neural scaling law

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 $N,\; D,\; C,\; L$ (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.

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.{{Cite journal |last1=Rajbhandari |first1=Samyam |last2=Li |first2=Conglong |last3=Yao |first3=Zhewei |last4=Zhang |first4=Minjia |last5=Aminabadi |first5=Reza Yazdani |last6=Awan |first6=Ammar Ahmad |last7=Rasley |first7=Jeff |last8=He |first8=Yuxiong |date=2022-06-28 |title=DeepSpeed-MoE: Advancing Mixture-of-Experts Inference and Training to Power Next-Generation AI Scale |url=https://proceedings.mlr.press/v162/rajbhandari22a.html |journal=Proceedings of the 39th International Conference on Machine Learning |language=en |publisher=PMLR |pages=18332–18346|arxiv=2201.05596 }} 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.

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.Goodfellow, I., Bengio, Y., & Courville, A. (2016). Deep Learning. MIT Press. 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.{{Cite arXiv|last1=Zhou |first1=Chunting |last2=Liu |first2=Pengfei |last3=Xu |first3=Puxin |last4=Iyer |first4=Srini |last5=Sun |first5=Jiao |last6=Mao |first6=Yuning |last7=Ma |first7=Xuezhe |last8=Efrat |first8=Avia |last9=Yu |first9=Ping |last10=Yu |first10=Lili |last11=Zhang |first11=Susan |last12=Ghosh |first12=Gargi |last13=Lewis |first13=Mike |last14=Zettlemoyer |first14=Luke |last15=Levy |first15=Omer |date=2023-05-01 |title=LIMA: Less Is More for Alignment |class=cs.CL |eprint=2305.11206 }}

In some cases, a small amount of high quality data suffices for finetuning, and more data does not necessarily improve performance.

File:Amortized_hardware_and_energy_cost_to_train_frontier_AI_models_over_time.png

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").

File:MMLU_performance_vs_AI_scale.pngFile:Big_Bench_Hard_performance_vs_AI_scale.png{{See also|List of language model benchmarks}}

File:Densing_Law_of_LLMs,_Figure_2.png

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{{cite arXiv | eprint=2104.03113 | last1=Jones | first1=Andy L. | title=Scaling Scaling Laws with Board Games | date=2021 | class=cs.LG }} or preference by a human judge.[https://huggingface.co/spaces/lmsys/chatbot-arena-leaderboard LMSYS Chatbot leaderboard]

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 $[0,\; 1]$, such as accuracy, precision, etc., it often scales as a sigmoid function of cost, as seen in the figures.

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 $L\; \backslash propto\; D^\{-\backslash alpha\}$, with $\backslash alpha\; \backslash in\; \backslash \{0.5,\; 1,\; 2\backslash \}$, the paper found that $\backslash alpha\; \backslash in\; [0.07,\; 0.35]$.

Of the factors they varied, only task can change the exponent $\backslash alpha$. 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 $L\; =\; 1000\; D^\{-0.3\}$ while another might have $L\; =\; 500\; D^\{-0.3\}$. 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 $N\; \backslash propto\; D^\{\backslash beta\}$ for another exponent $\backslash beta$.

They studied machine translation with LSTM ($\backslash alpha\; \backslash sim\; 0.13$), generative language modelling with LSTM ($\backslash alpha\; \backslash in\; [0.06,\; 0.09],\; \backslash beta\; \backslash approx\; 0.7$), ImageNet classification with ResNet ($\backslash alpha\; \backslash in\; [0.3,\; 0.5],\; \backslash beta\; \backslash approx\; 0.6$), and speech recognition with two hybrid (LSTMs complemented by either CNNs or an attention decoder) architectures ($\backslash alpha\; \backslash approx\; 0.3$).

