Because they produce output probabilistically, when multiplication is deterministic. Why is this so hard for everyone?
Why can't transformers learn multiplication?
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Re: Why can't transformers learn multiplication?
#12I tried to ask a model to tell me what is the "long multiplication algorithm". It gave it to me. I asked it to follow that algorithm to solve eg. 12987318927 * 12098102983, and it followed the algorithm, and it got the right answer. It DOES fail more when the numbers are longer (because it results with more text in the context), but that can be improved by having the model focus on the right subset of the text, right?
Re: Why can't transformers learn multiplication?
#13Earlier quoted context omitted.
Language _is_ the symbolic manipulation system par excellence though.
There's equivocation in that statement, though, whether you meant there to be or not. There is clearly a difference in how we manipulate English words for normal human activities and the symbolic manipulation with very strict rules we today associate with mathematics and computer science. Human language goes back thousands of years, into the indefinite past we can't track past. Symbolic manipulation is a much, much m…
Doing multiplication in your head isn't the point though, you can externalise language and use it to do things you can't do in your head by writing it down.
Mathematics was born out of very careful reasoning that we do through language, we only use formalisms as they allow us to avoid the massive ambiguities that exist in natural language. Formal symbolic manipulation came out of our already existing abilities of symbolic manipulation through language.
Re: Why can't transformers learn multiplication?
#14Would love to see an architecture that learned more like humans. Start with just imitating one letter, then a few more, than some syllables, then full words, then sentences, etc. Progressively adding on top of previous knowledge Also, it’s interesting that one of the big goals/measures of models is their capacity to “generalize”, but the training methods optimize for loss/accuracy, and only after training test for ge…
Yes, I also wonder about this! Progress from children books to scientific papers etc.
Could it learn e.g. language structure faster in a pre-training stage?
Also somehow one needs to define a proxy to generalization to compute a loss and do backpropagation.
Re: Why can't transformers learn multiplication?
#15This is a gut impression and I don't deny it, but LLMs are Large Language Models, and in my own brain, my Language Model isn't doing large-scale multiplication. I have a language-based intuition for the sigle-digit multiplication table and a touch beyond (and based on my observations that's already above average for a human Language Model, at least in my age peer group), but it's not my Language Model doing 283 times…
A lot of savants that are able to do really cool calculations, or even people that have synesthesia seeing numbers as colors, don't actually do "real" calculations.
I think most humans that do math aren't actually literally computing things as some kind of logic machine.
We can produce logic, and follow the steps of using that logic, but it doesn't seem to me that our cognition is some kind of logic machine itself.
Re: Why can't transformers learn multiplication?
#16Because they produce output probabilistically, when multiplication is deterministic. Why is this so hard for everyone?
Not true though. Internally they can “shell out” to sub-tasks that know how to do specific things. The specific things don’t have to be models.
(I’m specifically talking about commercial hosted ones that have the capability i describe - obviously your run of the mill one downloaded off of the internet cannot do this).
Re: Why can't transformers learn multiplication?
#17This is a gut impression and I don't deny it, but LLMs are Large Language Models, and in my own brain, my Language Model isn't doing large-scale multiplication. I have a language-based intuition for the sigle-digit multiplication table and a touch beyond (and based on my observations that's already above average for a human Language Model, at least in my age peer group), but it's not my Language Model doing 283 times…
Hmm, I wonder what happens if you let them manipulate their own context symbolically, maybe something like a stack machine. Perhaps all you need is a "delete" token, or a "replace" flag. That way you don't have context full of irrelevant information.
I guess the challenge is, where would the training data come from? Data on the internet is in its final form so "next token" is never a delete.
Edit: I guess in essence, that's what reasoning LLMs already do. IIUC the thought blocks are ephemeral, and only the response is maintained for the chat. Maybe there'd be some benefit of doing this recursively? But that's also kind of what subagents are for. So, perhaps nothing new here.
Re: Why can't transformers learn multiplication?
#18Would love to see an architecture that learned more like humans. Start with just imitating one letter, then a few more, than some syllables, then full words, then sentences, etc. Progressively adding on top of previous knowledge Also, it’s interesting that one of the big goals/measures of models is their capacity to “generalize”, but the training methods optimize for loss/accuracy, and only after training test for ge…
Yes, I also wonder about this! Progress from children books to scientific papers etc. Could it learn e.g. language structure faster in a pre-training stage? Also somehow one needs to define a proxy to generalization to compute a loss and do backpropagation.
This field of study is known as "Curriculum Learning" for your Googling pleasure (or I guess ChatGPT Deep Research now).
Re: Why can't transformers learn multiplication?
#19I think it should be able to learn multiplication with chain of thought. Without it, it's probably really difficult to generalize the multiplication of two n-digit integers when you have to accumulate up to n products of digits and handle carrying for each output digit.
Re: Why can't transformers learn multiplication?
#20They're not any better at addition, are they? If they are, I wonder how good they are at adding numbers in log space.
The paper uses a number representation that is designed to make attention easy to learn: each digit is a separate token and the least significant digit is put first, so that the first digit of the output is simply the sum of the first digits of the inputs and the second digit is the sum of the second digits plus an optional carry from the first digits and so on.
If the numbers are represented with the most significant digit first as usual, you need a bunch of intermediate steps before outputting even the first digit just to determine whether it is affected by a carry or not.
The paper looks at multiplication of numbers represented with the least significant digit first as a toy task requiring several additions as intermediate steps to study why a model large enough to perform those additions in principle fails to learn to do so in practice.
They compare with a model that is first trained to produce the intermediate additions explicitly (as a "chain of thought" with a specific format) and then has this CoT progressively shortened during training until there's nothing left of it. But that second model successfully multiplies.
The difference appears to be that the presence of the intermediate results induces a better number representation in latent space, whereas the model without CoT gets stuck in a less efficient local minimum.
So the answer to the question "Why can't transformers learn multiplication?" is that the training process is insufficient for the model to discover the best intermediate steps on its own.
You could do a similar experiment where the CoT involves first taking the logarithm, adding, and then exponentiating to get the final result, but I think logarithms are probably another computation that's too difficult to learn without additional hints for intermediate steps.