containing up to 1.5 billion parameters. Though some subsequent works attempt to unveil these details and scale to larger models, they often neglect the training dependency of important factors such as the learning rate, context length and batch size, leading to their failure to establish a reliable formula for predicting the test loss trajectory. In this technical report, we confirm that the scaling law formulations proposed in the original OpenAI paper remain valid when scaling the model size up to 33 billion, but the constant coefficients in these formulas vary significantly with the experiment setup. We meticulously identify influential factors and provide transparent, step-by-step instructions to estimate all constant terms in scaling-law formulas by training on models with only 1M~60M parameters. Using these estimated formulas, we showcase the capability to accurately predict various attributes for models with up to 33B parameters before their training, including (1) the minimum possible test loss; (2) the minimum required training steps and processed tokens to achieve a specific loss; (3) the critical batch size with an optimal time/computation trade-off at any loss value; and (4) the complete test loss trajectory with arbitrary batch size.

Large Language Models (LLMs), like LLaMA, have exhibited remarkable performance across various tasks. Nevertheless, when deployed to specific domains such as law or medicine, the models still confront the challenge of a deficiency in domain-specific knowledge and an inadequate capability to leverage that knowledge to resolve domain-related problems. In this paper, we propose a new framework to adapt LLMs to specific domains and build Lawyer LLaMA, a legal domain LLM, based on this framework. Specifically, we inject domain knowledge during the continual training stage and teach the model to learn professional skills using properly designed supervised fine-tuning tasks. Moreover, to alleviate the hallucination problem during the model's generation, we add a retrieval module and extract relevant legal articles before the model answers any queries. When learning domain-specific skills, we find that experts' experience is much more useful than experiences distilled from ChatGPT, where hundreds of expert-written data outperform tens of thousands of ChatGPT-generated ones. We will release our model and data.

There have been a lot of interest in the scaling properties of Transformer models. However, not much has been done on the front of investigating the effect of scaling properties of different inductive biases and model architectures. Do model architectures scale differently? If so, how does inductive bias affect scaling behaviour? How does this influence upstream (pretraining) and downstream (transfer)? This paper conducts a systematic study of scaling behaviour of ten diverse model architectures such as Transformers, Switch Transformers, Universal Transformers, Dynamic convolutions, Performers, and recently proposed MLP-Mixers. Via extensive experiments, we show that (1) architecture is an indeed an important consideration when performing scaling and (2) the best performing model can fluctuate at different scales. We believe that the findings outlined in this work has significant implications to how model architectures are currently evaluated in the community.

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 the third 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(log P) while showing superior inference efficiency. For example, ParScale can use up to 22times less memory increase and 6times 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.

We propose a general two-stage algorithm that enjoys a provable scaling law for the test-time compute of large language models (LLMs). Given an input problem, the proposed algorithm first generates N candidate solutions, and then chooses the best one via a multiple-round knockout tournament where each pair of candidates are compared for K times and only the winners move on to the next round. In a minimalistic implementation, both stages can be executed with a black-box LLM alone and nothing else (e.g., no external verifier or reward model), and a total of N times (K + 1) highly parallelizable LLM calls are needed for solving an input problem. Assuming that a generated candidate solution is correct with probability p_{gen} > 0 and a comparison between a pair of correct and incorrect solutions identifies the right winner with probability p_{comp} > 0.5 (i.e., better than a random guess), we prove theoretically that the failure probability of the proposed algorithm decays to zero exponentially with respect to N and K: $P(final output is incorrect) le (1 - p_{gen})^N + lceil log_2 N rceil e^{-2 K (p_{comp} - 0.5)^2}.$ Our empirical results with the challenging MMLU-Pro benchmark validate the technical assumptions, as well as the efficacy of the proposed algorithm and the gains from scaling up its test-time compute.

Widely observed neural scaling laws, in which error falls off as a power of the training set size, model size, or both, have driven substantial performance improvements in deep learning. However, these improvements through scaling alone require considerable costs in compute and energy. Here we focus on the scaling of error with dataset size and show how in theory we can break beyond power law scaling and potentially even reduce it to exponential scaling instead if we have access to a high-quality data pruning metric that ranks the order in which training examples should be discarded to achieve any pruned dataset size. We then test this improved scaling prediction with pruned dataset size empirically, and indeed observe better than power law scaling in practice on ResNets trained on CIFAR-10, SVHN, and ImageNet. Next, given the importance of finding high-quality pruning metrics, we perform the first large-scale benchmarking study of ten different data pruning metrics on ImageNet. We find most existing high performing metrics scale poorly to ImageNet, while the best are computationally intensive and require labels for every image. We therefore developed a new simple, cheap and scalable self-supervised pruning metric that demonstrates comparable performance to the best supervised metrics. Overall, our work suggests that the discovery of good data-pruning metrics may provide a viable path forward to substantially improved neural scaling laws, thereby reducing the resource costs of modern deep learning.