A 2020 analysis {{Cite book |last1=Henighan |first1=Tom |title=Scaling Laws for Autoregressive Generative Modeling |last2=Kaplan |first2=Jared |last3=Katz |first3=Mor |last4=Chen |first4=Mark |last5=Hesse |first5=Christopher |last6=Jackson |first6=Jacob |last7=Heewoo |first7=Jun |last8=Brown |first8=Tom B. |last9=Dhariwal |first9=Prafulla |date=2020-10-27 |arxiv=2010.14701 |oclc=1228442047 |last10=Mann |first10=Chris |last11=Radford |first11=Alec |last12=Ramesh |first12=Aditya |last13=Ryder |first13=Nick |last14=Ziegler |first14=Daniel M. |last15=Schulman |first15=John |last16=Gray |first16=Scott |last17=Hallacy |first17=Chris |last18=Amodei |first18=Dario |last19=McCandlish |first19=Sam}} studied statistical relations between $C,\; N,\; D,\; L$ over a wide range of values and found similar scaling laws, over the range of $N\; \backslash in\; [10^3,\; 10^9]$, $C\backslash in\; [10^\{12\},\; 10^\{21\}]$, 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 $C,\; N$, and varying the other one ($D$ is varied along using $D\; =\; C/6N$), the achievable test loss satisfies$$L\; =\; L\_0\; +\; \backslash left(\; \backslash frac\{x\_0\}\{x\}\backslash right)^\backslash alpha$$where $x$ is the varied variable, and $L\_0,\; x\_0,\; \backslash alpha$ are parameters to be found by statistical fitting. The parameter $\backslash alpha$ is the most important one.
• When $N$ is the varied variable, $\backslash alpha$ ranges from $0.037$ to $0.24$ depending on the model modality. This corresponds to the $\backslash alpha\; =\; 0.34$ from the Chinchilla scaling paper.
• When $C$ is the varied variable, $\backslash alpha$ ranges from $0.048$ to $0.19$ depending on the model modality. This corresponds to the $\backslash beta\; =\; 0.28$ from the Chinchilla scaling paper.
• Given fixed computing budget, optimal model parameter count is consistently around$$N\_\{opt\}(C)\; =\backslash left(\backslash frac\{C\}\{5\backslash times\; 10^\{-12\}\backslash text\{petaFLOP-day\}\}\backslash right)^\{0.7\}\; =\; 9.0\backslash times\; 10^\{-7\}\; C^\{0.7\}$$The parameter $9.0\; \backslash times\; 10^\{-7\}$ varies by a factor of up to 10 for different modalities. The exponent parameter $0.7$ varies from $0.64$ to $0.75$ for different modalities. This exponent corresponds to the $\backslash approx\; 0.5$ from the Chinchilla scaling paper.
• It's "strongly suggested" (but not statistically checked) that $D\_\{opt\}(C)\; \backslash propto\; N\_\{opt\}(C)^\{0.4\}\; \backslash propto\; C^\{0.28\}$. This exponent corresponds to the $\backslash approx\; 0.5$ from the Chinchilla scaling paper.

The scaling law of $L\; =\; L\_0\; +\; (C\_0/C)^\{0.048\}$ was confirmed during the training of GPT-3 (Figure 3.1 {{Cite arXiv |last1=Brown |first1=Tom B. |last2=Mann |first2=Benjamin |last3=Ryder |first3=Nick |last4=Subbiah |first4=Melanie |last5=Kaplan |first5=J. |last6=Dhariwal |first6=Prafulla |last7=Neelakantan |first7=Arvind |last8=Shyam |first8=Pranav |last9=Sastry |first9=Girish |last10=Askell |first10=Amanda |last11=Agarwal |first11=Sandhini |last12=Herbert-Voss |first12=Ariel |last13=Krueger |first13=Gretchen |last14=Henighan |first14=T. |last15=Child |first15=Rewon |date=2020-05-28 |title=Language Models are Few-Shot Learners |class=cs.CL |eprint=2005.14165 }}).

File:Optimal_ratio_of_training_tokens_to_model_parameters_for_Chinchilla_scaling_law.png

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:{{Cite arXiv |last1=Hoffmann |first1=Jordan |last2=Borgeaud |first2=Sebastian |last3=Mensch |first3=Arthur |last4=Buchatskaya |first4=Elena |last5=Cai |first5=Trevor |last6=Rutherford |first6=Eliza |last7=Casas |first7=Diego de Las |last8=Hendricks |first8=Lisa Anne |last9=Welbl |first9=Johannes |last10=Clark |first10=Aidan |last11=Hennigan |first11=Tom |last12=Noland |first12=Eric |last13=Millican |first13=Katie |last14=Driessche |first14=George van den |last15=Damoc |first15=Bogdan |date=2022-03-29 |title=Training Compute-Optimal Large Language Models |class=cs.CL |eprint=2203.15556 }}$$\backslash begin\{cases\}$$

C = C_0 ND\\

L = \frac{A}{N^\alpha} + \frac{B}{D^{\beta}} + L_0

\end{cases}where the variables are

• $C$ is the cost of training the model, in FLOPS.
• $N$ is the number of parameters in the model.
• $D$ is the number of tokens in the training set.
• $L$ is the average negative log-likelihood loss per token (nats/token), achieved by the trained LLM on the test dataset.
• $L\_0$ represents the loss of an ideal generative process on the test data
• $\backslash frac\{A\}\{N^\backslash alpha\}$ captures the fact that a Transformer language model with $N$ parameters underperforms the ideal generative process
• $\backslash frac\{B\}\{D^\backslash beta\}$ captures the fact that the model trained on $D$ tokens underperforms the ideal generative process

