Short Remark on the (subjective) mathematical 'naturalness' of the Nanda--Lieberum addition modulo 113 algorithm

Thanks for the comment Rohin, that's interesting (though I haven't looked at the paper you linked).

I'll just record some confusion I had after reading your comment that stopped me replying initially: I was confused by the distinction between modular and non-modular because I kept thinking: If I add a bunch of numbers $x$ and $y$ and don't do any modding, then it is equivalent to doing modular addition modulo some large number (i.e. at least as large as the largest sum you get). And otoh if I tell you I'm doing 'addition modulo 113', but I only ever use inputs that add up to 112 or less, then you never see the fact that I was secretly intending to do modular addition. And these thoughts sort of stopped me having anything more interesting to add.

 

Hi Garrett, 

OK so just being completely honest, I don't know if it's just me but I'm getting a slightly weird or snarky vibe from this comment? I guess I will assume there is a good faith underlying point being made to which I can reply. So just to be clear:

• I did not use any words such as "trivial", "obvious" or "simple". Stories like the one you recount are obviously making fun of mathematicians, some of whom do think its cool to say things are trivial/simple/obvious after they understand them. I often strongly disagree and generally dislike this behaviour and think there are many normal mathematicians who don't engage in this sort of thing.  In particular sometimes the most succinct insights are the hardest ones to come by (this isn't a reference to my post; just a general point).  And just because such insights are easily expressible once you have the right framing and the right abstractions, they should by no means be trivialized. 
• I deliberately emphasized the subjectivity of making the sorts of judgements that I am making. Again this kinda forms part of the joke of the story.
• I have indeed been aware of the work since when it was first posted 10 months ago or so and have given it some thought on and off for a while (in the first sentence of the post I was just saying that I didn't spend long writing the post, not that these thoughts were easily arrived-at).
• I do not claim to have explained the entire algorithm, only to shed some light on why it might actually be a more natural thing to do than some people seem to have appreciated.
• I think the original work is of a high quality and one might reasonably say 'groundbreaking'.
   

In another one of my posts [LW · GW] I discuss at more length the kind of thing you bring up in the last sentence of your comment, e.g.

it can feel like the role that serious mathematics has to play in interpretability is primarily reactive, i.e. consists mostly of activities like 'adding' rigour after the fact or building narrow models to explain specific already-observed phenomena. 

....[but]... one of the most lauded aspects of mathematics is a certain inevitability with which our abstractions take on a life of their own and reward us later with insight, generalization, and the provision of predictions. Moreover - remarkably - often those abstractions are found in relatively mysterious, intuitive ways: i.e. not as the result of us just directly asking "What kind of thing seems most useful for understanding this object and making predictions?" but, at least in part, as a result of aesthetic judgement and a sense of mathematical taste.

And e.g. I talk about how this sort of thing has been the case in areas like mathematical physics for a long time. Part of the point is that (in my opinion, at least) there isn't any neat shortcut to the kind of abstract thinking that lets you make the sort of predictions you are making reference to. It is very typical that you have to begin by reacting to existing empirical phenomena and using it as scaffolding. But I think, to me, it has come across as that you are being somewhat dismissive of this fact? As if, when B might well follow from A and someone actually starts to do A, you say "I would be far more impressed if B" instead of "maybe that's progress towards B"?

(Also FWIW,  Neel claims here that regarding the algorithm itself, another researcher he knows "roughly predicted this".)

My submission: when we teach modular arithmetic to people, we do it using the metaphor of clock arithmetic. Well, if you ignore the multiple frequencies and argmax weirdness, clock arithmetic is exactly what this network is doing! Find the coordinates of rotating the hour hand (on a 113-hour clock) x hours, then y hours, use trig identities to work out what it would be if you rotated x+y hours, then count how many steps back you have to rotate to get to 0 to tell where you ended up. In fairness, the final step is a little bit different than the usual imagined rule of "look at the hour mark where the hand ends up", but not so different that clock arithmetic counts as a bad prediction IMO.

Here's another way of looking at it which could be said to make it more trivial:

We can transform addition into multiplication by taking the exponential, i.e. x+y=z is equivalent to 10^x * 10^y = 10^z.

But if we unfold the digits into separate axes rather than as a single number, then 10^n is just a one-hot encoding of the integer n.

Taking the Fourier transform of the digits to do convolutions is a well-known fast multiplication algorithm.