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Transformers Can Do Arithmetic with the Right Embeddings

arxiv.org

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Re: Transformers Can Do Arithmetic with the Right Embeddings

#111

Post by refulgentis »

vessenes wrote: post #57

Wow, a lot of grumpiness in here. If it's true that adding like 20 or so tokens to encode column location / decimal spot triples math performance in out of band tasks, that's a big deal. It's a simple fix, it improves performance A LOT, and they even indicate it's not just a party trick, in that the LLM can use the information to do better on related tasks like sorting and list making. This is basically free to add,…

> This is basically free to add, and there's no reason it shouldn't be made part of standard tokenization.

This is muchhhhh different from how tokenization works today. Adding tokens to the vocabulary is free, everything outside that (i.e. string -> tokens) is going to be a major pain in the ass. Doable but annoying and error prone

Re: Transformers Can Do Arithmetic with the Right Embeddings

#112

Post by zacksiri »

I think the problem here is that 'understanding' is not the same as curve fitting.

If all one is doing is giving a model lots of data and fitting curves it's not really 'understanding' but brute forcing it's way (with gradient descent) and then storing the weights and finally approximate the solution when a query is passed in.

This is not the same as understanding. Human intelligence can operate deterministically as well as non-deterministically. We can listen to language, which is by it's nature non-deterministic and convert that into deterministic operations and vice a versa. IE we can operate on some logic and explain it in multiple ways to other people.

Understanding requires much less data than brute forcing your way into pattern recognition.

When you see a simple number like this 2 * 4 you are able to understand that it's equivalent to 2 + 2 + 2 + 2 and that in turn means 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 Because you 'understand' this basic concept and all the operations in between you are able to compute more examples. But you only need to understand it once. Once you understand multiplications and additions and all the tricks in between you are able to compute 23 * 10 without being fed 23 * 10 as prior data. Understanding is very different from fitting a curve. You can reach conclusions and understanding through pattern recognition, but it's important to differentiate 'approximation' from 'calculation'. If you understand something in it's entirety you should be able to calculate an outcome deterministically.

Right now LLMs lack 'understanding', and seems to only 'approximate' which may seem like 'understanding' but is actually not.

Re: Transformers Can Do Arithmetic with the Right Embeddings

#113

Post by Filligree »

refulgentis wrote: post #111
vessenes wrote: post #57

Wow, a lot of grumpiness in here. If it's true that adding like 20 or so tokens to encode column location / decimal spot triples math performance in out of band tasks, that's a big deal. It's a simple fix, it improves performance A LOT, and they even indicate it's not just a party trick, in that the LLM can use the information to do better on related tasks like sorting and list making. This is basically free to add,…

> This is basically free to add, and there's no reason it shouldn't be made part of standard tokenization. This is muchhhhh different from how tokenization works today. Adding tokens to the vocabulary is free, everything outside that (i.e. string -> tokens) is going to be a major pain in the ass. Doable but annoying and error prone

Doesn’t seem as complicated as, say, coding a lexer for C. And why shouldn’t tokenisation use lexers or an equivalent?

Re: Transformers Can Do Arithmetic with the Right Embeddings

#114

Post by spencerchubb »

torginus wrote: post #34

I just wonder if numbers were written right to left, llms would be much better at arithmetic. You can 'predict' the least significant digit by reusing the already written digits in the computation, but to generate most significant ones, you generally need to do the entire computation in one go.

They do that in the paper. Least significant digit on the left

Re: Transformers Can Do Arithmetic with the Right Embeddings

#115

Post by spencerchubb »

matrix2596 wrote: post #72
infogulch wrote: post #2

The other day I was wondering if LLMs are bad at at maths because they don't have readily apparent access to the concept of "columns". Apparently the answer is yes. Vertical alignment across lines is pretty important for humans to learn operations on digits, but the way we encode lines with a \n separator doesn't really help. In a recent codebullet video gpt really struggled with any kind of vertical alignment task.…

wouldnt presenting numbers in reverse order, with the least significant digit on the left and most significant on the right help with the reasoning?