and the statistical parameters are

• $C\_0\; =\; 6$, meaning that it costs 6 FLOPs per parameter to train on one token. This is estimated by Kaplan et al.{{Cite journal |last1=Kaplan |first1=Jared |last2=McCandlish |first2=Sam |last3=Henighan |first3=Tom |last4=Brown |first4=Tom B. |last5=Chess |first5=Benjamin |last6=Child |first6=Rewon |last7=Gray |first7=Scott |last8=Radford |first8=Alec |last9=Wu |first9=Jeffrey |last10=Amodei |first10=Dario |title=Scaling Laws for Neural Language Models |journal=CoRR |volume=abs/2001.08361 |year=2020 |arxiv=2001.08361}} Note that training cost is much higher than inference cost, as training entails both forward and backward passes, whereas inference costs 1 to 2 FLOPs per parameter to infer on one token.
• $\backslash alpha\; =\; 0.34,\; \backslash beta\; =\; 0.28,\; A\; =\; 406.4,\; B\; =\; 410.7,\; L\_0\; =\; 1.69$.

Although Besiroglu et al.{{cite arXiv |last1=Besiroglu |first1=Tamay |title=Chinchilla Scaling: A replication attempt |date=2024-04-15 |eprint=2404.10102 |last2=Erdil |first2=Ege |last3=Barnett |first3=Matthew |last4=You |first4=Josh|class=cs.AI }} claims that the statistical estimation is slightly off, and should be $\backslash alpha\; =\; 0.35,\; \backslash beta\; =\; 0.37,\; A\; =\; 482.01,\; B\; =\; 2085.43,\; L\_0\; =\; 1.82$.

The statistical laws were fitted over experimental data with $N\backslash in\; [7\backslash times\; 10^7,\; 1.6\; \backslash times\; 10^\{10\}],\; D\; \backslash in\; [5\backslash times\; 10^9,\; 5\backslash times\; 10^\{11\}],\; C\; \backslash in\; [10^\{18\},\; 10^\{24\}]$.

Since there are 4 variables related by 2 equations, imposing 1 additional constraint and 1 additional optimization objective allows us to solve for all four variables. In particular, for any fixed $C$, we can uniquely solve for all 4 variables that minimizes $L$. This provides us with the optimal $D\_\{opt\}(C),\; N\_\{opt\}(C)$ for any fixed $C$:$$N\_\{o\; p\; t\}(C)=G\backslash left(\backslash frac\{C\}\{6\}\backslash right)^a,\; \backslash quad\; D\_\{o\; p\; t\}(C)=G^\{-1\}\backslash left(\backslash frac\{C\}\{6\}\backslash right)^b,\; \backslash quad\; \backslash text\; \{\; where\; \}\; \backslash quad\; G=\backslash left(\backslash frac\{\backslash alpha\; A\}\{\backslash beta\; B\}\backslash right)^\{\backslash frac\{1\}\{\backslash alpha+\backslash beta\}\},\; \backslash quad\; a=\backslash frac\{\backslash beta\}\{\backslash alpha+\backslash beta\}\; \backslash text\; \{,\; and\; \}\; b=\backslash frac\{\backslash alpha\}\{\backslash alpha+\backslash beta\}\; \backslash text\; \{.\; \}$$Plugging in the numerical values, we obtain the "Chinchilla efficient" model size and training dataset size, as well as the test loss achievable:$$\backslash begin\{cases\}$$

N_{opt}(C) = 0.6 \; C^{0.45} \\

D_{opt}(C) = 0.3 \; C^{0.55} \\

L_{opt}(C) = 1070 \; C^{-0.154} + 1.7

\end{cases}Similarly, we may find the optimal training dataset size and training compute budget for any fixed model parameter size, and so on.

There are other estimates for "Chinchilla efficient" model size and training dataset size. The above is based on a statistical model of $L\; =\; \backslash frac\{A\}\{N^\backslash alpha\}\; +\; \backslash frac\{B\}\{D^\{\backslash beta\}\}\; +\; L\_0$. One can also directly fit a statistical law for $D\_\{opt\}(C),\; N\_\{opt\}(C)$ without going through the detour, for which one obtains:$$\backslash begin\{cases\}$$

N_{opt}(C) = 0.1 \; C^{0.5}\\

D_{opt}(C) = 1.7 \; C^{0.5}

\end{cases}or as tabulated:

The Chinchilla scaling law analysis for training transformer language models suggests that for a given training compute budget ($C$), to achieve the minimal pretraining loss for that budget, the number of model parameters ($N$) and the number of training tokens ($D$) should be scaled in equal proportions, $N\_\{opt\}(C)\; \backslash propto\; C^\{0.5\},\; D\_\{opt\}(C)\; \backslash propto\; C^\{0.5\}$.

This conclusion differs from analysis conducted by Kaplan et al., which found that $N$ should be increased more quickly than $D$, $N\_\{opt\}(C)\; \backslash propto\; C^\{0.73\},\; D\_\{opt\}(C)\; \backslash propto\; C^\{0.27\}$.

