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The root-case of skepticism has always been that while Transformers do exceptionally well on finite-sized tasks, they lack any fully recursive understanding of

The root-case of skepticism has always been that while Transformers do exceptionally well on finite-sized tasks, they lack any fully recursive understanding of the concepts.[0] A human can learn basic arithmetic, then generalize those principles to bigger number arithmetic, then go from there to algebra, then calculus, then so. Successively building on previously learned concepts in a fully recursive manner. Transformers are limited by the exponential size of their network. So GPT-3 does very well with 2-digit addition and okay with 2-digit multiplication, but can't abstract to 6-digit arithmetic. DALL-E is an incredible achievement, but doesn't really do anything to change this fact. GPT-3 can have an excellent understanding of a finite sized concept space, yet it's still architecturally limited at building recursive abstractions. So maybe it can understand "green block on a red block". But try to give it something like "a 32x16 checkerboard of green and red blocks surrounded by a gold border frame studded with blue triangles". I guarantee the architecture can't get that exactly correct. The point is that, in some sense, GPT-3 is a technical dead-end. We've had to exponentially scale up the size of the network (12B parameters) to make the same complexity gains that humans make with linear training. The fact that we've managed to push it this far is an incredible technical achievement, but it's pretty clear that we're still missing something fundamental. [0] https://arxiv.org/pdf/1906.06755.pdf https://arxiv.org/pdf/1906.06755.pdf

codetrotter 6y ago
> So GPT-3 does very well with 2-digit addition and okay with 2-digit multiplication, but can't abstract to 6-digit arithmetic. That sounds disappointing but what if instead of trying to teach it to do addition one would teach it to write source code for making addition and other maths operations instead? Then you can ask it to solve a problem but instead of it giving you the answer it would give you source code for finding the answer. So for example you ask it “what is the square root of five?” then it responds: fn main () { println!("{}", 5f64.sqrt()); }
Veedrac 6y ago
> So GPT-3 does very well with 2-digit addition and okay with 2-digit multiplication, but can't abstract to 6-digit arithmetic. This is false, GPT-3 can do 10-digit addition with ~60% accuracy, with comma separators. Without BPEs it would doubtlessly manage much better.
- dcolkitt 6y ago
  The accuracy largely comes from the fact that addition rarely requires carrying more than a single digit. So it's easy to pattern match from single digit problems that it was previously trained on. With multiplication, which requires much more extensive cross-column interaction, accuracy falls off a cliff with anything more than a few digits.
  - Veedrac 6y ago
    > The accuracy largely comes from the fact that addition rarely requires carrying more than a single digit. So it's easy to pattern match from single digit problems that it was previously trained on. Again, not at all true due to BPEs. [12] [65] [25] [42] [185] + [580] [22] [75] [80] [93] --------------------------- [706] [75] [300] [278] (note that GPT-3 is never told that the BPE [580] is composed of the digits [5], [8], and [0]. It has to guess this from the contexts [580] occurs in.) > With multiplication, which requires much more extensive cross-column interaction, accuracy falls off a cliff with anything more than a few digits. You couldn't learn long multiplication if you had to use BPEs, were never told how BPEs worked or corresponded to sane encodings, were basically never shown how to do multiplication, and were forced to do it without any working out. Quick, what's 542983 * 39486? No writing anything down, you have to output the numbers in order, and a single wrong digit is a fail. (That's easy mode, I won't even bother asking you to do BPEs.) ML models can learn multiplication, obviously they can learn multiplication, they just can't do it in this absurd adversarial context. GPT-f[1] was doing Metamath proofs on 9-digit division (again, a vastly harder context, they involve ~10k proof steps) with 50-50 accuracy, and we have a toy proof of concept[2] for straight multiplication. [1] https://arxiv.org/abs/2009.03393 https://arxiv.org/abs/2009.03393 [2] https://github.com/Thopliterce/transformer-arithmetic https://github.com/Thopliterce/transformer-arithmetic
    - Veedrac 6y ago
      Correction/clarification: 10k proof steps was the number of steps in the training dataset, which was for 100 such proofs.
    - 6y ago

codetrotter 6y ago

> So GPT-3 does very well with 2-digit addition and okay with 2-digit multiplication, but can't abstract to 6-digit arithmetic. That sounds disappointing but what if instead of trying to teach it to do addition one would teach it to write source code for making addition and other maths operations instead? Then you can ask it to solve a problem but instead of it giving you the answer it would give you source code for finding the answer. So for example you ask it “what is the square root of five?” then it responds: fn main () { println!("{}", 5f64.sqrt()); }

Veedrac 6y ago

> So GPT-3 does very well with 2-digit addition and okay with 2-digit multiplication, but can't abstract to 6-digit arithmetic. This is false, GPT-3 can do 10-digit addition with ~60% accuracy, with comma separators. Without BPEs it would doubtlessly manage much better.

dcolkitt 6y ago
The accuracy largely comes from the fact that addition rarely requires carrying more than a single digit. So it's easy to pattern match from single digit problems that it was previously trained on. With multiplication, which requires much more extensive cross-column interaction, accuracy falls off a cliff with anything more than a few digits.
- Veedrac 6y ago
  > The accuracy largely comes from the fact that addition rarely requires carrying more than a single digit. So it's easy to pattern match from single digit problems that it was previously trained on. Again, not at all true due to BPEs. [12] [65] [25] [42] [185] + [580] [22] [75] [80] [93] --------------------------- [706] [75] [300] [278] (note that GPT-3 is never told that the BPE [580] is composed of the digits [5], [8], and [0]. It has to guess this from the contexts [580] occurs in.) > With multiplication, which requires much more extensive cross-column interaction, accuracy falls off a cliff with anything more than a few digits. You couldn't learn long multiplication if you had to use BPEs, were never told how BPEs worked or corresponded to sane encodings, were basically never shown how to do multiplication, and were forced to do it without any working out. Quick, what's 542983 * 39486? No writing anything down, you have to output the numbers in order, and a single wrong digit is a fail. (That's easy mode, I won't even bother asking you to do BPEs.) ML models can learn multiplication, obviously they can learn multiplication, they just can't do it in this absurd adversarial context. GPT-f[1] was doing Metamath proofs on 9-digit division (again, a vastly harder context, they involve ~10k proof steps) with 50-50 accuracy, and we have a toy proof of concept[2] for straight multiplication. [1] https://arxiv.org/abs/2009.03393 https://arxiv.org/abs/2009.03393 [2] https://github.com/Thopliterce/transformer-arithmetic https://github.com/Thopliterce/transformer-arithmetic
  - Veedrac 6y ago
    Correction/clarification: 10k proof steps was the number of steps in the training dataset, which was for 100 such proofs.
  - 6y ago