In machine learning, a neural scaling law is an empirical scaling law that describes how neural network performance changes as key factors are scaled up or down. These factors typically include the number of parameters, training dataset size, and training cost. Some models also exhibit performance gains by scaling inference through increased test-time compute (TTC), extending neural scaling laws beyond training to the deployment phase. == Introduction == In general, a deep learning model can be characterized by four parameters: model size, training dataset size, training cost, and the post-training error rate (e.g., the test set error rate). Each of these variables can be defined as a real number, usually written as N , D , C , L {\displaystyle N,D,C,L} (respectively: parameter count, dataset size, computing cost, and loss). A neural scaling law is a theoretical or empirical statistical law between these parameters. There are also other parameters with other scaling laws. === Size of the model === In most cases, the model's size is simply the number of parameters. However, one complication arises with the use of sparse models, such as mixture-of-expert models. With sparse models, during inference, only a fraction of their parameters are used. In comparison, most other kinds of neural networks, such as transformer models, always use all their parameters during inference. === Size of the training dataset === The size of the training dataset is usually quantified by the number of data points within it. Larger training datasets are typically preferred, as they provide a richer and more diverse source of information from which the model can learn. This can lead to improved generalization performance when the model is applied to new, unseen data. However, increasing the size of the training dataset also increases the computational resources and time required for model training. With the "pretrain, then finetune" method used for most large language models, there are two kinds of training dataset: the pretraining dataset and the finetuning dataset. Their sizes have different effects on model performance. Generally, the finetuning dataset is less than 1% the size of pretraining dataset. In some cases, a small amount of high quality data suffices for finetuning, and more data does not necessarily improve performance. Many scaling laws, due to their inherent diminishing returns nature, value data based on a submodular set function which was shown in a paper on this topic. === Cost of training === Training cost is typically measured in terms of time (how long it takes to train the model) and computational resources (how much processing power and memory are required). It is important to note that the cost of training can be significantly reduced with efficient training algorithms, optimized software libraries, and parallel computing on specialized hardware such as GPUs or TPUs. The cost of training a neural network model is a function of several factors, including model size, training dataset size, the training algorithm complexity, and the computational resources available. In particular, doubling the training dataset size does not necessarily double the cost of training, because one may train the model for several times over the same dataset (each being an "epoch"). === Performance === The performance of a neural network model is evaluated based on its ability to accurately predict the output given some input data. Common metrics for evaluating model performance include: Negative log-likelihood per token (logarithm of perplexity) for language modeling; Accuracy, precision, recall, and F1 score for classification tasks; Mean squared error (MSE) or mean absolute error (MAE) for regression tasks; Elo rating in a competition against other models, such as gameplay or preference by a human judge. Performance can be improved by using more data, larger models, different training algorithms, regularizing the model to prevent overfitting, and early stopping using a validation set. When the performance is a number bounded within the range of [ 0 , 1 ] {\displaystyle [0,1]} , such as accuracy, precision, etc., it often scales as a sigmoid function of cost, as seen in the figures. == Examples == === (Hestness, Narang, et al, 2017) === The 2017 paper is a common reference point for neural scaling laws fitted by statistical analysis on experimental data. Previous works before the 2000s, as cited in the paper, were either theoretical or orders of magnitude smaller in