This discrepancy can primarily be attributed to the two studies using different methods for measuring model size. Kaplan et al.:{{Citation |last1=Pearce |first1=Tim |title=Reconciling Kaplan and Chinchilla Scaling Laws |date=2024 |arxiv=2406.12907 |last2=Song |first2=Jinyeop}}

• did not count the parameters in the token embedding layer, which when analyzed at smaller model sizes leads to biased coefficients;
• studied smaller models than the Chinchilla group, magnifying the effect;
• assumed that $L\_\backslash infty\; =\; 0$.

Secondary effects also arise due to differences in hyperparameter tuning and learning rate schedules. Kaplan et al.:{{Citation |last1=Porian |first1=Tomer |title=Resolving Discrepancies in Compute-Optimal Scaling of Language Models |date=2024-07-25 |arxiv=2406.19146 |last2=Wortsman |first2=Mitchell |last3=Jitsev |first3=Jenia |last4=Schmidt |first4=Ludwig |last5=Carmon |first5=Yair}}

• used a warmup schedule that was too long for smaller models, making them appear less efficient;
• did not fully tuning optimization hyperparameters.

As Chinchilla scaling has been the reference point for many large-scaling training runs, there had been a concurrent effort to go "beyond Chinchilla scaling", meaning to modify some of the training pipeline in order to obtain the same loss with less effort, or deliberately train for longer than what is "Chinchilla optimal".

Usually, the goal is to make the scaling law exponent larger, which means the same loss can be trained for much less compute. For instance, filtering data can make the scaling law exponent larger.{{cite arXiv |last1=Sorscher |first1=Ben |title=Beyond neural scaling laws: beating power law scaling via data pruning |date=2023-04-21 |eprint=2206.14486 |last2=Geirhos |first2=Robert |last3=Shekhar |first3=Shashank |last4=Ganguli |first4=Surya |last5=Morcos |first5=Ari S.|class=cs.LG }}

Another strand of research studies how to deal with limited data, as according to Chinchilla scaling laws, the training dataset size for the largest language models already approaches what is available on the internet.{{cite arXiv |last1=Tay |first1=Yi |title=Transcending Scaling Laws with 0.1% Extra Compute |date=2022-11-16 |eprint=2210.11399 |last2=Wei |first2=Jason |last3=Chung |first3=Hyung Won |last4=Tran |first4=Vinh Q. |last5=So |first5=David R. |last6=Shakeri |first6=Siamak |last7=Garcia |first7=Xavier |last8=Zheng |first8=Huaixiu Steven |last9=Rao |first9=Jinfeng|class=cs.CL }} found that augmenting the dataset with a mix of "denoising objectives" constructed from the dataset improves performance.{{Cite journal |last1=Muennighoff |first1=Niklas |last2=Rush |first2=Alexander |last3=Barak |first3=Boaz |last4=Le Scao |first4=Teven |last5=Tazi |first5=Nouamane |last6=Piktus |first6=Aleksandra |last7=Pyysalo |first7=Sampo |last8=Wolf |first8=Thomas |last9=Raffel |first9=Colin A. |date=2023-12-15 |title=Scaling Data-Constrained Language Models |url=https://proceedings.neurips.cc/paper_files/paper/2023/hash/9d89448b63ce1e2e8dc7af72c984c196-Abstract-Conference.html |journal=Advances in Neural Information Processing Systems |language=en |volume=36 |pages=50358–50376|arxiv=2305.16264 }} studies optimal scaling when all available data is already exhausted (such as in rare languages), so one must train multiple epoches over the same dataset (whereas Chinchilla scaling requires only one epoch). The Phi series of small language models were trained on textbook-like data generated by large language models, for which data is only limited by amount of compute available.{{cite arXiv |last1=Li |first1=Yuanzhi |title=Textbooks Are All You Need II: phi-1.5 technical report |date=2023-09-11 |eprint=2309.05463 |last2=Bubeck |first2=Sébastien |last3=Eldan |first3=Ronen |last4=Del Giorno |first4=Allie |last5=Gunasekar |first5=Suriya |last6=Lee |first6=Yin Tat|class=cs.CL }}

Chinchilla optimality was defined as "optimal for training compute", whereas in actual production-quality models, there will be a lot of inference after training is complete. "Overtraining" during training means better performance during inference.{{cite arXiv |last1=Sardana |first1=Nikhil |title=Beyond Chinchilla-Optimal: Accounting for Inference in Language Model Scaling Laws |date=2023-12-31 |eprint=2401.00448 |last2=Frankle |first2=Jonathan|class=cs.LG }} LLaMA models were overtrained for this reason. Subsequent studies discovered scaling laws in the overtraining regime, for dataset sizes up to 32x more than Chinchilla-optimal.{{cite arXiv |last1=Gadre |first1=Samir Yitzhak |title=Language models scale reliably with over-training and on downstream tasks |date=2024-03-13 |eprint=2403.08540 |last2=Smyrnis |first2=Georgios |last3=Shankar |first3=Vaishaal |last4=Gururangan |first4=Suchin |last5=Wortsman |first5=Mitchell |last6=Shao |first6=Rulin |last7=Mercat |first7=Jean |last8=Fang |first8=Alex |last9=Li |first9=Jeffrey|class=cs.CL }}

