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I said “WOW!” out loud. An LLM can discover a new solution in high dimensional geometry that hasn’t advanced in 20 years!? That goes way beyond glueing little

I said “WOW!” out loud. An LLM can discover a new solution in high dimensional geometry that hasn’t advanced in 20 years!? That goes way beyond glueing little bits of plagiarized training data together in a plausible way. This suggest that there are hidden depths to LLMs’ capabilities if we can just figure out how to prompt and evaluate them correctly. This significantly broke my expectations. Who knows what discovery could be hiding behind the next prompt and random seed.

bendergarcia 3y ago
It’s kind of like humans: seed of two people and a random interest to pursue, what could they do?!? It makes poverty and children dying unnecessarily even more depressing.
riku_iki 3y ago
> An LLM can discover a new solution in high dimensional geometry that hasn’t advanced in 20 years!? LLM + brute force coded by humans.
- dougmwne 3y ago
  And that not only created a human interpretable method, but beat out 20 years of mathematicians working on a high profile open question. Who’s to say our brains themselves aren’t brute force solution searchers?
  - riku_iki 3y ago
    > Who’s to say our brains themselves aren’t brute force solution searchers? yes, its just much slower (many trillions times) for numbers crunching.
nybsjytm 3y ago
> An LLM can discover a new solution in high dimensional geometry that hasn’t advanced in 20 years!? That goes way beyond glueing little bits of plagiarized training data together in a plausible way. The first advance in 20 years was by Fred Tyrrell last year https://arxiv.org/abs/2209.10045 https://arxiv.org/abs/2209.10045, who showed that the combinatorics quantity in question is between 2.218 and 2.756, improving the previous lower bound of 2.2174. DeepMind has now shown that the number is between 2.2202 and 2.756. That's why the DeepMind authors describe it as "the largest increase in the size of cap sets in the past 20 years," not as the only increase. The arithmetic is that 2.2202-2.218 is larger than 2.218-2.2174. (If you consider this a very meaningful comparison to make, you might work for DeepMind.)
a_wild_dandan 3y ago
> That goes way beyond glueing little bits of plagiarized training data together Moreover, LLMs almost ALWAYS extrapolate, and never interpolate. They don't regurgitate training data. Doing so is virtually impossible. An LLM's input (AND feature) space is enormous. Hundreds or thousands of dimensions. 3D space isn't like 50D or 5,000D space. The space is so combinatorially vast that basically no two points are neighbors. You cannot take your input and "pick something nearby" to a past training example. There IS NO nearby. No convex hull to walk around in. This "curse of dimensionality" wrecks arguments that these models only produce "in distribution" responses. They overwhelmingly can't! (Check out the literature of LeCun et al. for more rigor re. LLM extrapolation.) LLMs are creative. They work. They push into new areas daily. This reality won't change regardless of how weirdly, desperately the "stochastic parrot" people wish it were otherwise. At this point they're just denialists pushing goalposts around. Don't let 'em get to you!
- jcgrillo 3y ago
  Do you have a specific reference? I've mostly ignored LLMs until now because it seemed like the violent failure mode (confident + competent + wrong) renders them incapable of being a useful tool[1]. However this application, combined with the dimensionality idea, has me interested. I do wish the authors of the work referenced here made it more clear what, if anything, the LLM is doing here. It's not clear to me it confers some advantage over a more normal genetic programming approach to these particular problems. [1] in the sense that useful, safe tools degrade predictably. An airplane which stalls violently and in an unrecoverable manner doesn't get mass-produced. A circular saw which disintegrates when the blade binds throwing shrapnel into its operator's body doesn't pass QA. Etc.
  - a_wild_dandan 3y ago
    "Learning in High Dimension Always Amounts to Extrapolation" [1] [1] https://arxiv.org/abs/2110.09485 https://arxiv.org/abs/2110.09485
    - jcgrillo 3y ago

bendergarcia 3y ago

It’s kind of like humans: seed of two people and a random interest to pursue, what could they do?!? It makes poverty and children dying unnecessarily even more depressing.

riku_iki 3y ago

> An LLM can discover a new solution in high dimensional geometry that hasn’t advanced in 20 years!? LLM + brute force coded by humans.

dougmwne 3y ago
And that not only created a human interpretable method, but beat out 20 years of mathematicians working on a high profile open question. Who’s to say our brains themselves aren’t brute force solution searchers?
- riku_iki 3y ago
  > Who’s to say our brains themselves aren’t brute force solution searchers? yes, its just much slower (many trillions times) for numbers crunching.

nybsjytm 3y ago

> An LLM can discover a new solution in high dimensional geometry that hasn’t advanced in 20 years!? That goes way beyond glueing little bits of plagiarized training data together in a plausible way. The first advance in 20 years was by Fred Tyrrell last year https://arxiv.org/abs/2209.10045 https://arxiv.org/abs/2209.10045, who showed that the combinatorics quantity in question is between 2.218 and 2.756, improving the previous lower bound of 2.2174. DeepMind has now shown that the number is between 2.2202 and 2.756. That's why the DeepMind authors describe it as "the largest increase in the size of cap sets in the past 20 years," not as the only increase. The arithmetic is that 2.2202-2.218 is larger than 2.218-2.2174. (If you consider this a very meaningful comparison to make, you might work for DeepMind.)

a_wild_dandan 3y ago

> That goes way beyond glueing little bits of plagiarized training data together Moreover, LLMs almost ALWAYS extrapolate, and never interpolate. They don't regurgitate training data. Doing so is virtually impossible. An LLM's input (AND feature) space is enormous. Hundreds or thousands of dimensions. 3D space isn't like 50D or 5,000D space. The space is so combinatorially vast that basically no two points are neighbors. You cannot take your input and "pick something nearby" to a past training example. There IS NO nearby. No convex hull to walk around in. This "curse of dimensionality" wrecks arguments that these models only produce "in distribution" responses. They overwhelmingly can't! (Check out the literature of LeCun et al. for more rigor re. LLM extrapolation.) LLMs are creative. They work. They push into new areas daily. This reality won't change regardless of how weirdly, desperately the "stochastic parrot" people wish it were otherwise. At this point they're just denialists pushing goalposts around. Don't let 'em get to you!

jcgrillo 3y ago
Do you have a specific reference? I've mostly ignored LLMs until now because it seemed like the violent failure mode (confident + competent + wrong) renders them incapable of being a useful tool[1]. However this application, combined with the dimensionality idea, has me interested. I do wish the authors of the work referenced here made it more clear what, if anything, the LLM is doing here. It's not clear to me it confers some advantage over a more normal genetic programming approach to these particular problems. [1] in the sense that useful, safe tools degrade predictably. An airplane which stalls violently and in an unrecoverable manner doesn't get mass-produced. A circular saw which disintegrates when the blade binds throwing shrapnel into its operator's body doesn't pass QA. Etc.
- a_wild_dandan 3y ago
  "Learning in High Dimension Always Amounts to Extrapolation" [1] [1] https://arxiv.org/abs/2110.09485 https://arxiv.org/abs/2110.09485
  - jcgrillo 3y ago