Yet another Hacker News

A look at OpenAI’s math that’s claimed computes faster Fourier transforms

57 points mochomocha 2 hours ago 36 comments

p11p 1 hour ago | parent

This is a bad take. I will happily acknowledge the new term "slopository" which I had not heard before, while taking the author to task for malice towards theorists. STOC/FOCS/SODA, etc are not about "doing practical algorithms". If you want that, you go to ESA, ALENEX, or VLDB/SIGMOD, or any of the other more applied venues featuring this work. I don't think it's fair to just shit on theory because they don't do experiments---they're mathematicians, and it is fine to have absurd constants in your big-O as long as they are constants.

PS. I can have this view while simultaneously thinking that Ben is a badass.

skybrian 1 hour ago | parent

If algorithmic theorists don't care about practical performance then why should we care about what they're doing? It's just a strange game they're playing, of no interest to anyone else.

It seems like rather than coming up with more "galactic" algorithms, a more useful thing to do might be to come up with a replacement for big-O that's not so exploitable, and to convince theoreticians to play that game instead.

frotaur 57 minutes ago | parent

I think you are free not to care about what they are doing. It turns out that big O notation has some useful properties, and in particular it is a useful definition as it allows you to define complexity classes which are a very useful tool for analyzing algorithms in general.

Much of our 'practical' understanding of algorithms also in part came from theoretical advances.

adastra22 42 minutes ago | parent

That’s sloppy reasoning. There might be notations that allow for classifying complexity classes that don’t suffer this problem.

ToValueFunfetti 33 minutes ago | parent

That may well be the case, in theory.

eru 51 minutes ago | parent

> If algorithmic theorists don't care about practical performance then why should we care about what they're doing? It's just a strange game they're playing, of no interest to anyone else.

You don't need to care. But every so often breakthroughs in theory can be turned into breakthroughs in practice.

> It seems like rather than coming up with more "galactic" algorithms, a more useful thing to do might be to come up with a replacement for big-O that's not so exploitable, and to convince theoreticians to play that game instead.

There's plenty of replacements.

Btw, we already have the optimal (in big-O terms) algorithm for every NP hard problem. We just don't know whether it runs in polynomial time or takes longer. And, yes, that algorithm called Levin's Universal Search is a galactic algorithm. Simplifying a lot, Levin's Universal Search does some clever interleaving to essentially try all possible algorithms in sort-of parallel.

Lo and behold, even the theorists you complain about don't see that 'loophole' as a solution, declare victory and go home. They still keep researching.

skybrian 19 minutes ago | parent

If Big-O lets you run every possible algorithm in "parallel" while still being in some sense "optimal," that seems like yet another exploit?

cpa 48 minutes ago | parent

You care because Pierre de Fermat 16th century work on modular arithmetic was of “no intest to anyone else” except other theorists but it got to shine in the twentienth century in modern cryptography.

Doing theory is what gives you a chance to push the envelope to places you can’t even imagine when you’re knee deep in real world problems. And proving something is possible (if impractical) gives an incentive to find a better algorithm and may attract others.

For example see the first graph here: http://klotza.blogspot.com/2016/06/the-fall-of-twin-prime-ga...

adastra22 43 minutes ago | parent

No, I don’t care. Until there is a real, practical result, I don’t care. There are plenty of examples of math discovered in service of solving a problem. There is nothing that says math has to be purely abstract first.

asdff 38 minutes ago | parent

You can't imagine how that severely limits the potential size of the discovery space shackling it to present day problems?

cpa 26 minutes ago | parent

Because clearly we’d have invented GPS if we had asked ourselves “how do I create a global positioning system with centimeter accuracy”?

ViscountPenguin 41 minutes ago | parent

Considering this line:

" In 1973, Jacques Morgenstern proved that you couldn’t get a Fourier transform faster than ½ n log n if you required that complex coefficients used in the computation had bounded magnitude"

I highly doubt this is going to somehow inspire people to make faster algorithms than the mathematical bound...

I have no problem with pure mathematics, my training was in it; but it seems very possible to me that most people funding theoretical computer science (as opposed to pure maths) probably assume the work has a more realistic computational model backing it up.

tomrod 15 minutes ago | parent

Does the apply to representation of complex numbers, which is not uncountably infinite and represented by floats?

modeless 38 minutes ago | parent

"Malicious" is an overstatement, but "exploit" is completely fair in my view. Theory with no possible practical relevance is useless, and I think a computational complexity result requiring arbitrary-precision arithmetic that is impossible to implement using any physical system qualifies.

nullc 36 minutes ago | parent

Plenty of mathematics was practically useless for long periods of time, waiting for us to invent computers, cryptography, error correcting codes, etc.

