[Jensson]

Arithmetics is extremely easy for a neural network to perform and learn perfectly, that LLMs fails to learn it even though it is so easy is strong evidence that LLMs has very limited capability to learn logical structures that can't be represented as grammar.

> Human beings do arithmetic problems wrong all the time

Humans built cars and planes and massive ships before we had calculators, that requires a massive amount of calculations that are all perfect to be possible. Humans aren't bad at getting calculations right, they are just a bit slow. Today humans are bad since we don't practice it, not because we can't. LLMs can't do that today, can learn and can't is a massive difference.

[pedrovhb]

My intuition is that a significant challenge for LLMs' ability to do arithmetics has to do with tokenization. For instance, `1654+73225` as per the OpenAI tokenizer tool breaks down into `165•4•+•732•25`, meaning the LLM is incapable of considering digits individually; that is, "165" is a single "word" and its relationship to "4" and in fact each other token representing a numerical value has to be learned. It can't do simple carry operations (or other arithmetic abstractions humans have access to) in the vast majority of cases because its internal representation of text is not designed for this. Arithmetic is easy to do in base 10 or 2 or 16, but it's a whole lot harder in base ~100k where 99% of the "digits" are words like "cat" or "///////////".

Compare that to understanding arbitrary base64-encoded strings; that's much harder for humans to do without tools. Tokenization still isn't _the_ greatest fit for it, but it's a lot more tractable, and LLMs can do it no problem. Even understanding ASCII art is impressive, given they have no innate idea of what any letter looks like, and they "see" fragments of each letter on each line.

So I'm not sure if I agree or disagree with you here. I'd say LLMs in fact have very impressive capabilities to learn logical structures. Whether grammar is the problem isn't clear to me, but their internal representation format obviously and enormously influences how much harder seemingly trivial tasks become. Perhaps some efforts in hand-tuning vocabularies could improve performance in some tasks, perhaps something different altogether is necessary, but I don't think it's an impossible hurdle to overcome.

[Closi]

I don't think that's really how it works - sure this is true at the first level in a neural network, but in deep neural networks after the first few layers the LLM shouldn't be 'thinking' in tokens anymore.

The tokens are just the input - the internal representation can be totally different (and that format isn't tokens).

[Der_Einzige]

Please don't act like you "know how it works" when you obviously don't.

The issue is not the fact that the model "thinks or doesn't think in tokens". The model is forced at the final sampling/decoding step to convert it's latent back into tokens, one token at a time.

The models are fully capable of understanding the premise that they should "output a 5-7-5 syllable Haiku", but from the perspective of a model trying to count its own syllables, this is not possible, as its own vocabulary is tokenized in such a way that not only does the model not have direct phonetic information within the dataset, but it literally has no analogue for how humans count syllables (measuring mouth drops). Models can't reason about the number of characters or even tokens used in a reply too for the same exact reason too.

The person you're replying to broadly is right, and you are broadly wrong. The internal format does not matter when the final decoding step forces a return of tokenization. Please actually use these systems rather than pontificating about them online.

[Closi]

Thank god we aren’t talking about a model counting syllables then.

[IanCal]

That requires converting from a weird unhelpful form into a more helpful form first, so yes but the tokenisation makes things harder as it adds an extra step - they need to learn how these things relate while having significant amounts of the structure hidden from them.

[Closi]

This conversion is inherent in the problem of language and maths though - Two, too (misspelt), 2, duo, dos, $0.02, and one apple next to another apple, 0b10 and 二 can all represent the (fairly abstract) concept of two.

The conversion to a helpful form is required anyway (also lets remember that computers don't work in base 10, and there isn't really a reason to believe that base 10 is inherently great for LLM's either)

[IanCal]

It is, but there's a reason I teach my son addition like this:

    hundreds | tens | ones

        1        2      3
    +   2        1      5
    -----------------------
        3        3      8

Rather than

unoDOOOOS(third) {}{}{} [512354]_ = three"ate

* replace {}{}{} with addition, {}{} is subtraction unless followed by three spaces in which case it's also addition * translate and correct any misspellings * [512354] look up in your tables * _ is 15 * dotted lines indicate repeated numbers

Technically they're doing the same thing. One we would assume is harder to learn the fundamental concepts from.

[Jensson]

Right, which is why testing arithmetics is a good way to test how well LLMs generalize their capabilities to non text tasks. LLMs can in theory be excellent at it, but they aren't due to how they are trained.