A 2022 analysis{{cite arXiv | eprint=2210.14891 | last1=Caballero | first1=Ethan | last2=Gupta | first2=Kshitij | last3=Rish | first3=Irina | last4=Krueger | first4=David | title=Broken Neural Scaling Laws | date=2022 | class=cs.LG }} found that many scaling behaviors of artificial neural networks follow a smoothly broken power law functional form:

$y\; =\; a\; +\; \backslash bigg(bx^\{-c\_0\}\backslash bigg)\; \backslash prod\_\{i=1\}^n\; \backslash left(1\; +\; \backslash left(\backslash frac\{x\}\{d\_i\}\backslash right)^\{1/f\_i\}\backslash right)^\{-c\_i\; *\; f\_i\}$

in which $x$ refers to the quantity being scaled (i.e. $C$, $N$, $D$, number of training steps, number of inference steps, or model input size) and $y$ refers to the downstream (or upstream) performance evaluation metric of interest (e.g. prediction error, cross entropy, calibration error, AUROC, BLEU score percentage, F1 score, reward, Elo rating, solve rate, or FID score) in zero-shot, prompted, or fine-tuned settings. The parameters $a,\; b,\; c\_0,\; c\_1\; ...\; c\_n,\; d\_1\; ...\; d\_n,\; f\_1\; ...\; f\_n$ are found by statistical fitting.

On a log–log plot, when $f\_i$ is not too large and $a$ is subtracted out from the y-axis, this functional form looks like a series of linear segments connected by arcs; the $n$ transitions between the segments are called "breaks", hence the name broken neural scaling laws (BNSL).

The scenarios in which the scaling behaviors of artificial neural networks were found to follow this functional form include large-scale vision, language, audio, video, diffusion, generative modeling, multimodal learning, contrastive learning, AI alignment, AI capabilities, robotics, out-of-distribution (OOD) generalization, continual learning, transfer learning, uncertainty estimation / calibration, out-of-distribution detection, adversarial robustness, distillation, sparsity, retrieval, quantization, pruning, fairness, molecules, computer programming/coding, math word problems, arithmetic, emergent abilities, double descent, supervised learning, unsupervised/self-supervised learning, and reinforcement learning (single agent and multi-agent).

The architectures for which the scaling behaviors of artificial neural networks were found to follow this functional form include residual neural networks, transformers, MLPs, MLP-mixers, recurrent neural networks, convolutional neural networks, graph neural networks, U-nets, encoder-decoder (and encoder-only) (and decoder-only) models, ensembles (and non-ensembles), MoE (mixture of experts) (and non-MoE) models, and sparse pruned (and non-sparse unpruned) models.

File:Trading_Off_Compute_in_Training_and_Inference_in_the_game_of_Hex_for_AlphaZero_agents.png{{Main|Reasoning language model}}

Other than scaling up training compute, one can also scale up inference compute (or "test-time compute"{{Citation |last1=Cobbe |first1=Karl |title=Training Verifiers to Solve Math Word Problems |date=2021-11-18 |arxiv=2110.14168 |last2=Kosaraju |first2=Vineet |last3=Bavarian |first3=Mohammad |last4=Chen |first4=Mark |last5=Jun |first5=Heewoo |last6=Kaiser |first6=Lukasz |last7=Plappert |first7=Matthias |last8=Tworek |first8=Jerry |last9=Hilton |first9=Jacob}}). As an example, the Elo rating of AlphaGo improves steadily as it is allowed to spend more time on its Monte Carlo Tree Search per play.{{Cite journal |last1=Silver |first1=David |last2=Huang |first2=Aja |last3=Maddison |first3=Chris J. |last4=Guez |first4=Arthur |last5=Sifre |first5=Laurent |last6=van den Driessche |first6=George |last7=Schrittwieser |first7=Julian |last8=Antonoglou |first8=Ioannis |last9=Panneershelvam |first9=Veda |last10=Lanctot |first10=Marc |last11=Dieleman |first11=Sander |last12=Grewe |first12=Dominik |last13=Nham |first13=John |last14=Kalchbrenner |first14=Nal |last15=Sutskever |first15=Ilya |date=January 2016 |title=Mastering the game of Go with deep neural networks and tree search |url=https://www.nature.com/articles/nature16961 |journal=Nature |language=en |volume=529 |issue=7587 |pages=484–489 |doi=10.1038/nature16961 |pmid=26819042 |bibcode=2016Natur.529..484S |issn=1476-4687}}{{Pg|location=Fig 4}} For AlphaGo Zero, increasing Elo by 120 requires either 2x model size and training, or 2x test-time search.{{Cite AV media |url=https://www.youtube.com/watch?v=eaAonE58sLU |title=Parables on the Power of Planning in AI: From Poker to Diplomacy: Noam Brown (OpenAI) |date=2024-09-17 |last=Noam |first=Brown |type=Video |language=en |access-date=2024-09-24 |via=YouTube}}

