There are some neural scaling laws: https://en.wikipedia.org/wiki/Neural_scaling_law

And you can in practice often see a large network follow an expected training trajectory for loss/entropy.

In practice data is getting more sparse than compute, so you are usually overtraining using more tokens and being suboptimal.

It should be noted that these laws aren't, well, laws; "a neural scaling law is an empirical scaling law that describes how neural network performance changes as key factors are scaled up or down.". Notice the word empirical here.

These empirical observations on quantities which aren't "intelligence" since no one has yet defined intelligence.

Another dumb question: can the model be smarter than the data?

I wonder about this a lot.

There are all these publications from OpenAI and Anthropic that are solving complex math & science problems.

Are they just "brute forcing" it? Able to try a lot more "known" or reasonable approaches given existing knowledge?

Or are they showing "emergent intelligence" and able to try genuinely new approaches that can't be expected from the knowledge in their dataset?

> Another dumb question: can the model be smarter than the data?

There's a mathematical sense in which it cannot be: the data-processing inequality (https://en.wikipedia.org/wiki/Data_processing_inequality). The extent to which that sense is totally counterintuitive to how things normally work in ML nowadays has led to investigating different foundations for information theory: https://arxiv.org/abs/2601.03220.

Data isn't "smart", it doesn't contain intelligence. I would argue that LLMs are functionally intelligent, they have learned to what functionally amounts to thinking by training on text (and images, in some cases). They are capable of performing experiments autonomously and make decisions based on out-of-distribution data. Besides (current) lack of online learning, they are limited by their biases and working memory - but so are humans, and I'd like to see a convincing argument on why these mathematical laws should not also apply to humans, which are arguably intelligent.

"Smarter" is ill-defined.

Are you smart if you see that cognitive scientists had something figured out, connected it to something in your field of chemistry and made a breakthrough treatment for some disease?

Does any of our existing intelligence build on completely unknown information? You could argue that any new solution/emergent intelligence is an association of existing information in new ways.

It also brings us to what is brute forcing? Humans are brute forcing stuff most of the time. You will have 10 stupid thoughts for every good thought. 10 stupid ideas for every good idea. It's very rare, if not near impossible, that a non-mathematician with no prior knowledge, goes into a room on day 1 of deciding to think about a conjecture they don't know anything about, and solve it on the blackboard. They need to use their collect associations, formalize their axioms, think of first principles (known information), and associate it.

So to answer your question you need to answer:

1) What does "smarter" refer to? Is it accumulation of information about the universe? Is it solving tasks useful for the human in the world? Something else? Is it acting as an agent with it's own goals and interests in the universe? What if the goals are hard for humans to understand?

2) What does "genuinely new approaches" refer to? Is it fine to use math axioms to prove things? Does interpolation between two existing solutions count as new? At what granularity of concept does novelty happen (Does using two higher abstraction concepts, like combining some new, modern neural network architecture with a database count as a new thing), even if the underlying lower abstraction of information as storage, latency and compute is not novel?

Yes, the Chinchilla scaling laws from 2022: https://arxiv.org/abs/2203.15556 More parameters directly and mechanically means less prediction error for each token.

(This moves the argument to "is being really good at predicting the next token the same as being intelligent?" which has happily diverted many minds for years now)

From your link:

> for every doubling of model size the number of training tokens should also be doubled

Well, I don't think many claim that increasing the model size while also increasing the training tokens doesn't make sense.

The problem is how to get the training tokens...

Interestingly enough labs do train with 100s or even 1000s times more tokens than what's proposed by the paper as an optimal value.

And the answer to your question is basically just “synthetic data”.

The Chinchilla paper is probably the poster child of “paper that aged like milk” in LLM field though. Everyone over the past 4 years have trained with several orders of magnitude more data than the Chinchilla optimal number.

If you don’t know which data has the highest value, though more at it. Chinchilla is probably still true, but with that caveat. And it could be like advertising… you know a lot of it is worthless, but you don’t know which part.

Your phrasing (_observed_ scaling _part of_ some equation) might indicate you assume / think there’s some theoretical equation that relate intelligence to those parameters and the we observe the data to confirm it. As far as I know, there isn’t - it’s purely empirical curve fitting. Train a model with different amount of compute / data / parameters, measure performance, fit curve.

'Scaling law' is the prediction that increasing the training data and model size will cause the result to be better. It's not a law, but it was the prediction that OpenAI staked their fledgling startup on to get to the ChatGPT moment.