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.

Would like to see a car that moved like a horse.

There's an interesting question here.

Would a single human/entity learn more in ..say.. three million years or would short lived ones evolving over three million years and then ~20 years of education learn more?

The current AI tech cycle is focusing on the first, but we don't really know if there are benefits of both.

There's no obvious way to combine these yet.

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

"an architecture that learned more like humans"

i.e. enduring countless generations of evolutionary selection and cross breeding, then fine-tuning a bit?

although it could be interesting, i don't think training on progressively complex strings entirely recapitulates this.

"Would love to see an architecture that learned"

Would be a far more accurate statement. Training != Learning.

Take your damned upvote, and go away.

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.

But it's very hard to define "real-world CoT" -- think about human, we learn multiplications by vertical calculation and we learn division in a similar way -- all these learning process requires an "information dense" tools (calculation process) with intrinsic math rules in it. Isn't that an adapted way of CoT?

> 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 to any kind of problem that fundamentally reduces to model-checking behind the scenes.. we would be talking about exploring a state-space with potentially many thousands of state-transitions for simple stuff. If each one even has a small chance of crapping out due to hallucination, the chance of encountering errors at the macro-scale is going to be practically guaranteed.

Everyone will say, "but you want tool-use or code-gen for this anyway". Sure! But carry-digits or similar is just one version of "correct matters" and putting some non-local kinds of demands on attention, plus it's easier to check than code. So tool-use or code-gen is just pushing the same problem somewhere else to hide it.. there's still a lot of steps involved, and each one really has to be correct if the macro-layer is going to be correct and the whole thing is going to be hands-off / actually automated. Maybe that's why local-models can still barely handle nontrivial tool-calling.

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.

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.

I agree with you, seems like we are trying to make the shoe fit. Not only are we missing the understanding of what is happening inside transformers, but now we are trying to teach them and see how they respond and then interpret it. That seems fine with viruses and animals, but we are talking about a piece of software here. Shouldn't we know what's happening inside? Maybe these kinds of papers can shine more light and give us better understanding though, still it feels backwards to me...Regarding the multiplication itself, shouldn't pure understanding of the meaning of multiplication(it's a summation basically) be enough for 'AI' to call it a day? If AI or human understands that, then the rest is computation part. We already got that covered, so instead of having 'AI' learn it on its own on crazy amount of data and get it right 99% of time, shouldn't we just give it a calculator? Somebody PLEEAASE give this AI a calculator :-)

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.

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.

Language _is_ the symbolic manipulation system par excellence though.

If being probabilistic prevented learning deterministic functions, transformers couldn’t learn addition either. But they can, so that can't be the reason.

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).

Then your transformer would need to know Python.

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.

Math is just symbol manipulation with a set of rules, no?