Lecture at Paul G. Allen School on Thursday, May 23, 2024, 3:30 pm Similarly, a language model for solving competition-level coding challenges, AlphaCode, consistently improved (log-linearly) in performance with more search time.{{Cite journal |last1=Li |first1=Yujia |last2=Choi |first2=David |last3=Chung |first3=Junyoung |last4=Kushman |first4=Nate |last5=Schrittwieser |first5=Julian |last6=Leblond |first6=Rémi |last7=Eccles |first7=Tom |last8=Keeling |first8=James |last9=Gimeno |first9=Felix |last10=Dal Lago |first10=Agustin |last11=Hubert |first11=Thomas |last12=Choy |first12=Peter |last13=de Masson d’Autume |first13=Cyprien |last14=Babuschkin |first14=Igor |last15=Chen |first15=Xinyun |date=2022-12-09 |title=Competition-level code generation with AlphaCode |url=https://www.science.org/doi/10.1126/science.abq1158 |journal=Science |language=en |volume=378 |issue=6624 |pages=1092–1097 |doi=10.1126/science.abq1158 |pmid=36480631 |arxiv=2203.07814 |bibcode=2022Sci...378.1092L |issn=0036-8075}}

For Hex, 10x training-time compute trades for 15x test-time compute. For Libratus for heads up no-limit Texas hold 'em, and Cicero for Diplomacy, and many other abstract games of partial information, inference-time searching improves performance at a similar tradeoff ratio, for up to 100,000x effective increase in training-time compute.

In 2024, the OpenAI o1 report documented that o1's performance consistently improved with both increased train-time compute and test-time compute, and gave numerous examples of test-time compute scaling in mathematics, scientific reasoning, and coding tasks.{{Cite web |last=Villalobos |first=Pablo |date=2023-07-28 |title=Trading Off Compute in Training and Inference |url=https://epochai.org/blog/trading-off-compute-in-training-and-inference |access-date=2024-09-24 |website=Epoch AI |language=en}}{{cite web |title=Learning to Reason with LLMs |url=https://openai.com/index/learning-to-reason-with-llms/ |access-date=2024-09-16 |publisher=OpenAI}}

One method for scaling up test-time compute is process-based supervision, where a model generates a step-by-step reasoning chain to answer a question, and another model (either human or AI) provides a reward score on some of the intermediate steps, not just the final answer. Process-based supervision can be scaled arbitrarily by using synthetic reward score without another model, for example, by running Monte Carlo rollouts and scoring each step in the reasoning according to how likely it leads to the right answer. Another method is by revision models, which are models trained to solve a problem multiple times, each time revising the previous attempt.{{Citation |last1=Snell |first1=Charlie |title=Scaling LLM Test-Time Compute Optimally can be More Effective than Scaling Model Parameters |date=2024-08-06 |arxiv=2408.03314 |last2=Lee |first2=Jaehoon |last3=Xu |first3=Kelvin |last4=Kumar |first4=Aviral}}

Vision transformers, similar to language transformers, exhibit scaling laws. A 2022 research trained vision transformers, with parameter counts $N\backslash in\; [5\backslash times\; 10^6,\; 2\backslash times\; 10^9]$, on image sets of sizes $D\; \backslash in\; [3\backslash times\; 10^\{7\},\; 3\backslash times\; 10^\{9\}]$, for computing $C\backslash in\; [0.2,\; 10^4]$ (in units of TPUv3-core-days).{{Cite journal |last1=Zhai |first1=Xiaohua |last2=Kolesnikov |first2=Alexander |last3=Houlsby |first3=Neil |last4=Beyer |first4=Lucas |date=2022 |title=Scaling Vision Transformers |url=https://openaccess.thecvf.com/content/CVPR2022/html/Zhai_Scaling_Vision_Transformers_CVPR_2022_paper.html |journal=CVPR |language=en |pages=12104–12113}}

After training the model, it is finetuned on ImageNet training set. Let $L$ be the error probability of the finetuned model classifying ImageNet test set. They found $\backslash min\_\{N,\; D\}\; L\; =\; 0.09\; +\; \backslash frac\; \{0.26\}\{(C\; +\; 0.01)^\{0.35\}\}$.

