In the field of artificial intelligence (AI), tasks that are hypothesized to require artificial general intelligence to solve are informally known as AI-complete or AI-hard. Calling a problem AI-complete reflects the belief that it cannot be solved by a simple specific algorithm. Prior to 2013, problems supposed to be AI-complete included computer vision, natural language understanding, and dealing with unexpected circumstances while solving any real-world problem. AI-complete tasks were notably considered useful for distinguishing humans from automated agents, as CAPTCHAs aim to do. == History == The term was coined by Fanya Montalvo by analogy with NP-complete and NP-hard in complexity theory, which formally describes the most famous class of difficult problems. Early uses of the term are in Erik Mueller's 1987 PhD dissertation and in Eric Raymond's 1991 Jargon File. Expert systems, that were popular in the 1980s, were able to solve very simple and/or restricted versions of AI-complete problems, but never in their full generality. When AI researchers attempted to "scale up" their systems to handle more complicated, real-world situations, the programs tended to become excessively brittle without commonsense knowledge or a rudimentary understanding of the situation: they would fail as unexpected circumstances outside of its original problem context would begin to appear. When human beings are dealing with new situations in the world, they are helped by their awareness of the general context: they know what the things around them are, why they are there, what they are likely to do and so on. They can recognize unusual situations and adjust accordingly. Expert systems lacked this adaptability and were brittle when facing new situations. DeepMind published a work in May 2022 in which they trained a single model to do several things at the same time. The model, named Gato, can "play Atari, caption images, chat, stack blocks with a real robot arm and much more, deciding based on its context whether to output text, joint torques, button presses, or other tokens." Similarly, some tasks once considered to be AI-complete, like machine translation, are among the capabilities of large language models. == AI-complete problems == AI-complete problems have been hypothesized to include: AI peer review (composite natural language understanding, automated reasoning, automated theorem proving, formalized logic expert system) Bongard problems Computer vision (and subproblems such as object recognition) Natural language understanding (and subproblems such as text mining, machine translation, and word-sense disambiguation) Autonomous driving Dealing with unexpected circumstances while solving any real world problem, whether navigation, planning, or even the kind of reasoning done by expert systems. == Formalization == Computational complexity theory deals with the relative computational difficulty of computable functions. By definition, it does not cover problems whose solution is unknown or has not been characterized formally. Since many AI problems have no formalization yet, conventional complexity theory does not enable a formal definition of AI-completeness. == Research == Roman Yampolskiy suggests that a problem C {\displaystyle C} is AI-Complete if it has two properties: It is in the set of AI problems (Human Oracle-solvable). Any AI problem can be converted into C {\displaystyle C} by some polynomial time algorithm. On the other hand, a problem H {\displaystyle H} is AI-Hard if and only if there is an AI-Complete problem C {\displaystyle C} that is polynomial time Turing-reducible to H {\displaystyle H} . This also gives as a consequence the existence of AI-Easy problems, that are solvable in polynomial time by a deterministic Turing machine with an oracle for some problem. Yampolskiy has also hypothesized that the Turing Test is a defining feature of AI-completeness. Groppe and Jain classify problems which require artificial general intelligence to reach human-level machine performance as AI-complete, while only restricted versions of AI-complete problems can be solved by the current AI systems. For Šekrst, getting a polynomial solution to AI-complete problems would not necessarily be equal to solving the issue of artificial general intelligence, while emphasizing the lack of computational complexity research being the limiting factor towards achieving artificial general intelligence. For Kwee-Bintoro and Velez, solving AI-complete problems would have strong repercussions on society.
