Smallest transformer that can add two 10-digit numbers

One major limitation of the LLM architecture is that even the failure mode varies unpredictably between inputs.

The set of 11-digit numbers with any given failure mode (or even successful output) has no discernable pattern, merely whatever randomness the training process baked into the model.

You can't predict ahead of time when they will fail spectacularly, nor draw a clear boundary around the failure cases. And early major example of this were the "glitch tokens" introduced into most LLMs by training on reddit data.

But there is an "in general"/"average failure rate across all inputs of a given size" answer: LLMs performance drops off a cliff once the input reaches too much complexity. (A "┐" shaped curve) In contrast to humans, where you can ask a child to add two N-digit numbers and the error rate will be approximately linear to N.

Depends on how the transformer has been trained. If it has seen 11 digit examples while training it might work, else the input will be out of distribution and it will respond with a nonsensical number.

For instance the current high score model (311 params [0]), when given 12345678900 + 1, responds with 96913456789.

An interesting experiment would be: what's the minimum number of parameters required to handle unbounded addition (without offloading it to tool calls).

Of course memory constraints would preclude such an experiment. And so a sensible proxy would be: what kind of neural-net architecture and training would allow a model to handle numbers lengths it hasn't been trained on. I suspect, this may be not be possible.

[0] https://github.com/rezabyt/digit-addition-311p

I didn't look at all the details, but wanted to see how you did the initial embedding and see you do have a 14x5 matrix there. I guess when you are setting things by-hand (rather than learning), the definition of counting "parameters" is a bit unclear. One could say all those are parameters! even if setting in a straight-forward way.

I ask this question as someone who can't do much more than confirm that your blog post is written in English by someone who knows math.

Does this result suggest that if we had N clever humans manually building an LLM, they might come up with something as smart as a frontier model, but potentially 45 times smaller? (1644 / 36 ~= 45, N = very large, time not specified)

> So far it seems to me that self-attention really brought new capabilities to a network

Do we have a layman explanation for what makes self-attention so uniquely powerful? Something more than "it lets you do self-attention".

Writing code by hand, what is this, 6 months ago??

Good question.

It might work, I considered running a test like this. But it does demand certain things.

The subnetwork has to be either crafted as "gradient resistant" or remain frozen. Not all discovered or handcrafted circuits would survive gradient pressure as is. Especially the kind of gradients that fly in early pre-training.

It has to be able to interface with native representations that would form in a real LLM during pre-training, which is not trivial. This should happen early enough in pre-training. Gradients must start routing through our subnetwork. We can trust "rich get richer" dynamics to take over from there, but for that, we need the full network to discover the subnetwork and start using it.

And finally, it has to start being used for what we want it to be used for. It's possible that an "addition primitive" structure would be subsumed for something else, if you put it into the training run early enough, when LLM's native circuitry is nonexistent.

Overall, for an early test, I'd spray 200 frozen copies of the same subnetwork into an LLM across different layers and watch the dynamics as it goes through pre-training. Roll extra synthetic addition problems into the pre-training data to help discovery along. Less of a principled solution and more of an engineering solution.

I had that in mind too. What if you handcraft a subnetwork with (some subset of) Turing machine capability? Do those kinds of circuits emerge naturally during training? Can reasoning use them for complex computation?

You might find https://corticallabs.com/cl1.html interesting (that's of course assuming this is not a scam, which I'm unable to assess).

I don't think we have good tools for formally proving that a transformer's output will match a more traditionally-defined function. But the leading transformers are small enough that formal verification may be possible.

Without any formal verification: The input space of two 10-digit numbers is a bit bigger than 64-bits, so exhaustively verifying all possible inputs doesn't sound practical. Using the same subset of the input space for verifying each submission seems like the easiest way to be fair, and not disclosing that subset to the competitors is obviously necessary.

No. You cannot. It's the wrong tool for the problem.

That little "add" of yours has the overhead of: having an LLM emit it as a tool call, having to pause the LLM inference while waiting for it to resolve, then having to encode the result as a token to feed it back.

At the same time, a "transformer-native" addition circuit? Can be executed within a single forward pass at a trivial cost, generate transformer-native representations, operate both in prefill and in autoregressive generation, and more. It's cheaper.

"smallest supercomputing cluster that can add two numbers"

I mean, yeah, no need to put a bunch of high powered cars in a circular track to watch them race really close to each other at incredible speeds, causing various hazards, either. Especially since city buses have been around for ages.

So can you take an arbitrary transformer and somehow turn it into a compact set of low-power fast gates by some algorithm?

My reaction was the same as I expected there to be a fancy analogue computer build mainly with transformers.

For one the specific 36 parameter version is impossible without float64 so you might guess the corollary that it is not exactly amenable to being found by gradient descent. I think the question of how you can structure transformers and neural nets in general so that they can both very parsimoniously represent things like this and have it be amenible to learning by gradient descent.

"Minsky, why did you close your eyes?"

"So that the room will be empty."

Can you do basic addition of 2 arbitrary numbers with 100% accuracy (no tools) ? No you can't. You will make mistakes for a sufficiently large N even with pen and paper, and a very small N without. Are you no longer generally intelligent ?

LLMs should use tool calling (which is 100% reliable) instead of doing math internally. But in general it would be nice to be able to teach a process and have the AI execute it deterministically. In some sense, reliability between 99% and 100% is the worst because you still can't trust the output but the verification feels like wasted effort. Maybe code gen and execution will get us there.

And npm install llm-is-odd to divide and conquer!

Yes, but it's interesting that you can teach it to do arithmetic, don't you think? Most things can't be taught to do arithmetic, making this "transformer" thing slightly magical. And so then it seems interesting to investigate exactly how much magic is needed to achieve this.

What would be an acceptable amount of energy to spend on something that someone has done in a different manner before? Would you rather we stick with all of the current known ways to do things.

Does this boil down to a condemnation of all scientific endeavours if they use resources?

Would it change things if the people who did it enjoyed themselves? Would they have spent more energy playing a first person shooter to get the same degree of enjoyment?

How do you make the calculation of the worth of a human endeavour? Perhaps the greater question is why are you making a calculation of the worth of a human endeavour.

Because it's fun. Life is meant to be enjoyed.

Those who worry about an imaginary risk and live their lives in constant fear have turned into nothing more than machines enslaved by propaganda.

>Hacker News

not any more, eh?

> the sheer amount of energy and associated global warming risk

I think that's one very good reason to make them more efficient, and that's part of the point of contests like this one.

Wait until you see the quantum computer that it takes to factor the integer 15.

You need to recalibrate your sense of scale if you think that this is a geologically relevant usage of energy.