Ghorbani, Behrooz et al.{{Cite arXiv |last1=Ghorbani |first1=Behrooz |last2=Firat |first2=Orhan |last3=Freitag |first3=Markus |last4=Bapna |first4=Ankur |last5=Krikun |first5=Maxim |last6=Garcia |first6=Xavier |last7=Chelba |first7=Ciprian |last8=Cherry |first8=Colin |date=2021-09-01 |title=Scaling Laws for Neural Machine Translation |class=cs.LG |eprint=2109.07740}} studied scaling laws for neural machine translation (specifically, English as source, and German as target) in encoder-decoder Transformer models, trained until convergence on the same datasets (thus they did not fit scaling laws for computing cost $C$ or dataset size $D$). They varied $N\; \backslash in\; [10^8,\; 3.5\; \backslash times\; 10^9]$ They found three results:

• $L$ is a scaling law function of $N\_E,\; N\_D$, where $N\_E,\; N\_D$ are encoder and decoder parameter count. It is not simply a function of total parameter count $N\; =\; N\_E\; +\; N\_D$. The function has form $L\backslash left(N\_e,\; N\_d\backslash right)=\backslash alpha\backslash left(\backslash frac\{\backslash bar\{N\}\_e\}\{N\_e\}\backslash right)^\{p\_e\}\backslash left(\backslash frac\{\backslash bar\{N\}\_d\}\{N\_d\}\backslash right)^\{p\_d\}+L\_\{\backslash infty\}$, where $\backslash alpha,\; p\_e,\; p\_d,\; L\_\{\backslash infty\},\; \backslash bar\; N\_e,\; \backslash bar\; N\_d$ are fitted parameters. They found that $N\_d/N\; \backslash approx\; 0.55$ minimizes loss if $N$ is held fixed.
• $L$ "saturates" (that is, it reaches $L\_\backslash infty$) for smaller models when the training and testing datasets are "source-natural" than "target-natural". A "source-natural" data point means a pair of English-German sentences, and the model is asked to translate the English sentence into German, and the English sentence is written by a natural English writer, while the German sentence is translated from the English sentence by a machine translator.{{Cite journal |last1=Chen |first1=Mia Xu |last2=Firat |first2=Orhan |last3=Bapna |first3=Ankur |last4=Johnson |first4=Melvin |last5=Macherey |first5=Wolfgang |last6=Foster |first6=George |last7=Jones |first7=Llion |last8=Schuster |first8=Mike |last9=Shazeer |first9=Noam |last10=Parmar |first10=Niki |last11=Vaswani |first11=Ashish |last12=Uszkoreit |first12=Jakob |last13=Kaiser |first13=Lukasz |last14=Chen |first14=Zhifeng |last15=Wu |first15=Yonghui |date=July 2018 |title=The Best of Both Worlds: Combining Recent Advances in Neural Machine Translation |url=https://aclanthology.org/P18-1008 |journal=Proceedings of the 56th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers) |location=Melbourne, Australia |publisher=Association for Computational Linguistics |pages=76–86 |doi=10.18653/v1/P18-1008|arxiv=1804.09849 }} To construct the two kinds of datasets, the authors collected natural English and German sentences online, then used machine translation to generate their translations.
• As models grow larger, models trained on source-original datasets can achieve low loss but bad BLEU score. In contrast, models trained on target-original datasets achieve low loss and good BLEU score in tandem (Figure 10, 11 ).

The authors hypothesize that source-natural datasets have uniform and dull target sentences, and so a model that is trained to predict the target sentences would quickly overfit.

{{Cite book |last1=Gordon |first1=Mitchell A |last2=Duh |first2=Kevin |last3=Kaplan |first3=Jared |title=Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing |chapter=Data and Parameter Scaling Laws for Neural Machine Translation |date=2021 |pages=5915–5922 |location=Stroudsburg, PA, USA |publisher=Association for Computational Linguistics |doi=10.18653/v1/2021.emnlp-main.478|doi-access=free }} trained Transformers for machine translations with sizes $N\; \backslash in\; [4\; \backslash times\; 10^5\; ,\; 5.6\; \backslash times\; 10^7]$ on dataset sizes $D\; \backslash in\; [6\backslash times\; 10^5,\; 6\; \backslash times\; 10^9]$. They found the Kaplan et al. (2020) scaling law applied to machine translation: $L(N,\; D)=\backslash left[\backslash left(\backslash frac\{N\_C\}\{N\}\backslash right)^\{\backslash frac\{\backslash alpha\_N\}\{\backslash alpha\_D\}\}+\backslash frac\{D\_C\}\{D\}\backslash right]^\{\backslash alpha\_D\}$. They also found the BLEU score scaling as $BLEU\; \backslash approx\; C\; e^\{-kL\}$.

