Formally, what hypotheses are you comparing? What do you think the specific hypothesis of the "AI = stats" person is? It isnt that the NN literally remembers data tokens, right?
In any case:
The issue with forcing NNs to model mathematical features is that the structure of the data itself has those properties. So the distributional hypothesis is true for sorting ordinals.
But it's really obviously false for natural language. The properties of the world are not the properties of word order... being red isnt "red follows words like...".
Let's not be so hasty. I think I do put it as clearly as possible. I'm comparing essentially your Hyp1 and Hyp2, where Hyp1 (aka the stochastic parrot) is expressed a little bit more clearly as the LLM is learning an n-gram that produces correct sorts through rote memorization of statistical correlations in the training data, like that sorted lists tend to start with '0', end with '99', and increase monotonically; and Hyp2 is that the LLM's training molds it into representing an actual sorting algorithm that would correctly generalize to any input list.
> But it's really obviously false for natural language. The properties of the world are not the properties of word order... being red isnt "red follows words like..."
This is not really obviously false. Yes, being red isn't "red follows words like...". But a word order should still map to properties of the world, especially if those words are to be meaningful to a listener. Being red is "a surface reflects or transmits most of the light in the 600-800 nm spectrum and absorbs most of the rest". Of course, it won't do to just echo those tokens; once you've nailed down the concept of "red", you need to make sure that concepts like "reflects", "light", and "spectrum" are represented as well. It's an open question as to whether this sort of knowledge graph can be properly bootstrapped from a large volume of text descriptions, but I am strongly inclined to believe it can. If you dismiss it outright you're just begging the question.
Redness is not in the structure of those sentences. And there will always be an infinity of sentences which are True but cannot be infered by an LLM -- but can be so, trivially, by a person acquainted with redness.
In any case,
I'd need more time than I have at the moment to seriously state Hyp1 for your case -- but atm, I can say that because the data itself has the property, Hyp1 becomes much harder to state and the argument much subtler.
Since what is a "statistical distribution" of "ordinals" anyway? And how much memory is required to represent it? My sense is this distribution has highly redundant features which will be trivially compressible without learning any "sorting algorithm".
At a quick glance of your article it feels like you havent formulated Hyp1 correctly -- P(CorrectSort | f(HistoricalCases)) is perhaps arbitrarily high if some statistical f() can be chosen well.
Which is exactly how the set of sentences actually written encodes in it the idea of "Redness". It's the "actually written" part that carries information about the real world.
> And there will always be an infinity of sentences which are True but cannot be infered by an LLM -- but can be so, trivially, by a person acquainted with redness.
That's cheating, because "a person acquainted with redness" presumably learned it by sight, which LLMs can't do just yet (at least the widely accessible ones can't). Would you also say that a person born blind also cannot infer those True sentences about redness? Because if they can, that means the concept of redness is capable of being taught through language, and so there's no reason LLMs couldn't pick up on it too.
Sure; it's in the spectrum of reflected light. (Or perhaps, the retina's trichromal responsivity). But that physical concept can be meaningfully described by sentences. It doesn't require an infinite number of them to create a coherent world-model, which can do things like predicting that a blue object will become red if it moves away from you at a high enough speed. Which is something a human might be surprised by even after many years of visual experience with red objects -- unless they've read sentences about the Doppler effect in a physics textbook.
If you can manage to trick GPT-4 into revealing that it doesn't have a world-model of the concept of 'red', please show us!
> At a quick glance of your article it feels like you havent formulated Hyp1 correctly -- P(CorrectSort | f(HistoricalCases)) is perhaps arbitrarily high if some statistical f() can be chosen well.
Keep in mind, the LLM's structure was not hand-crafted to do well on this mathematical task. It was built to be good at language modelling, and initialized with essentially a uniform prior over all token sequences. Even if a dataset is efficiently compressible, that's no guarantee that the LLM will be able to compress it efficiently. In fact, many people would probably be surprised to learn that it can do this problem at all, let alone so well with so little training. But do think about the statistics of sorting a bit more. I think it's not as easily compressible as you think it is, except by an actual sorting algorithm. Again, you can compress it a bit with monotonicity and so on, but nowhere near the amount you'd need to sort a long list without errors, using so few parameters. I compute the number of sorted and unsorted lists in the footnotes.
One of the things that makes sorting tricky for an LLM is you always need to look at every item in the input list. Even if the previous output token was '99', you can't be sure you're now at the end of the list; you still need to count how many '99's were output already and how many are needed.
(The dataset itself, of course, does not contain the notion of sorting, a description of sorting, a test for sortedness, or any algorithm for sorting. It only contains a large but finite number of examples of sorted and unsorted lists. It's up to the LLM, and its training process, to discover the mechanism that generated these results.)
Where did you find this terminology?
EDIT:
>> and Hyp2 is that the LLM's training molds it into representing an actual sorting algorithm that would correctly generalize to any input list.
Btw, you have not shown anything like that. You trained and tested on lists of two-digit positive integers expressible in 128 characters. That's not "any input list". As a for instance, what do you think would happen if you gave your model an alphanumeric list to sort? Did you try that?
Your model also doesn't correctly generalise, not even to its own training set that you tested it on. There's plenty of error in the figure where you show its accuracy (not clear if that's training or test accuracy).
It's not clear to me how you account for those obvious limitations of your model (it's a toy model after all) when you claim that it "learned to implement a sorting algorithm" etc. It would be great if you could clarify that.
The tokenizer would throw an exception, because it doesn't have any tokens to represent alphabetical characters. But you tell me - if I had tokenized alphabetical characters and defined an ordering, would you expect the results to be any different?
> You say e.g. that "LLM is learning an n-gram"[...] you can't "learn an n-gram".
Where do I say that? I don't think I make any reference to "learning an n-gram", which is a relief because I don't know what it would mean to "learn an n-gram".
> There's plenty of error in the figure where you show its accuracy (not clear if that's training or test accuracy).
Test accuracy between training iterations (not part of the training process itself, which uses its own separate validation set which is split from the training set). And yes, I agree, it is not error-free, and I wouldn't expect it to be, especially after so little training. What the figure shows is the percentage of sorts that were error-free, and how rapidly that decreases. I've since repeated the test with finer resolution, and the fraction of imperfect sorts continues to decrease about as you expect, which is enough to satisfy my curiosity, although I'm a little curious to see if there is some point where it falls completely to zero.