They do that in the paper

Re: Transformers Can Do Arithmetic with the Right Embeddings

#116

Post by gwern »

torginus wrote: post #34

I just wonder if numbers were written right to left, llms would be much better at arithmetic. You can 'predict' the least significant digit by reusing the already written digits in the computation, but to generate most significant ones, you generally need to do the entire computation in one go.

Yes. This has already been demonstrated by "Teaching Arithmetic to Small Transformers" https://arxiv.org/abs/2307.03381 , I'm not sure what OP adds except demonstrating that you can do that via the embedding itself rather than the tokenization.

> We first demonstrate that conventional training data is not the most effective for arithmetic learning, and simple formatting changes can significantly improve accuracy. This leads to sharp phase transitions as a function of training data scale, which, in some cases, can be explained through connections to low-rank matrix completion. Building on prior work, we then train on chain-of-thought style data that includes intermediate step results. Even in the complete absence of pretraining, this approach significantly and simultaneously improves accuracy, sample complexity, and convergence speed. We also study the interplay between arithmetic and text data during training and examine the effects of few-shot prompting, pretraining, and model scale. Additionally, we discuss length generalization challenges.

Re: Transformers Can Do Arithmetic with the Right Embeddings

#117

Post by zyklu5 »

zacksiri wrote: post #112

I think the problem here is that 'understanding' is not the same as curve fitting. If all one is doing is giving a model lots of data and fitting curves it's not really 'understanding' but brute forcing it's way (with gradient descent) and then storing the weights and finally approximate the solution when a query is passed in. This is not the same as understanding. Human intelligence can operate deterministically as…

I think you are mixing layers of abstraction. To make a crude but I think not unhelpful analogy: 'Understanding' is a natural language concept that is our way to describe whats happening in our heads, and like most other such concepts is resistant to any clear definition and will exhibit sorites type paradoxes when one is attempted. It belongs to the presentation layer of the stack. While the process of curve fitting, however it is implemented, with whatever NN structure (like transformers) or maybe something else entirely belongs to the physical layer of the stack -- akin to frequency modulation.

While I am unsure whether LLMs are really understanding, whatever that means, I think it is not difficult to believe that any form of understanding we implement will involve 'curve fitting' as a central part.

Re: Transformers Can Do Arithmetic with the Right Embeddings

#118

Post by naasking »

catapart wrote: post #102
lapitopi wrote: post #99

Earlier quoted context omitted.

That’s why I am asking a computer.

I'm with you. I get that this is akin to asking a human, because we're trying to reason, so we will bring along (assumedly) unavoidable deficiencies of human reasoning. But if I were to ask a human genius this question, ne would grab a calculator and employ it as ne did the rest of ner reasoning. So it seems like we should probably teach LLMs to "use a calculator", rather than try to get them to be more right when do…

Indeed, "use a calculator" is "just a trick"!

Re: Transformers Can Do Arithmetic with the Right Embeddings

#119

Post by HarHarVeryFunny »

0-_-0 wrote: post #66
Havoc wrote: post #43

For things like this where we have computationally cheap, well understood, reliable tools available (aka calculator) it seems better to train the model in tool use. I guess perhaps the techniques could be generalized though?

To me this finding shows how transformers don't generalise, since they need specialised embeddings to handle a problem

I think this is more a matter of how numbers are input and lack of specific training, including visual training.

For example, the number 12,345,678 is input to ChatGPT as the three tokens "123" "456" "78", which isn't the best place to start to learn that this is an 8 digit number with specific digit positions!

https://platform.openai.com/tokenizer

As a human child you learn about numbers largely visually by pointing to units, tens, hundreds etc, visually aligning them to add, etc. Maybe a multi-modal model, if it was visually trained on chalkboard primary school math, would do better in learning the concept of position based powers of 10, etc.

Re: Transformers Can Do Arithmetic with the Right Embeddings

#120

Post by mikeocool »

olalonde wrote: post #65
psychoslave wrote: post #38

Earlier quoted context omitted.

I didn’t test with all LLM out there, but all of thus I tested failed with something as basic as "What is the number of words in the sentence coming before the next one? Please answer."

GPT-4 (OpenAI): The sentence you're referring to is "What is the number of words in the sentence coming before the next one? Please answer." It contains 14 words.

Interestingly, chat gpt 4o gave me the answer 15.
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