scale. Whereas previous works generally found the scaling exponent to scale like L ∝ D − α {\displaystyle L\propto D^{-\alpha }} , with α ∈ { 0.5 , 1 , 2 } {\displaystyle \alpha \in \{0.5,1,2\}} , the paper found that α ∈ [ 0.07 , 0.35 ] {\displaystyle \alpha \in [0.07,0.35]} . Of the factors they varied, only task can change the exponent α {\displaystyle \alpha } . Changing the architecture optimizers, regularizers, and loss functions, would only change the proportionality factor, not the exponent. For example, for the same task, one architecture might have L = 1000 D − 0.3 {\displaystyle L=1000D^{-0.3}} while another might have L = 500 D − 0.3 {\displaystyle L=500D^{-0.3}} . They also found that for a given architecture, the number of parameters necessary to reach lowest levels of loss, given a fixed dataset size, grows like N ∝ D β {\displaystyle N\propto D^{\beta }} for another exponent β {\displaystyle \beta } . They studied machine translation with LSTM ( α ∼ 0.13 {\displaystyle \alpha \sim 0.13} ), generative language modelling with LSTM ( α ∈ [ 0.06 , 0.09 ] , β ≈ 0.7 {\displaystyle \alpha \in [0.06,0.09],\beta \approx 0.7} ), ImageNet classification with ResNet ( α ∈ [ 0.3 , 0.5 ] , β ≈ 0.6 {\displaystyle \alpha \in [0.3,0.5],\beta \approx 0.6} ), and speech recognition with two hybrid (LSTMs complemented by either CNNs or an attention decoder) architectures ( α ≈ 0.3 {\displaystyle \alpha \approx 0.3} ). === (Henighan, Kaplan, et al, 2020) === A 2020 analysis studied statistical relations between C , N , D , L {\displaystyle C,N,D,L} over a wide range of values and found similar scaling laws, over the range of N ∈ [ 10 3 , 10 9 ] {\displaystyle N\in [10^{3},10^{9}]} , C ∈ [ 10 12 , 10 21 ] {\displaystyle C\in [10^{12},10^{21}]} , and over multiple modalities (text, video, image, text to image, etc.). In particular, the scaling laws it found are (Table 1 of ): For each modality, they fixed one of the two C , N {\displaystyle C,N} , and varying the other one ( D {\displaystyle D} is varied along using D = C / 6 N {\displaystyle D=C/6N} ), the achievable test loss satisfies L = L 0 + ( x 0 x ) α {\displaystyle L=L_{0}+\left({\frac {x_{0}}{x}}\right)^{\alpha }} where x {\displaystyle x} is the varied variable, and L 0 , x 0 , α {\displaystyle L_{0},x_{0},\alpha } are parameters to be found by statistical fitting. The parameter α {\displaystyle \alpha } is the most important one. When N {\displaystyle N} is the varied variable, α {\displaystyle \alpha } ranges from 0.037 {\displaystyle 0.037} to 0.24 {\displaystyle 0.24} depending on the model modality. This corresponds to the α = 0.34 {\displaystyle \alpha =0.34} from the Chinchilla scaling paper. When C {\displaystyle C} is the varied variable, α {\displaystyle \alpha } ranges from 0.048 {\displaystyle 0.048} to 0.19 {\displaystyle 0.19} depending on the model modality. This corresponds to the β = 0.28 {\displaystyle \beta =0.28} from the Chinchilla scaling paper. Given fixed computing budget, optimal model parameter count is consistently around N o p t ( C ) = ( C 5 × 10 − 12 petaFLOP-day ) 0.7 = 9.0 × 10 − 7 C 0.7 {\displaystyle N_{opt}(C)=\left({\frac {C}{5\times 10^{-12}{\text{petaFLOP-day}}}}\right)^{0.7}=9.0\times 10^{-7}C^{0.7}} The parameter 9.0 × 10 − 7 {\displaystyle 9.0\times 10^{-7}} varies by a factor of up to 10 for different modalities. The exponent parameter 0.7 {\displaystyle 0.7} varies from 0.64 {\displaystyle 0.64} to 0.75 {\displaystyle 0.75} for different modalities. This exponent corresponds to the ≈ 0.5 {\displaystyle \approx 0.5} from the Chinchilla scaling paper. It's "strongly suggested" (but not statistically checked) that D o p t ( C ) ∝ N o p t ( C ) 0.4 ∝ C 0.28 {\displaystyle D_{opt}(C)\propto N_{opt}(C)^{0.4}\propto C^{0.28}} . This exponent corresponds to the ≈ 0.5 {\displaystyle \approx 0.5} from the Chinchilla scaling paper. The scaling law of L = L 0 + ( C 0 / C ) 0.048 {\displaystyle L=L_{0}+(C_{0}/C)^{0.048}} was confirmed during the training of GPT-3 (Figure 3.1 ). === Chinchilla scaling (Hoffmann, et al, 2022) === One particular scaling law ("Chinchilla scaling") states that, for a large language model (LLM) autoregressively trained for one epoch, with a cosine learning rate schedule, we have: { C = C 0 N D L = A N α + B D β + L 0 {\displaystyle {\begin{cases}C=C_{0}ND\\L={\frac {A}{N^{\alpha }}}+{\frac {B}{D^{\beta }}}+L_{0}\end{cases}}} where the variables are C {\displaystyle C} is the cost o
Artificial brain