While self-supervised learning has made rapid advances in natural language processing, it remains unclear when researchers should engage in resource-intensive domain-specific pretraining (domain pretraining). The law, puzzlingly, has yielded few documented instances of substantial gains to domain pretraining in spite of the fact that legal language is widely seen to be unique. We hypothesize that these existing results stem from the fact that existing legal NLP tasks are too easy and fail to meet conditions for when domain pretraining can help. To address this, we first present CaseHOLD (Case Holdings On Legal Decisions), a new dataset comprised of over 53,000+ multiple choice questions to identify the relevant holding of a cited case. This dataset presents a fundamental task to lawyers and is both legally meaningful and difficult from an NLP perspective (F1 of 0.4 with a BiLSTM baseline). Second, we assess performance gains on CaseHOLD and existing legal NLP datasets. While a Transformer architecture (BERT) pretrained on a general corpus (Google Books and Wikipedia) improves performance, domain pretraining (using corpus of approximately 3.5M decisions across all courts in the U.S. that is larger than BERT's) with a custom legal vocabulary exhibits the most substantial performance gains with CaseHOLD (gain of 7.2% on F1, representing a 12% improvement on BERT) and consistent performance gains across two other legal tasks. Third, we show that domain pretraining may be warranted when the task exhibits sufficient similarity to the pretraining corpus: the level of performance increase in three legal tasks was directly tied to the domain specificity of the task. Our findings inform when researchers should engage resource-intensive pretraining and show that Transformer-based architectures, too, learn embeddings suggestive of distinct legal language.

Scaling the capacity of language models has consistently proven to be a reliable approach for improving performance and unlocking new capabilities. Capacity can be primarily defined by two dimensions: the number of model parameters and the compute per example. While scaling typically involves increasing both, the precise interplay between these factors and their combined contribution to overall capacity remains not fully understood. We explore this relationship in the context of sparse Mixture-of-Experts (MoEs), which allow scaling the number of parameters without proportionally increasing the FLOPs per example. We investigate how varying the sparsity level, i.e., the fraction of inactive parameters, impacts model's performance during pretraining and downstream few-shot evaluation. We find that under different constraints (e.g., parameter size and total training compute), there is an optimal level of sparsity that improves both training efficiency and model performance. These results provide a better understanding of the impact of sparsity in scaling laws for MoEs and complement existing works in this area, offering insights for designing more efficient architectures.

Mixture-of-Experts (MoE) has become a dominant architecture for scaling Large Language Models (LLMs) efficiently by decoupling total parameters from computational cost. However, this decoupling creates a critical challenge: predicting the model capacity of a given MoE configurations (e.g., expert activation ratio and granularity) remains an unresolved problem. To address this gap, we introduce Efficiency Leverage (EL), a metric quantifying the computational advantage of an MoE model over a dense equivalent. We conduct a large-scale empirical study, training over 300 models up to 28B parameters, to systematically investigate the relationship between MoE architectural configurations and EL. Our findings reveal that EL is primarily driven by the expert activation ratio and the total compute budget, both following predictable power laws, while expert granularity acts as a non-linear modulator with a clear optimal range. We integrate these discoveries into a unified scaling law that accurately predicts the EL of an MoE architecture based on its configuration. To validate our derived scaling laws, we designed and trained Ling-mini-beta, a pilot model for Ling-2.0 series with only 0.85B active parameters, alongside a 6.1B dense model for comparison. When trained on an identical 1T high-quality token dataset, Ling-mini-beta matched the performance of the 6.1B dense model while consuming over 7x fewer computational resources, thereby confirming the accuracy of our scaling laws. This work provides a principled and empirically-grounded foundation for the scaling of efficient MoE models.