It's extremely hard to know what mathematics may be useful in the future or not.

modeless 33 minutes ago | parent

It's hard to know for certain, but that doesn't mean we should throw up our hands and say anything goes. We can still say that some things are more likely to be useful than others.

winwang 23 minutes ago | parent

The idea that FFT's decomposition is non-optimal is surprising. What the new bound means is we basically were looking at the problem and its structure incorrectly, and therefore implicates other problems for which we have used recursion to get to our currently-best bounds. Almost no interesting math problems are just sitting there alone in their little corner. Getting a hint like that is a big deal.

scuppernong 1 hour ago | parent

Did not realize that the model of computation for all of the banner computational complexity results from last week was "exact complex arithmetic." Is this and/or "exact real arithmetic" a common premise for papers in this field? My assumption as an outsider was that people mostly care about things equivalent to Turing machines.

edit: by "things equivalent to" I really mean universal register machines, something that operates on integers and not directly on real numbers.

zeroonetwothree 56 minutes ago | parent

No, it’s unusual to use Turing machines as the model.

eru 55 minutes ago | parent

Almost no one uses Turing machines for complexity theory.

At most you use them for stuff like P vs NP, because Turing machines are 'only' polynomially slower than register machines, so it doesn't matter.

But when you are talking about linear vs quadratic on something resembling real computers, Turing machines are the wrong model.

LPisGood 48 minutes ago | parent

Turing machines are good for computability but not great for complexity.

agrm 42 minutes ago | parent

This kind of this is pretty common (see, for example https://en.wikipedia.org/wiki/Blum%E2%80%93Shub%E2%80%93Smal...).

There is, of course, the very rich field of Numerical Analysis which deals with how we actually represent these things in real life, but for a starting point assuming this sort of thing is par for the course.

jiggawatts 1 hour ago | parent

> the algorithm requires you to multiply complex numbers whose magnitudes grow superpolynomially with n.

Err... doesn't that completely invalidate the big-O analysis!?

I suspected that these Open AI "proofs" of sub-nlogn times were cheating somehow, by sweeping something under a rug. This seems to be it!

LPisGood 47 minutes ago | parent

You can be linear in the bit length of the input number, but exponential in the magnitude.

This is a common thing in complexity, and I think even the abstract says it is talking about “length n” and not “magnitude n”

LegionMammal978 28 minutes ago | parent

I can't speak for the FFT claim, but least the sub-O(n log n) multiplication claim is for the multitape Turing machine model with a fixed number of tapes, which if anything is more restrictive than actual random-access memory. Of course, it's still galactic just like the O(n log n) algorithm, but it definitely seems like there's something there.

itissid 52 minutes ago | parent

There is currently an effort underway to improve the bound. They have reached at a much larger kappa around 0.001635527380375392

https://github.com/CrocSwap/integer-mult-bounds

loufe 40 minutes ago | parent

I am astounded by the activity in this repo... what is going on? 508 PRs on 408 commits..? It's getting additions at a frenetic pace by a large number of contributors.

AnotherGoodName 21 minutes ago | parent

It’s unintentionally making a mockery of OpenAIs results by highlighting Blums speedup theorem in action. Namely that you can always reduce time complexity with a small move into space complexity.

That unintentional mockery of a repo is 262million lines of such moves and growing. I don’t think for a second anyone involved has a clue about complexity theory though or they’d just jump to the obvious “O(1) time complexity using an O(2^n) lookup table to cover all possible n bits of input” which is the final end state of Blums speedup theorem.

dyauspitr 45 minutes ago | parent

I’ve made some half hearted attempts to try and understand even the framing of some of these problems that were solved, and my god theoretical math is hard. It’s completely impenetrable with unknown symbols and hard to comprehend bounds and several orders of derivatives. It’s all unintuitive to me.

agnishom 43 minutes ago | parent

I am as upset as the slew of math slop from OpenAI as the next guy, but I do not understand this article. Are they saying that this was not a true improvement because the algorithm is _galactic_? Human algorithm researchers have been publishing those sorts of results for decades now

adastra22 41 minutes ago | parent

A lot of people are emphatically NOT upset about the OpenAI math dump.

Glyptodon 42 minutes ago | parent

To be honest this reminds me of interview leetcode. Like I remember one where you could get constant time if you made nested hashtables covering character sequences that might have made sense in USCII but probably would be impossible to maintain in UTF. In general leetcode problems love goading you to treat the ASCII character space as constant time to deal with and it's stupid. And it seems very similar to this.

Regex777 42 minutes ago | parent

I Fairly know about FFT, its an algorithm of DFT.

As the argmin suggests Writing computational complexity in Big O isn't great here.

since we generally see the interms of no.of complex additions and no.of Multiplications in the FFT algorithm.

traes 13 minutes ago | parent

An unpleasant part of the AI math revolution is that people's hatred of AI is bringing to bear their hatred of pure mathematics in general. I've always known in the back of my mind that mathematicians were secretly despised and looked down upon by many, but it's a bad feeling to see it become so mainstream.

omoikane 8 minutes ago | parent

> This new OpenAI result is thus less a mathematical breakthrough and more an oddity of computer scientists being bad at formalizing their design specifications.

Related, another article about proofs that may depend on bad formalizations:

https://news.ycombinator.com/item?id=49994145 - Navier-Stokes Lost in Translation (5 days ago, 237 comments)

AI seems to be very good at getting answers, but not always answers to the correct questions. These remind me of Deep Thought from The Hitchhiker's Guide to the Galaxy, where they got an answer to the ultimate question before actually knowing what the ultimate question is.