# Scaling laws for very large neural nets

Compute/size/data tradeoffs

January 14, 2021 — February 16, 2023

Got good behaviour from a million parameter model? Want to see if stuff gets weirder as we hit a billion parameters? Turns out it does!

Brief links on the theme of scaling in the *extremely* large model/large data limit and what that does to the behaviour of the models. A new front in the complexity, and/or statistical mechanics of statistics.

As to *how* to scale up these models in practice, see distributed gradient descent.

## 1 Side note: The bitter, better lesson

See optimal cleverness.

## 2 Big transformers

One fun result comes from Transformer language models. An interesting observation way back in 2020 was that there seemed to be an unexpected trade-off where you can go faster by training a bigger network. Indeed, there is a whole family of observations in this vein trying to identify actual scaling behaviour.

nostalgebraist summarises Henighan et al. (2020);Kaplan et al. (2020):

## 2.1 L(D): information

OpenAI derives a scaling law called L(D). This law is the best you could possibly do – even with arbitrarily large compute/models – if you are only allowed to train on D data points.

No matter how good your model is, there is only so much it can learn from a finite sample. L(D) quantifies this intuitive fact (if the model is an autoregressive transformer).

## 2.2 L(C): budgeting

OpenAI also derives another a scaling law called L(C). This is the best you can do with compute C, if you spend it optimally.

What does optimal spending look like? Remember, you can spend a unit of compute on

- a bigger model (N), or
- training the same model for longer (S)
…In the compute regime we are currently in, making the model bigger is way more effective than taking more steps.

Controversy! the scaling laws have been revised.

## 3 Incoming

- Zhang et al. (2020) (how do NNs learn from language as
*n*increases? - DeepSpeed Compression: A composable library for extreme compression and zero-cost quantization (targeting large language models)

## 4 References

*arXiv:2010.14701 [Cs]*.

*arXiv:2001.08361 [Cs, Stat]*.

*arXiv:2110.04374 [Cs]*.

*arXiv:2011.03641 [Cs]*.

*arXiv:2004.10802 [Cs, Stat]*.

*arXiv:2011.04946 [Cs]*.