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Why can't transformers learn multiplication?

arxiv.org

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Re: Why can't transformers learn multiplication?

#41

Post by skinner_ »

smartmic wrote: post #32

Yesterday, I learned the opposite. Simon Willison demonstrated in another thread how this works out … see https://news.ycombinator.com/item?id=45686295

That's very cool, but it's not an apples to apples comparison. The reasoning model learned how to do long multiplication. (Either from the internet, or from generated examples of long multiplication that were used to sharpen its reasoning skills. In principle, it might have invented it on its own during RL, but no, I don't think so.)

In this paper, the task is to learn how to multiply, strictly from AxB=C examples, with 4-digit numbers. Their vanilla transformer can't learn it, but the one with (their variant of) chain-of-thought can. These are transformers that have never encountered written text, and are too small to understand any of it anyway.

Re: Why can't transformers learn multiplication?

#42

Post by lossolo »

kovek wrote: post #40
photonthug wrote: post #35

Earlier quoted context omitted.

> It DOES fail more when the numbers are longer (because it results with more text in the context), I tried to raise this question yesterday. https://news.ycombinator.com/item?id=45683113#45687769 Declaring victory on "reasoning" based on cherry-picking a correct result about arithmetic is, of course, very narrow and absurdly optimistic. Even if it correctly works for all NxM calculations. Moving on from arithmetic t…

Well, if the model can reliably keep in context CPU cache plus CPU registers plus CPU instructions and is able to do operations based on those, then we pretty much solved computation using LLMs, right? It could use RAG to operate on RAM and SSD. Here we can see the amount of data a high end traditional non-SOC CPU holds: > For a recent high-end non-SoC desktop CPU: > Cache: ~40-100 MB total (L1 + L2 + shared L3) > Re…

It doesn't work like mapping CPU caches/registers into an LLM context. Transformers have no mutable registers, they attend over past tokens and can't update prior state. RAG isn't RAM. Even with huge context, you still can't step CPU style instructions without an external, read/write memory/tooling.

And temperature 0 makes outputs deterministic, not magically correct.

Re: Why can't transformers learn multiplication?

#43

Post by photonthug »

lossolo wrote: post #42
kovek wrote: post #40

Earlier quoted context omitted.

Well, if the model can reliably keep in context CPU cache plus CPU registers plus CPU instructions and is able to do operations based on those, then we pretty much solved computation using LLMs, right? It could use RAG to operate on RAM and SSD. Here we can see the amount of data a high end traditional non-SOC CPU holds: > For a recent high-end non-SoC desktop CPU: > Cache: ~40-100 MB total (L1 + L2 + shared L3) > Re…

It doesn't work like mapping CPU caches/registers into an LLM context. Transformers have no mutable registers, they attend over past tokens and can't update prior state. RAG isn't RAM. Even with huge context, you still can't step CPU style instructions without an external, read/write memory/tooling. And temperature 0 makes outputs deterministic, not magically correct.

> And temperature 0 makes outputs deterministic, not magically correct.

For reasons I don't claim to really understand, I don't think it even makes them deterministic. Floating point something something? I'm not sure temperature even has a static technical definition or implementation everywhere at this point. I've been ignoring temperature and using nucleus sampling anywhere that's exposed and it seems to work better.

Random but typical example.. pydantic-ai has a caveat that doesn't reference any particular model: "Note that even with temperature of 0.0, the results will not be fully deterministic". And of course this is just the very bottom layer of model-config and in a system of diverse agents using different frameworks and models, it's even worse.

Re: Why can't transformers learn multiplication?

#44

Post by kovek »

lossolo wrote: post #42
kovek wrote: post #40

Earlier quoted context omitted.

Well, if the model can reliably keep in context CPU cache plus CPU registers plus CPU instructions and is able to do operations based on those, then we pretty much solved computation using LLMs, right? It could use RAG to operate on RAM and SSD. Here we can see the amount of data a high end traditional non-SOC CPU holds: > For a recent high-end non-SoC desktop CPU: > Cache: ~40-100 MB total (L1 + L2 + shared L3) > Re…

It doesn't work like mapping CPU caches/registers into an LLM context. Transformers have no mutable registers, they attend over past tokens and can't update prior state. RAG isn't RAM. Even with huge context, you still can't step CPU style instructions without an external, read/write memory/tooling. And temperature 0 makes outputs deterministic, not magically correct.

Well, the LLM may re-infer the whole state fully on every instruction. Temperature 0 is deterministic and that's what we are looking for. If the model is trained properly on how the CPU state + instructions should be handled, then it should be able to produce the next state.

Re: Why can't transformers learn multiplication?

#45

Post by Nifty3929 »

Numbers aren't language, or even sequences of tokens, or vectors.

There is an inherent numeric-ness and logic to math that I don't think we can represent well using LLMs and transformers.

3 isn't about the word "three" - it is a quantity or a measurement. And 3x4 is a specific numerical operation that is not really contained in that sequence of symbols.

Re: Why can't transformers learn multiplication?

#46

Post by hodgehog11 »

CaveTech wrote: post #36
ddingus wrote: post #31

A while back I saw a post where people ran a model over and over to accomplish a code base port from one language to another. In their prompt, they told it to leave itself a note and to accomplish something each time. Then they put the model in a loop and it worked. In one instance, a model removed itself from the loop by editing a file or some other basic means. To me, iterative tasks like like multiply and long div…

The amount of paths in the wrong direction are infinitely more than then number in the right direction. You'll quickly realize this doesn't actually scale.

I'm a bit confused by this; are you referring to vanishing/exploding gradients during training or iteration at inference? If the former, this is only true if you take too many steps. If the latter, we already know this works and scales well.

Re: Why can't transformers learn multiplication?

#47

Post by hodgehog11 »

jerf wrote: post #5

This 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…

I think you might be missing some appropriate context. I agree that it is ridiculous to expect a language model to be good at symbolic manipulation; that is best done with tool use. However, there is a significant line of work dedicated to algorithm discovery for mathematical problems using neural networks. Transformers are used here due to their popularity, but also some theoretical analysis to suggest that they are the among the most efficient architecture for learning automata. It's still unclear whether this is truly sound though, which is where this kind of research matters.

Re: Why can't transformers learn multiplication?

#48

Post by ares623 »

nico wrote: post #10

Would 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…

Isn’t that what all the hundreds of billions are banking on? “General” intelligence.

Re: Why can't transformers learn multiplication?

#49

Post by onlyrealcuzzo »

ares623 wrote: post #48
nico wrote: post #10

Would 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…

Isn’t that what all the hundreds of billions are banking on? “General” intelligence.

You don't need general intelligence to make good memes to keep people scrolling through Instagram.

You don't need general intelligence to make a decent coding tool like Cursor.

You don't need general intelligence to improve SERPs.

You don't need general intelligence to sell a subscription for a decent AI assistant.

There's tons of value already added without anything general.

Re: Why can't transformers learn multiplication?

#50

Post by akomtu »

IMO, the mystery has a simple explanation: addition is mostly local in nature, when the 5th digit in the input impacts only 5th or 4th digits in the output, while multiplication is not. That being said, LLMs don't understand addition either: the illusion will break down on very large inputs.
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