Hernandez, Danny et al.{{Cite arXiv |last1=Hernandez |first1=Danny |last2=Kaplan |first2=Jared |last3=Henighan |first3=Tom |last4=McCandlish |first4=Sam |date=2021-02-01 |title=Scaling Laws for Transfer |class=cs.LG |eprint=2102.01293}} studied scaling laws for transfer learning in language models. They trained a family of Transformers in three ways:

• pretraining on English, finetuning on Python
• pretraining on an equal mix of English and Python, finetuning on Python
• training on Python

The idea is that pretraining on English should help the model achieve low loss on a test set of Python text. Suppose the model has parameter count $N$, and after being finetuned on $D\_F$ Python tokens, it achieves some loss $L$. We say that its "transferred token count" is $D\_T$, if another model with the same $N$ achieves the same $L$ after training on $D\_F\; +\; D\_T$ Python tokens.

They found $D\_T=1.9\; e\; 4\backslash left(D\_F\backslash right)^\{.18\}(N)^\{.38\}$ for pretraining on English text, and $D\_T=2.1\; e\; 5\backslash left(D\_F\backslash right)^\{.096\}(N)^\{.38\}$ for pretraining on English and non-Python code.

Kumar et al.{{Citation |last1=Kumar |first1=Tanishq |title=Scaling Laws for Precision |date=2024-11-30 |arxiv=2411.04330 |last2=Ankner |first2=Zachary |last3=Spector |first3=Benjamin F. |last4=Bordelon |first4=Blake |last5=Muennighoff |first5=Niklas |last6=Paul |first6=Mansheej |last7=Pehlevan |first7=Cengiz |last8=Ré |first8=Christopher |last9=Raghunathan |first9=Aditi}} study scaling laws for numerical precision in the training of language models. They train a family of language models with weights, activations, and KV cache in varying numerical precision in both integer and floating-point type to measure the effects on loss as a function of precision. For training, their scaling law accounts for lower precision by wrapping the effects of precision into an overall "effective parameter count" that governs loss scaling, using the parameterization $N\; \backslash mapsto\; N\_\backslash text\{eff\}(P)\; =\; N(1-e^\{-P/\backslash gamma\})$. This illustrates how training in lower precision degrades performance by reducing the true capacity of the model in a manner that varies exponentially with bits.

For inference, they find that extreme overtraining of language models past Chinchilla-optimality can lead to models being more sensitive to quantization, a standard technique for efficient deep learning. This is demonstrated by observing that the degradation in loss due to weight quantization increases as an approximate power law in the token/parameter ratio $D/N$ seen during pretraining, so that models pretrained on extreme token budgets can perform worse in terms of validation loss than those trained on more modest token budgets if post-training quantization is applied. Other work examining the effects of overtraining include Sardana et al.{{Citation |last1=Sardana |first1=Nikhil |title=Beyond Chinchilla-Optimal: Accounting for Inference in Language Model Scaling Laws |date=2024-07-18 |arxiv=2401.00448 |last2=Portes |first2=Jacob |last3=Doubov |first3=Sasha |last4=Frankle |first4=Jonathan}} and Gadre et al.{{Citation |last1=Gadre |first1=Samir Yitzhak |title=Language models scale reliably with over-training and on downstream tasks |date=2024-06-14 |arxiv=2403.08540 |last2=Smyrnis |first2=Georgios |last3=Shankar |first3=Vaishaal |last4=Gururangan |first4=Suchin |last5=Wortsman |first5=Mitchell |last6=Shao |first6=Rulin |last7=Mercat |first7=Jean |last8=Fang |first8=Alex |last9=Li |first9=Jeffrey}}

Xiao et al.{{Citation |last1=Xiao |first1=Chaojun |title=Densing Law of LLMs |date=2024-12-06 |arxiv=2412.04315 |last2=Cai |first2=Jie |last3=Zhao |first3=Weilin |last4=Zeng |first4=Guoyang |last5=Lin |first5=Biyuan |last6=Zhou |first6=Jie |last7=Zheng |first7=Zhi |last8=Han |first8=Xu |last9=Liu |first9=Zhiyuan}} considered the parameter efficiency ("density") of models over time. The idea is that over time, researchers would discover models that use their parameters more efficiently, in that models with the same performance can have fewer parameters.

A model can have an actual parameter count $N$, defined as the actual number of parameters in the model, and an "effective" parameter count $\backslash hat\; N$, defined as how many parameters it would have taken a previous well-known model to reach he same performance on some benchmarks, such as MMLU. $\backslash hat\; N$ is not measured directly, but rather by measuring the actual model performance $S$, then plugging it back to a previously fitted scaling law, such as the Chinchilla scaling law, to obtain what $\backslash hat\; N$ would be required to reach that performance $S$, according to that previously fitted scaling laws.

A densing law states that $\backslash ln\; \backslash left(\backslash frac\{\backslash hat\; N\}\{N\}\backslash right)\_\{max\}\; =\; At\; +\; B$, where $t$ is real-world time, measured in days.

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