The optimal bit-width for achieving the best trade-off between quantized model size and accuracy has been a subject of ongoing debate. While some advocate for 4-bit quantization, others propose that 1.58-bit offers superior results. However, the lack of a cohesive framework for different bits has left such conclusions relatively tenuous. We present ParetoQ, the first unified framework that facilitates rigorous comparisons across 1-bit, 1.58-bit, 2-bit, 3-bit, and 4-bit quantization settings. Our findings reveal a notable learning transition between 2 and 3 bits: For 3-bits and above, the fine-tuned models stay close to their original pre-trained distributions, whereas for learning 2-bit networks or below, the representations change drastically. By optimizing training schemes and refining quantization functions, ParetoQ surpasses all previous methods tailored to specific bit widths. Remarkably, our ParetoQ ternary 600M-parameter model even outperforms the previous SoTA ternary 3B-parameter model in accuracy, using only one-fifth of the parameters. Extensive experimentation shows that ternary, 2-bit, and 3-bit quantization maintains comparable performance in the size-accuracy trade-off and generally exceeds 4-bit and binary quantization. Considering hardware constraints, 2-bit quantization offers promising potential for memory reduction and speedup.

Quantization methods reduce the number of bits required to represent each parameter in a model, trading accuracy for smaller memory footprints and inference latencies. However, the final model size depends on both the number of parameters of the original model and the rate of compression. For example, a 30B 8-bit model and a 60B 4-bit model have the same number of bits but may have very different zero-shot accuracies. In this work, we study this trade-off by developing inference scaling laws of zero-shot performance in Large Language Models (LLMs) to determine the bit-precision and model size that maximizes zero-shot performance. We run more than 35,000 experiments with 16-bit inputs and k-bit parameters to examine which zero-shot quantization methods improve scaling for 3 to 8-bit precision at scales of 19M to 176B parameters across the LLM families BLOOM, OPT, NeoX/Pythia, and GPT-2. We find that it is challenging to improve the bit-level scaling trade-off, with the only improvements being the use of a small block size -- splitting the parameters into small independently quantized blocks -- and the quantization data type being used (e.g., Int vs Float). Overall, our findings show that {4-bit} precision is almost universally optimal for total model bits and zero-shot accuracy.

Annotation and classification of legal text are central components of empirical legal research. Traditionally, these tasks are often delegated to trained research assistants. Motivated by the advances in language modeling, empirical legal scholars are increasingly turning to prompting commercial models, hoping that it will alleviate the significant cost of human annotation. Despite growing use, our understanding of how to best utilize large language models for legal tasks remains limited. We conduct a comprehensive study of 260 legal text classification tasks, nearly all new to the machine learning community. Starting from GPT-4 as a baseline, we show that it has non-trivial but highly varied zero-shot accuracy, often exhibiting performance that may be insufficient for legal work. We then demonstrate that a lightly fine-tuned Llama 3 model vastly outperforms GPT-4 on almost all tasks, typically by double-digit percentage points. We find that larger models respond better to fine-tuning than smaller models. A few tens to hundreds of examples suffice to achieve high classification accuracy. Notably, we can fine-tune a single model on all 260 tasks simultaneously at a small loss in accuracy relative to having a separate model for each task. Our work points to a viable alternative to the predominant practice of prompting commercial models. For concrete legal tasks with some available labeled data, researchers are better off using a fine-tuned open-source model.

On a variety of tasks, the performance of neural networks predictably improves with training time, dataset size and model size across many orders of magnitude. This phenomenon is known as a neural scaling law. Of fundamental importance is the compute-optimal scaling law, which reports the performance as a function of units of compute when choosing model sizes optimally. We analyze a random feature model trained with gradient descent as a solvable model of network training and generalization. This reproduces many observations about neural scaling laws. First, our model makes a prediction about why the scaling of performance with training time and with model size have different power law exponents. Consequently, the theory predicts an asymmetric compute-optimal scaling rule where the number of training steps are increased faster than model parameters, consistent with recent empirical observations. Second, it has been observed that early in training, networks converge to their infinite-width dynamics at a rate 1/width but at late time exhibit a rate width^{-c}, where c depends on the structure of the architecture and task. We show that our model exhibits this behavior. Lastly, our theory shows how the gap between training and test loss can gradually build up over time due to repeated reuse of data.

Recent work in scientific machine learning (SciML) has focused on incorporating partial differential equation (PDE) information into the learning process. Much of this work has focused on relatively ``easy'' PDE operators (e.g., elliptic and parabolic), with less emphasis on relatively ``hard'' PDE operators (e.g., hyperbolic). Within numerical PDEs, the latter problem class requires control of a type of volume element or conservation constraint, which is known to be challenging. Delivering on the promise of SciML requires seamlessly incorporating both types of problems into the learning process. To address this issue, we propose ProbConserv, a framework for incorporating conservation constraints into a generic SciML architecture. To do so, ProbConserv combines the integral form of a conservation law with a Bayesian update. We provide a detailed analysis of ProbConserv on learning with the Generalized Porous Medium Equation (GPME), a widely-applicable parameterized family of PDEs that illustrates the qualitative properties of both easier and harder PDEs. ProbConserv is effective for easy GPME variants, performing well with state-of-the-art competitors; and for harder GPME variants it outperforms other approaches that do not guarantee volume conservation. ProbConserv seamlessly enforces physical conservation constraints, maintains probabilistic uncertainty quantification (UQ), and deals well with shocks and heteroscedasticities. In each case, it achieves superior predictive performance on downstream tasks.