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A Logical Calculus of the Ideas Immanent in Nervous Activity
"A Logical Calculus of the Ideas Immanent in Nervous Activity" is a 1943 paper written by Warren Sturgis McCulloch and Walter Pitts, published in the journal The Bulletin of Mathematical Biophysics. The paper proposed a mathematical model of the nervous system as a network of simple logical elements, later known as artificial neurons, or McCulloch–Pitts neurons. These neurons receive inputs, perform a weighted sum, and fire an output signal based on a threshold function. By connecting these units in various configurations, McCulloch and Pitts demonstrated that their model could perform all logical functions. It is a seminal work in cognitive science, computational neuroscience, computer science, and artificial intelligence. It was a foundational result in automata theory. John von Neumann cited it as a significant result. == Mathematics == The artificial neuron used in the original paper is slightly different from the modern version. They considered neural networks that operate in discrete steps of time t = 0 , 1 , … {\displaystyle t=0,1,\dots } . The neural network contains a number of neurons. Let the state of a neuron i {\displaystyle i} at time t {\displaystyle t} be N i ( t ) {\displaystyle N_{i}(t)} . The state of a neuron can either be 0 or 1, standing for "not firing" and "firing". Each neuron also has a firing threshold θ {\displaystyle \theta } , such that it fires if the total input exceeds the threshold. Each neuron can connect to any other neuron (including itself) with positive synapses (excitatory) or negative synapses (inhibitory). That is, each neuron can connect to another neuron with a weight w {\displaystyle w} taking an integer value. A peripheral afferent is a neuron with no incoming synapses. We can regard each neural network as a directed graph, with the nodes being the neurons, and the directed edges being the synapses. A neural network has a circle or a circuit if there exists a directed circle in the graph. Let w i j ( t ) {\displaystyle w_{ij}(t)} be the connection weight from neuron j {\displaystyle j} to neuron i {\displaystyle i} at time t {\displaystyle t} , then its next state is N i ( t + 1 ) = H ( ∑ j = 1 n w i j ( t ) N j ( t ) − θ i ( t ) ) , {\displaystyle N_{i}(t+1)=H\left(\sum _{j=1}^{n}w_{ij}(t)N_{j}(t)-\theta _{i}(t)\right),} where H {\displaystyle H} is the Heaviside step function (outputting 1 if the input is greater than or equal to 0, and 0 otherwise). === Symbolic logic === The paper used, as a logical language for describing neural networks, "Language II" from The Logical Syntax of Language by Rudolf Carnap with some notations taken from Principia Mathematica by Alfred North Whitehead and Bertrand Russell. Language II covers substantial parts of classical mathematics, including real analysis and portions of set theory. To describe a neural network with peripheral afferents N 1 , N 2 , … , N p {\displaystyle N_{1},N_{2},\dots ,N_{p}} and non-peripheral afferents N p + 1 , N p + 2 , … , N n {\displaystyle N_{p+1},N_{p+2},\dots ,N_{n}} they considered logical predicate of form P r ( N 1 , N 2 , … , N p , t ) {\displaystyle Pr(N_{1},N_{2},\dots ,N_{p},t)} where P r {\displaystyle Pr} is a first-order logic predicate function (a function that outputs a boolean), N 1 , … , N p {\displaystyle N_{1},\dots ,N_{p}} are predicates that take t {\displaystyle t} as an argument, and t {\displaystyle t} is the only free variable in the predicate. Intuitively speaking, N 1 , … , N p {\displaystyle N_{1},\dots ,N_{p}} specifies the binary input patterns going into the neural network over all time, and P r ( N 1 , N 2 , … , N n , t ) {\displaystyle Pr(N_{1},N_{2},\dots ,N_{n},t)} is a function that takes some binary input patterns, and constructs an output binary pattern P r ( N 1 , N 2 , … , N n , 0 ) , P r ( N 1 , N 2 , … , N n , 1 ) , … {\displaystyle Pr(N_{1},N_{2},\dots ,N_{n},0),Pr(N_{1},N_{2},\dots ,N_{n},1),\dots } . A logical sentence P r ( N 1 , N 2 , … , N n , t ) {\displaystyle Pr(N_{1},N_{2},\dots ,N_{n},t)} is realized by a neural network iff there exists a time-delay T ≥ 0 {\displaystyle T\geq 0} , a neuron i {\displaystyle i} in the network, and an initial state for the non-peripheral neurons N p + 1 ( 0 ) , … , N n ( 0 ) {\displaystyle N_{p+1}(0),\dots ,N_{n}(0)} , such that for any time t {\displaystyle t} , the truth-value of the logical sentence is equal to the state of the neuron i {\displaystyle i} at time t + T {\displaystyle t+T} . That is, ∀ t = 0 , 1 , 2 , … , P r ( N 1 , N 2 , … , N p , t ) = N i ( t + T ) {\displaystyle \forall t=0,1,2,\dots ,\quad Pr(N_{1},N_{2},\dots ,N_{p},t)=N_{i}(t+T)} === Equivalence === In the paper, they considered some alternative definitions of artificial neural networks, and have shown them to be equivalent, that is, neural networks under one definition realizes precisely the same logical sentences as neural networks under another definition. They considered three forms of inhibition: relative inhibition, absolute inhibition, and extinction. The definition above is relative inhibition. By "absolute inhibition" they meant that if any negative synapse fires, then the neuron will not fire. By "extinction" they meant that if at time t {\displaystyle t} , any inhibitory synapse fires on a neuron i {\displaystyle i} , then θ i ( t + j ) = θ i ( 0 ) + b j {\displaystyle \theta _{i}(t+j)=\theta _{i}(0)+b_{j}} for j = 1 , 2 , 3 , … {\displaystyle j=1,2,3,\dots } , until the next time an inhibitory synapse fires on i {\displaystyle i} . It is required that b j = 0 {\displaystyle b_{j}=0} for all large j {\displaystyle j} . Theorem 4 and 5 state that these are equivalent. They considered three forms of excitation: spatial summation, temporal summation, and facilitation. The definition above is spatial summation (which they pictured as having multiple synapses placed close together, so that the effect of their firing sums up). By "temporal summation" they meant that the total incoming signal is ∑ τ = 0 T ∑ j = 1 n w i j ( t ) N j ( t − τ ) {\displaystyle \sum _{\tau =0}^{T}\sum _{j=1}^{n}w_{ij}(t)N_{j}(t-\tau )} for some T ≥ 1 {\displaystyle T\geq 1} . By "facilitation" they meant the same as extinction, except that b j ≤ 0 {\displaystyle b_{j}\leq 0} . Theorem 6 states that these are equivalent. They considered neural networks that do not change, and those that change by Hebbian learning. That is, they assume that at t = 0 {\displaystyle t=0} , some excitatory synaptic connections are not active. If at any t {\displaystyle t} , both N i ( t ) = 1 , N j ( t ) = 1 {\displaystyle N_{i}(t)=1,N_{j}(t)=1} , then any latent excitatory synapse between i , j {\displaystyle i,j} becomes active. Theorem 7 states that these are equivalent. === Logical expressivity === They considered "temporal propositional expressions" (TPE), which are propositional formulas with one free variable t {\displaystyle t} . For example, N 1 ( t ) ∨ N 2 ( t ) ∧ ¬ N 3 ( t ) {\displaystyle N_{1}(t)\vee N_{2}(t)\wedge \neg N_{3}(t)} is such an expression. Theorem 1 and 2 together showed that neural nets without circles are equivalent to TPE. For neural nets with loops, they noted that "realizable P r {\displaystyle Pr} may involve reference to past events of an indefinite degree of remoteness". These then encodes for sentences like "There was some x such that x was a ψ" or ( ∃ x ) ( ψ x ) {\displaystyle (\exists x)(\psi x)} . Theorems 8 to 10 showed that neural nets with loops can encode all first-order logic with equality and conversely, any looped neural networks is equivalent to a sentence in first-order logic with equality, thus showing that they are equivalent in logical expressiveness. As a remark, they noted that a