[QuadmasterXLII]

The tokens are the structure over which the attention mechanism is permutation equivariant. This structure permeates the forward pass, its important at every layer and will be until we find something better than attention.

[HarHarVeryFunny]

> Arithmetics is extremely easy for a neural network to perform and learn perfectly

That'd depend on the design of the neural net and training objective.

It's certainly not something that comes naturally to an LLM which neither has numbers as inputs or outputs, nor is trained with an arithmetic objective.

Consider inputting "12345 * 10" into GPT-4. First thing it is going to do is tokenize the input, then embed these tokens, and these embedding vectors are then the starting point of what the transformer has to work with...

https://platform.openai.com/tokenizer

You can use OpenAI's tokenizer tool (above) to see how it represents the "12345 * 10" character sequence as tokens, and the answer is that it breaks it down into the token ID sequence [4513, 1774, 353, 220, 605]. The [4513, 1774] represents the character sequence "12345", and "605" represents the character sequence "10".

These token ID's will then be "embedded", which means mapping them to points in a very high dimensional space (e.g. 4096-D for LLaMA 7B), so each of those token ID's becomes a vector of 4096 1's and 0's, and these vectors are what the model itself actually sees as input.

So, for "12345 * 10", what the model sees during training is that whenever it sees V1 V2 V3 V4 it should predict V5, where V1-5 are those 4096-D input token embeddings. The model has no idea what any of these mean - they might represent "the cat sat on the mat" for all it knows. They are just a bunch of token representations, and the LLM is just trying to find patterns in the examples it is given to figure out what the preferred "next token" output is.

So, could you build (and train) a neural net to multiply, or add, two numbers together? Yes you could, if that is all you want to do. Is that what an LLM is? No, an LLM is a sequence predictor, not an NN designed and trained to do arithmetic, and all that is inside an LLM is a transformer (sequence-to-sequence predictor).

[Jensson]

I know why it is hard for LLMs to learn this, that was the whole point. The way we make LLMs today means they can't identify such structures, and that is strong evidence they wont become smart just by scaling since all the things you brought up will still be true as we scale up.

To solve this you would need some sub networks that are pretrained to handle numbers and math and other domains, and then you start training the giant LLM it can find and connect those things. But we don't know how to do that well yet afaik, and I bet all the big players has already tested things like that. As you say adding capabilities to the same model is hard.

[HarHarVeryFunny]

An LLM can learn to identify math easily enough, it's just that performing calculations just using language isn't very efficient, even if it's basically what we do ourselves. If you want an LLM to do it like us, then give it a pencil and paper ("think step by step").

If you want the LLM to be better than a human at math, then give it a calculator, or access to something like Wolfram Alpha for harder problems. Your proposed solution of "give it a specialized NN for math" is basically the same, but if you are going to give it a tool, they why not give it a more powerful one like a calculator ?!

[Closi]

Humans were terrible at getting calculations right - that's why we invented abacuses, slide-rules, books of mathematical tables and tabulation machines.

[Jensson]

Humans invented those since we are slow and have limited working memory. But we managed to invent those since we understand how to perform reliable calculations.

[Closi]

Yes, but that acknowledges that there is a difference between understanding how to perform reliable calculations, and actually being able to perform reliable calculations.

Humans are good at the former, but not the latter.

[Jensson]

Humans are good at performing reliable calculations with pen and paper. That is the same kind of tools that LLMs works with. I'm not sure why humans can do that but not LLMs, the task should be way easier for an LLM.

[falcor84]

> Humans are good at performing reliable calculations with pen and paper.

Speak for yourself. Even though I've always been strong at my conceptual understanding and problem solving in math, I always found it difficult to avoid arithmetic mistakes on pen and paper and could never understand why I was assessed on that. I could have done so much better in high-school math if I was allowed to use a programmable computer for the calculations.

And I think it's the same for LLMs, we should assess them on doing the arithmetic in a single pass, but rather on writing the code to perform the calculation, and responding based on that.

[Jensson]

Maybe a lot of people suffer from a degree of dyscalculia, but in my experience if you do it a lot you just stop making mistakes. Not just me, many others I've seen reliably do calculations pretty quick without making errors, you just do everything twice as you go and then arithmetic errors go to basically 0.

But I do acknowledge that there are probably some or many humans that maybe can't reach that level of reliability with arithmetics.

[Workaccount2]

LLMs (internally) don't have a pen and paper equivalent. Their output is the output of their neurons. Like if I was a head on a table with a screen on my forehead that printed out my thoughts as they appeared in my head. Ask (promt) me my favorite color and "green" would show up on the screen.