Data pruning algorithms are commonly used to reduce the memory and computational cost of the optimization process. Recent empirical results reveal that random data pruning remains a strong baseline and outperforms most existing data pruning methods in the high compression regime, i.e., where a fraction of 30% or less of the data is kept. This regime has recently attracted a lot of interest as a result of the role of data pruning in improving the so-called neural scaling laws; in [Sorscher et al.], the authors showed the need for high-quality data pruning algorithms in order to beat the sample power law. In this work, we focus on score-based data pruning algorithms and show theoretically and empirically why such algorithms fail in the high compression regime. We demonstrate ``No Free Lunch" theorems for data pruning and present calibration protocols that enhance the performance of existing pruning algorithms in this high compression regime using randomization.

Scaling laws are typically fit using a family of models with a narrow range of frozen hyper-parameter choices. In this work we study scaling laws using a wide range of architecture and hyper-parameter choices, and highlight their impact on resulting prescriptions. As a primary artifact of our research, we release the Gemstones: the most comprehensive open-source scaling law dataset to date, consisting of over 4000 checkpoints from transformers with up to 2 billion parameters; these models have been trained with different learning rates, cooldown schedules, and architectural shapes. Our checkpoints enable more complex studies of scaling, such as a law that predicts language modeling performance as a function of model width and depth. By examining the various facets of our model suite, we find that the prescriptions of scaling laws can be highly sensitive to the experimental design process and the specific model checkpoints used during fitting. Code: https://github.com/mcleish7/gemstone-scaling-laws

Large language models exhibit exceptional generalization capabilities, primarily attributed to the utilization of diversely sourced data. However, conventional practices in integrating this diverse data heavily rely on heuristic schemes, lacking theoretical guidance. This research tackles these limitations by investigating strategies based on low-cost proxies for data mixtures, with the aim of streamlining data curation to enhance training efficiency. Specifically, we propose a unified scaling law, termed BiMix, which accurately models the bivariate scaling behaviors of both data quantity and mixing proportions. We conduct systematic experiments and provide empirical evidence for the predictive power and fundamental principles of BiMix. Notably, our findings reveal that entropy-driven training-free data mixtures can achieve comparable or even better performance than more resource-intensive methods. We hope that our quantitative insights can shed light on further judicious research and development in cost-effective language modeling.

The ever-growing ecosystem of LLMs has posed a challenge in selecting the most appropriate pre-trained model to fine-tune amidst a sea of options. Given constrained resources, fine-tuning all models and making selections afterward is unrealistic. In this work, we formulate this resource-constrained selection task into predicting fine-tuning performance and illustrate its natural connection with scaling laws. Unlike pre-training, We find that the fine-tuning scaling curve includes not just the well-known "power phase" but also the previously unobserved "pre-power phase". We also explain why existing scaling laws fail to capture this phase transition phenomenon both theoretically and empirically. To address this, we introduce the concept of "pre-learned data size" into our rectified scaling law, which overcomes theoretical limitations and fits experimental results much better. By leveraging our law, we propose a novel LLM selection algorithm that selects the near-optimal model with hundreds of times less resource consumption, while other methods may provide negatively correlated selection.

We present the Large Language Model from Power Law Decoder Representations (PLDR-LLM), a language model that leverages non-linear and linear transformations through Power Law Graph Attention mechanism to generate well-defined deductive and inductive outputs. We pretrain the PLDR-LLMs of varying layer sizes with a small batch size of 32 and sim8B tokens from the RefinedWeb dataset, and show that they achieve competitive performance in zero-shot and few-shot settings compared to scaled dot-product LLMs of similar model size reported in the literature. We show that deductive outputs of PLDR-LLMs can be used to compare model characteristics or improve the performance by introducing the Directed Acyclic Graph (DAG) loss as a metric and regularizer. Our results indicate that the initial maximum learning rate and warm-up steps have a lasting impact on deductive outputs throughout the pretraining. We provide a detailed description of PLDR-LLM architecture, its implementation and the pretraining procedure.