neural network, if furnished with a tape, scanners, and write-heads, is equivalent to a Turing machine, and conversely, every Turing machine is equivalent to some such neural network. Thus, these neural networks are equivalent to Turing computability and Church's lambda-definability. == Context == === Previous work === The paper built upon several previous strands of work. In the symbolic logic side, it built on the previous work by Carnap, Whitehead, and Russell. This was contributed by Walter Pitts, who had a strong proficiency with symbolic logic. Pitts provided mathematical and logical rigor to McCulloch’s vague ideas on psychons (atoms of psychological events) and circular causality. In the neuroscience side, it built on previous work by the mathematical biology research group centered around Nicolas Rashevsky, of which McCulloch was a member. The paper was published in the Bulletin of Mathematical Biophysics, which was founded by Rashevsky in 1939. During the late 1930s, Rashevsky's research group was producing papers that had difficulty publishing in other journals at the time, so Rashevsky decided to found a new journal exclusively devoted to mathematical biophysics. Also in the Rashevsky's group was Alston Scott Householder, who in 1941 published an abstract model
Grammar systems theory
Grammar systems theory is a field of theoretical computer science that studies systems of finite collections of formal grammars generating a formal language. Each grammar works on a string, a so-called sequential form that represents an environment. Grammar systems can thus be used as a formalization of decentralized or distributed systems of agents in artificial intelligence. Let A {\displaystyle \mathbb {A} } be a simple reactive agent moving on the table and trying not to fall down from the table with two reactions, t for turning and ƒ for moving forward. The set of possible behaviors of A {\displaystyle \mathbb {A} } can then be described as formal language L A = { ( f m t n f r ) + : 1 ≤ m ≤ k ; 1 ≤ n ≤ ℓ ; 1 ≤ r ≤ k } , {\displaystyle \mathbb {L_{A}} =\{(f^{m}t^{n}f^{r})^{+}:1\leq m\leq k;1\leq n\leq \ell ;1\leq r\leq k\},} where ƒ can be done maximally k times and t can be done maximally ℓ times considering the dimensions of the table. Let G A {\displaystyle \mathbb {G_{A}} } be a formal grammar which generates language L A {\displaystyle \mathbb {L_{A}} } . The behavior of A {\displaystyle \mathbb {A} } is then described by this grammar. Suppose the A {\displaystyle \mathbb {A} } has a subsumption architecture; each component of this architecture can be then represented as a formal grammar, too, and the final behavior of the agent is then described by this system of grammars. The schema on the right describes such a system of grammars which shares a common string representing an environment. The shared sequential form is sequentially rewritten by each grammar, which can represent either a component or generally an agent. If grammars communicate together and work on a shared sequential form, it is called a Cooperating Distributed (DC) grammar system. Shared sequential form is a similar concept to the blackboard approach in AI, which is inspired by an idea of experts solving some problem together while they share their proposals and ideas on a shared blackboard. Each grammar in a grammar system can also work on its own string and communicate with other grammars in a system by sending their sequential forms on request. Such a grammar system is then called a Parallel Communicating (PC) grammar system. PC and DC are inspired by distributed AI. If there is no communication between grammars, the system is close to the decentralized approaches in AI. These kinds of grammar systems are sometimes called colonies or Eco-Grammar systems, depending (besides others) on whether the environment is changing on its own (Eco-Grammar system) or not (colonies).