An artificial brain (or artificial mind) is software and hardware with cognitive abilities similar to those of the animal or human brain. Research investigating "artificial brains" and brain emulation plays three important roles in science: An ongoing attempt by neuroscientists to understand how the human brain works, known as cognitive neuroscience. A thought experiment in the philosophy of artificial intelligence, demonstrating that it is possible, at least in theory, to create a machine that has all the capabilities of a human being. A long-term project to create machines exhibiting behavior comparable to those of animals with complex central nervous system such as mammals and most particularly humans. The ultimate goal of creating a machine exhibiting human-like behavior or intelligence is sometimes called strong AI. An example of the first objective is the project reported by Aston University in Birmingham, England where researchers are using biological cells to create "neurospheres" (small clusters of neurons) in order to develop new treatments for diseases including Alzheimer's, motor neurone and Parkinson's disease. The second objective is a reply to arguments such as John Searle's Chinese room argument, Hubert Dreyfus's critique of AI or Roger Penrose's argument in The Emperor's New Mind. These critics argued that there are aspects of human consciousness or expertise that can not be simulated by machines. One reply to their arguments is that the biological processes inside the brain can be simulated to any degree of accuracy. This reply was made as early as 1950, by Alan Turing in his classic paper "Computing Machinery and Intelligence". The third objective is generally called artificial general intelligence by researchers. However, Ray Kurzweil prefers the term "strong AI". In his book The Singularity is Near, he focuses on whole brain emulation using conventional computing machines as an approach to implementing artificial brains, and claims (on grounds of computer power continuing an exponential growth trend) that this could be done by 2025. Henry Markram, director of the Blue Brain project (which is attempting brain emulation), made a similar claim (2020) at the Oxford TED conference in 2009. == Approaches to brain simulation == W. Ross Ashby's pioneering work in cybernetics provided an early mathematical framework for understanding adaptive brain-like systems. In his 1952 book Design for a Brain, Ashby proposed that the brain could be modeled as an ultrastable system that maintains equilibrium through continuous adaptation to environmental perturbations. His approach used differential equations and state-space models to describe how neural systems could exhibit purposeful behavior through feedback mechanisms. Ashby's homeostat, a physical machine built in 1948, demonstrated these principles through an electromechanical device with four interconnected units that automatically adjusted their parameters to maintain stability when disturbed. The homeostat represented one of the first attempts to build an artificial system exhibiting brain-like adaptive behavior, influencing subsequent work in adaptive systems, neural networks, and artificial intelligence. Although direct human brain emulation using artificial neural networks on a high-performance computing engine is a commonly discussed approach, there are other approaches. An alternative artificial brain implementation could be based on Holographic Neural Technology (HNeT) non linear phase coherence/decoherence principles. The analogy has been made to quantum processes through the core synaptic algorithm which has strong similarities to the quantum mechanical wave equation. EvBrain is a form of evolutionary software that can evolve "brainlike" neural networks, such as the network immediately behind the retina. In November 2008, IBM received a US$4.9 million grant from the Pentagon for research into creating intelligent computers. The Blue Brain project is being conducted with the assistance of IBM in Lausanne. The project is based on the premise that it is possible to artificially link the neurons "in the computer" by placing thirty million synapses in their proper three-dimensional position. Some proponents of strong AI speculated in 2009 that computers in connection with Blue Brain and Soul Catcher may exceed human intellectual capacity by around 2015, and that it is likely that we will be able to download the human brain at some time around 2050. While Blue Brain is able to represent complex neural connections on the