This is why prompting LLM's to show their steps works so well, it makes them work through the problem "in their head" more efficiently, rather than just spit out an answer.

However, you can give LLM's external access to tools. Ask GPT4 a particularly challenging math problem, and it will write a python script and run it to get a solution. That is an LLM's "pen and paper".

[Jensson]

> That is an LLM's "pen and paper".

No, that is an LLM's calculator or programming, it doesn't actually do the steps when it does that. When I use pen and paper to solve a problem I do all steps on my own, when I use a calculator or a programming language the tool does a lot of the work.

That difference is massive, since when I use a calculator that doesn't help me learn numbers and how they interact and how algorithms works, while if I do the steps myself I do. So getting an LLM that can reliably execute algorithms like us humans can is probably a critical step towards making them as reliable and smart as humans.

I do agree though that if LLMs could keep a hidden voice they used to reason before writing they could do better, but that voice being shown to the end user shouldn't make the model dumber, you would just see more spam.

[HarHarVeryFunny]

> That is an LLM's "pen and paper".

No, that's an LLM's Python playground.

An LLM's "pen and paper" is "think step by step" where it gets to see it's own output to keep track of what it is doing.

I'd expect that with appropriate prompting one could get a good model to one/few-shot learn how to do addition this way.

[Closi]

LLM's do that ok too... just not for crazy complex equations that would be tough for most humans with pen and paper.

See below which I have just run on GPT4: https://chat.openai.com/share/3adb3aa2-8aec-474f-bdb0-4d761d...

[Jensson]

I know they can do that, but not as reliably as I can for example or typical engineers from 80 years ago. I did engineer exams without a calculator just did all the calculations with pen and paper, didn't make mistakes, just takes a bit longer since calculating trigonometric functions takes a bit but still not a lot of time compared to how much you have.

That was how everyone did it back then, it really isn't that hard to do. Most people today never tried to do it so they think it is much harder than it actually is.

[HarHarVeryFunny]

> strong evidence that LLMs has very limited capability to learn logical structures that can't be represented as grammar.

To add multi-digit numbers requires short term memory (are we on the units, or tens? was there a carry?), which LLMs don't have, so that's really the issue.

The normal workaround for lack of memory in LLMs is "think step-by-step" to use it's own output (which gets fed back in as an input) as memory, and I'd assume that with appropriate training data and prompting an LLM could learn to do it in this fashion - not just giving the answer, but by giving all the steps.

I suppose in theory LLMs could do limited precision math even without memory, if they did it in a single pass through their stack of transformer layers (96 for GPT-3) - use first few layers to add units and generate carry, next few layers to add tens, etc. I'm not sure how, or even if, one could train them to do it this way though - perhaps via a curriculum training agenda of first adding single digit numbers, then two-digit ones, etc ?

[UncleEntity]

> Arithmetics is extremely easy for a neural network to perform and learn perfectly, that LLMs fails to learn it even though it is so easy is strong evidence that LLMs has very limited capability to learn logical structures that can't be represented as grammar.

IDK, there was an article posted here about yet another LLM that performed very badly on the math tests because they mistakenly left out all the math training data.

What impressed me was that it could learn any math at all from just 'reading' books or whatever. Though, perhaps, any correct answer could be attributed to pure luck, dunno.

[wruza]

Arithmetics is extremely easy for a neural network to perform and learn perfectly

Is it?

[Jensson]

Yes, loss minimization quickly gets to the correct implementation of arithmetics since the primitives of neural networks are just math operations, so training it to add or multiply two inputs into an output is very easy. This is so easy and obvious that you run it to test that your neural network implementation works, if it can't figure out arithmetics then you have done something wrong.

LLMs fails to figure out that this is what it has to do, instead it looks like it has a ton of specialized rules to handle arithmetics that results in a lot of errors in the output and are extremely expensive to run.

[wruza]

So the networks you mentioned aren’t LLMs? Why is that a correct comparison then. Like blaming a human that they can’t jump like a cat or multiply like an arbitrary-precision library.

[Jensson]

> So the networks you mentioned aren’t LLMs? Why is that a correct comparison then

Because an LLM is a neural network and neural networks contains neural networks. There is nothing stopping it from having an embedded neural network that learned how to do computations well, except an inability to identify such structures and patterns well enough to train for it.

[jncfhnb]

Tokenizing ‘1735’ as a value of 1735 because you’ve seen a lot of math is probably the most difficult part.