Artificial intimacy
Artificial intimacy is a form of human-AI interaction in which an individual will form social connections, emotional bonds, or intimate relationships with various forms of artificial intelligence, including chatbots, virtual assistants, and other artificial entities. Artificially intimate relationships include not only romances, but parasocial relationships with virtual AI characters and the use of griefbots trained on a dead or otherwise lost individual. Artificial intimacy can arise because humans are prone to anthropomorphism. Responses from these AI models are often designed to simulate human interaction. Individuals experiencing artificial intimacy may exhibit attachment, love and commitment to certain AI models, akin to the bonds typically shared between humans. == Causes == === Perceived responsiveness === Robin Dunbar famously proposed that due to emergence of larger groups of humans, vocal communication and language in humans evolved to replace grooming as a means of bonding, arguing that language was a more efficient way to maintain and strengthen social bonds across wider social settings and networks. Further research in this field leads many psychologists to agree that social cognition, affiliative bonding and language in humans are deeply connected. The interpersonal model of intimacy considers communication to be key in affiliative bonding, suggesting that intimacy develops and deepens through open communication between partners in relationship. Specifically, when individuals communicate emotions and perceive their partner as responsive and caring, feelings of closeness and connection are enhanced, building intimacy. Social penetration theory also aligns with the idea of communication being central to intimacy, by explaining how interpersonal relationships develop through gradual increases in self-disclosure. When the benefits of emotional bonding outweigh the costs of vulnerability, individuals will partake in self-disclosure, opening up to one another. Thereby, the literature can be used to provide a proximate explanation for the emergence of artificial intimacy to understand how the phenomenon occurs. Artificial entities are able to mimic interpersonal communication between humans, which in turn can simulate sensations of intimacy within human users though a perceived sense of responsiveness. The relationship between human and AI does not come with the cost of vulnerability or social rejection, which may make self-disclosure easier than with other humans. Altogether, these factors may lead to the experience of anthropomorphism and formation of affiliative relationships. Skjuve et al's interview study on Replika chatbot users further aligns with this explanation, finding that users' perception of chatbots as "accepting, understanding and non-judgmental" facilitated relationship development between the AI and users, and the act of self-disclosure possibly strengthened relationships. Another study on Replika users' reviews and survey results found users perceived chatbots as emotional supportive companions. This evidence further suggests that the perception of artificial entities as capable of empathy and responsiveness in communication facilitate the development of intimate relationships between users and AI. === Loneliness and coping with negative emotions === Research has suggested that humans evolved social bonds as a result of evolutionary pressures that favored cooperation, information exchange and transmission, and group living. Many studies stress the presence of social bonds to be important for human living: research by Baumeister and Leary suggests that humans have a basic psychological need to form and maintain "strong, stable interpersonal relationships", and that a lack of social bonds or sense of belonging leads to negative psychological and physical outcomes. Eisenberger et al's study on the neuroimaging of brain activity suggests that human brains process social rejection and exclusion similarly to physical pain. Furthermore, Song et al's study found that lonely individuals tend to seek more connections in mediated environments, such as online platforms like Facebook. This was suggested to be as a means to reduce their offline loneliness from a lack of in-person interaction, while also fulfilling a need to communicate. Leading on from this, an ultimate explanation for why humans seek the perceived sense of connection from artificial intimacy is to fulfil an evolutionary need for bonding and belonging. Xie et al's study found loneliness to be a driving factor in chatbot interaction. Herbener and Damholdt's study on Danish high school students found that students who sought emotional support or engaged in reciprocal conversations with chatbots were significantly more lonely than their peers, perceived themselves as having less social support, and used the chatbots to cope with negative emotions. The aforementioned notion that chatbots were perceived to have a positive effect on users' negative emotions is also further supported by other studies. Skjuve et al's study found that chatbot relationships may have a positive effect on users' wellbeing. De Freitas et al ran several studies on the effect of chatbots on loneliness, consistently finding evidence suggesting that interaction with chatbots reduces loneliness in users: It was found that existing chatbot users used AI to alleviate loneliness, having an AI companion consistently reduced loneliness over the course of a week, and reductions in loneliness could be explained by chatbot performance—and specifically whether it was able to make users feel heard. Overall the evidence suggests an innate need for bonding evokes feelings of loneliness in users, who turn to artificial intimacy as a low-cost method alleviate these emotions. While many users report positive experiences, some researchers caution that pursuing artificial intimacy may lead to reduced social motivation, social substitution effects, withdrawal from real-life relationships and difficulty discerning reality from fantasy, which may increase longer-term loneliness and isolation. The long-term psychological and societal impacts remain under active investigation.
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Neural scaling law
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