large scale, the project does not achieve the link between brain activity and behaviors executed by the brain. In 2012, project Spaun (Semantic Pointer Architecture Unified Network) attempted to model multiple parts of the human brain through large-scale representations of neural connections that generate complex behaviors in addition to mapping. Spaun's design recreates elements of human brain anatomy. The model, consisting of approximately 2.5 million neurons, includes features of the visual and motor cortices, GABAergic and dopaminergic connections, the ventral tegmental area (VTA), substantia nigra, and others. The design allows for several functions in response to eight tasks, using visual inputs of typed or handwritten characters and outputs carried out by a mechanical arm. Spaun's functions include copying a drawing, recognizing images, and counting. There are good reasons to believe that, regardless of implementation strategy, the predictions of realising artificial brains in the near future are optimistic. In particular brains (including the human brain) and cognition are not currently well understood, and the scale of computation required is unknown. Another near term limitation is that all current approaches for brain simulation require orders of magnitude larger power consumption compared with a human brain. The human brain consumes about 20 W of power, whereas current supercomputers may use as much as 1 MW—i.e., an order of 100,000 more. == Artificial brain thought experiment == Some critics of brain simulation believe that it is simpler to create general intelligent action directly without imitating nature. Some commentators have used the analogy that early attempts to construct flying machines modeled them after birds, but that modern aircraft do not look like birds.
The Best Free AI Paragraph Rewriter for Beginners
Shopping for the best AI paragraph rewriter? An AI paragraph rewriter is software that uses machine learning to help you get more done — it keeps getting smarter as the underlying models improve. Pricing, accuracy, and the size of the model behind the tool are the three factors that most affect daily usefulness. Whether you are a beginner or a pro, the right AI paragraph rewriter slots into your workflow and pays for itself fast. We tested the leading options and ranked them by quality, value, and ease of use.
AI Chatbots: Free vs Paid (2026)
In search of the best AI chatbot? An AI chatbot is software that uses machine learning to help you get more done — it turns a rough idea into a polished result in seconds. When choosing one, weigh output quality, pricing, export formats, and how well it fits the tools you already use. Whether you are a beginner or a pro, the right AI chatbot slots into your workflow and pays for itself fast. We tested the leading options and ranked them by quality, value, and ease of use.
Best AI Art Generators in 2026
Curious about the best AI art generator? An AI art generator is software that uses machine learning to help you get more done — it combines speed, accuracy, and an interface that just works. Hands-on testing shows real-world results vary, so a short free trial is the smartest way to decide. Whether you are a beginner or a pro, the right AI art generator slots into your workflow and pays for itself fast. This guide breaks down the top picks, their pros and cons, and who each one is best for.
How to Choose an AI Copywriting Tool
Trying to pick the best AI copywriting tool? An AI copywriting tool is software that uses machine learning to help you get more done — it scales effortlessly from a single task to thousands. The best picks balance beginner-friendly simplicity with the depth power users need, and they ship updates often. Whether you are a beginner or a pro, the right AI copywriting tool slots into your workflow and pays for itself fast. Read on for hands-on impressions, pricing tiers, and the standout features that matter.
Is an AI Copywriting Tool Worth It in 2026?
Looking for the best AI copywriting tool? An AI copywriting tool is software that uses machine learning to help you get more done — it can save you hours every week by automating repetitive work. Most options offer a generous free tier, with paid plans unlocking higher limits, faster processing, and team features. Whether you are a beginner or a pro, the right AI copywriting tool slots into your workflow and pays for itself fast. Read on for hands-on impressions, pricing tiers, and the standout features that matter.