[Gisbitus]

Just like it's mentioned later in the article: it doesn't really matter if you get an addition mostly right. You either get it right or you don't. I still appreciate their effort though, because even after altering the grading system, there were still some emergent abilities.

[arka2147483647]

Assume we have a child, and we test him regularly:

- Test 1: First he can just draw squiggles on the math test

- Test 2: Then he can do arithmetic correctly

- Test 3: He fails on the last details on the algebraic calculation.

Now, event though he fails on all tests, any reasonable parent would see that he improving nicely, and would be able to work in his chosen field in a year or so.

Or alternatively, if we talk about AI, we can set the Test as a threshold, and we see the results are continuously trending upwards, and we can expect the curve to breach the threshold in the future.

That is; measuring improvement, instead of pass/fail, allows one to predict when we might be able to use the AI for something.

[londons_explore]

With AI you can do millions of tests. Some tests are easy by chance (eg. "Please multiply this list of numbers by zero"). Some tests are correct by chance alone, easy or hard.

When you actually do these millions of tests, I don't think it really matters what the exact success metric is - an AI which is 'closer to correct, but still wrong' on one test will still get more tests correct overall on the dataset of millions of tests.

[raincole]

Human beings do arithmetic problems wrong all the time so I'm not sure "doing addition 100% right" is a merit of intelligence.

I'm not saying LLM will achieve AGI (I don't know if it will, or when it does we'll even know). But somehow people seem to be judging AI's intelligence with this simple procedural:

1. Find a task that AI can't do perfectly. 2. Gotcha! AI isn't intelligent.

It just makes me question humans' intelligence if anything.

[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)

[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.

[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.

[hprotagonist]

> It just makes me question humans' intelligence if anything.

A more serious byproduct of the tendency to talk about machines in anthropomorphic terms is the companion phenomenon of talking about people in mechanistic terminology. The critical reading of articles about computer-assisted learning —excuse me: CAL for the intimi— leaves you no option: in the eyes of their authors, the educational process is simply reduced to a caricature, something like the building up of conditional reflexes. For those educationists, Pavlov’s dog adequately captures the essence of Mankind —while I can assure you, from intimate observations, that it only captures a minute fraction of what is involved in being a dog.

https://www.cs.utexas.edu/users/EWD/transcriptions/EWD09xx/E...

[advael]

I mean I could equally say that the opposing bias is

1. Choose a few good blunt instruments we use to gatekeep students on the premise that it tests their "intelligence" (or wait, do we mean subject matter comprehension with this one?)

2. Apply a big ol' machine learning model to those tests

3. Woa it's smarter than a third grader! OMG it's smarter than a lawyer! You guys this must be ASI already!

Rhetoric and selective rigor can justify any perspective. Smart and stupid arguments can be made for any position. Water is wet

I also can't claim to know with certainty whether transformers are going to end up being AGI in some meaningful sense, but I will definitely say that we've created a lot of rubrics for assessing human intelligence that mostly exist for expediency, and a cursory glance at education should tell you there's a lot of Goodhart's Law going on with all of 'em. I know for a fact I can do a damn good job on your average multiple choice test on knowing some etymology and being good at logical elimination, and I can bullshit my way through an essay, both without taking the class, and I view this more as a flaw in the instrument than evidence that I'm a godlike superintelligence that can just know anything without studying it. Humans make a lot of tests that are soft to bullshitting with a little pattern-recognition thrown in

[IanCal]

> it doesn't really matter if you get an addition mostly right.

This is definitely not true in the real world. Approximate solutions are often good enough to answer a question.

[RandomLensman]

But then it isn't "addition" but approximating "addition".

[Satam]

There's always some uncertainty to any kind of answer or computation, it's just at some threshold we unconsciously decide to take it for a fact.

[Jensson]

LLMs aren't wrong by a small percentage, they are wrong by a small number of tokens. They can miss a zero or be off by 100% and its just a token difference, to the LLM that is a minor mistake since everything else was right but it is a massive mistake in practice.

[wruza]

I watch math classes on youtube and some lecturers make symbolic mistakes all the time. Minus instead of a plus, missing exponents, saying x but writing y, etc. They only notice it when something unexpected contradicts down the line.

[Jensson]

They got it right as you said, it just took a bit longer. That doesn't contradict what I said, humans can get things right very reliably by looking over the answers especially if you have another human to help look at the answers. An AI isn't comparable to a human, it is comparable to a team of humans, two ChatGPTs can't get more accurate by correcting each others answers but two humans can.

[lewhoo]

So many things contribute to human error it is probably impossible to make a 1 to 1 parallel with LLM's. For instance, the fact that you are being recorded is in many cases a significant performance drop.

[RandomLensman]

What uncertainty and threshold is there in the addition of integers, for example (within mathematics and the usual definitions)? Or in Boolean logic with the "and" operation?

I don't think everything has uncertainty and thresholds to it, especially, when it actually resides outside of a technical implementation.

[Satam]

To verify the answer you'll always need to trust the technical implementation that's doing the computation. Doesn't matter if it's our brains or a calculator.

Somewhere between "it's always wrong" and "it's always right unless the bits got flipped by cosmic rays" we deem the accuracy to be good enough.

[RandomLensman]

Disagree, the theory exists outside of any specific technical implementation (every single one of those could be wrong, for example). You might not be able to verify something without being subject to random errors, but that doesn't mean the theory itself is subject to random errors.

Any implementation (or write down etc.) of something can have errors, but the errors are in the implementation and do not give rise to uncertainty outside of the implementation. There is no uncertainty as to what the sum of two integers should be (within the usual mathematics).

[t_mann]

> it doesn't really matter if you get an addition mostly right

Back-of-the-envelope / mental math often works like that, and it's something that humans regularly use, so clearly it has some use.

[nathan_compton]

Tell it to floating point numbers.