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Evidence of exponential speed-up in the solution of hard optimization problems?

arxiv.org

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Re: Evidence of exponential speed-up in the solution of hard optimization problems?

#11

Post by ColinWright »

Quoting from the abstract:

  > ... a non-combinatorial approach
  > to hard optimization problems that
  > ... finds better approximations than
  > the current state-of-the-art.
So it appears that they're finding approximate solutions, so already it's rather less significant than the title suggests.

Then:

  > We show empirical evidence that
  > our solver scales linearly with
  > the size of the problem, ...
We know that for most NP-Complete problems, most instances are easy. The question then is whether they are testing their algorithms on instances that are known to be hard. There's a chance they're doing something like finding factors of random integers, which we know is easy in almost every instance.

I'm deeply pessimistic.

Re: Evidence of exponential speed-up in the solution of hard optimization problems?

#12

Post by Animats »

marmaduke wrote: post #4
Animats wrote: post #2

This is really important if not bogus. Any comments from people in the field? It's like back-propagation for digital logic.

Did you look at the paper or just throwing the analogy out there? It seems the described problems are integer programming which wouldn’t have gradients as in back propagation. The magic seems to lie in the so called self organizing logic gates detailed https://arxiv.org/abs/1512.05064

Right. See the PDF of the paper at [1]. They're mapping Boolean logic into a kind of analog system.

"SOLGs can use any terminal simultaneously as input or output, i.e., signals can go in and out at the same time at any terminal resulting in a superposition of input and output signals ... The gate changes dynamically the outgoing components of the signals depending on the incoming components according to some rules aimed at satisfying the logic relations of the gate. ... A SOLG ... can have either stable configurations ... or unstable ... configurations. In the former case, the configuration of the signals at the terminals satisfies the logic relation required ... and the signals would then remain constant in time. Conversely, if the signals at a given time do not satisfy the logic relation, we have the unstable configuration: the SOLG drives the outgoing components of the signal to finally obtain a stable configuration."

This is sort of like supervised training of a neural net. I think. Test cases with both inputs and desired outputs are needed, and applying them to the network pushes the parameters towards values that yield the desired outputs. It's kind of like deliberately over-training a neural net to encode some explicit function.

The paper is vague about how you train this thing. It seems like it has to be driven by test cases, but they don't say much about that. It's not clear that this scales. It ought to work for small numbers of gates, but for large numbers, does this process converge?

[1] https://arxiv.org/pdf/1512.05064.pdf

Re: Evidence of exponential speed-up in the solution of hard optimization problems?

#13

Post by urgoroger »

ColinWright wrote: post #11

Quoting from the abstract: > ... a non-combinatorial approach > to hard optimization problems that > ... finds better approximations than > the current state-of-the-art. So it appears that they're finding approximate solutions, so already it's rather less significant than the title suggests. Then: > We show empirical evidence that > our solver scales linearly with > the size of the problem, ... We know that for most…

While I agree with you, you should note that even approximation within any degree of error is NP-complete for a large class of NP-complete problems (e.g. TSP, and the problem MAX-EkSAT used in the paper).

That is, a polynomial algorithm for the approximate problem would be just as significant as one for the exact version.

Re: Evidence of exponential speed-up in the solution of hard optimization problems?

#14

Post by petters »

zmonx wrote: post #7

The authors have built a start-up based on these ideas: http://memcpu.com/ They provide their SAT solver as a service that you can try. A related paper I recommend in this context is NP-complete Problems and Physical Reality by Scott Aaronson: https://www.scottaaronson.com/papers/npcomplete.pdf

Do I have to register for the SAT solver?

Their web page says: "We are currently in Alpha Test. Users can submit problems via our SaaS portal in Conjunctive Normal Form (CNF), which ultimately represents the normal form of Boolean propositions for the problem’s variables and constraints"

I have no problem providing real, hard SAT instances in CNF. It should be really easy to verify whether they have something good, unless they have a size restriction.

Re: Evidence of exponential speed-up in the solution of hard optimization problems?

#15

Post by ColinWright »

urgoroger wrote: post #13
ColinWright wrote: post #11

Quoting from the abstract: > ... a non-combinatorial approach > to hard optimization problems that > ... finds better approximations than > the current state-of-the-art. So it appears that they're finding approximate solutions, so already it's rather less significant than the title suggests. Then: > We show empirical evidence that > our solver scales linearly with > the size of the problem, ... We know that for most…

While I agree with you, you should note that even approximation within any degree of error is NP-complete for a large class of NP-complete problems (e.g. TSP, and the problem MAX-EkSAT used in the paper). That is, a polynomial algorithm for the approximate problem would be just as significant as one for the exact version.

But that's being NP-Complete, and so can be converted into exact solutions for other NP-Complete problems. Otherwise, by definition, it's not NP-Complete.

So it's not clear what they're actually doing, but if they can solve NPC problems they would say that. So I expect that they are getting approximate solutions that are not then NPC.

Re: Evidence of exponential speed-up in the solution of hard optimization problems?

#16

Post by ramgorur »

zitterbewegung wrote: post #5
Animats wrote: post #2

This is really important if not bogus. Any comments from people in the field? It's like back-propagation for digital logic.

Looking at the authors papers it doesn't look worth reading.

What do you mean? How does that help you to decide if a paper is worth reading or not?

Re: Evidence of exponential speed-up in the solution of hard optimization problems?

#17

Post by urgoroger »

ColinWright wrote: post #15
urgoroger wrote: post #13

Earlier quoted context omitted.

While I agree with you, you should note that even approximation within any degree of error is NP-complete for a large class of NP-complete problems (e.g. TSP, and the problem MAX-EkSAT used in the paper). That is, a polynomial algorithm for the approximate problem would be just as significant as one for the exact version.

But that's being NP-Complete, and so can be converted into exact solutions for other NP-Complete problems. Otherwise, by definition, it's not NP-Complete. So it's not clear what they're actually doing, but if they can solve NPC problems they would say that. So I expect that they are getting approximate solutions that are not then NPC.

That's correct. Given a polynomial time approximate algorithm for Ek-SAT (note, by approximate algorithm I mean along the lines of the formal definition of approximate algorithm, that the solution given by the algorithm falls in some fraction of the real answer for all instances, see https://en.wikipedia.org/wiki/Hardness_of_approximation), you would show P=NP.

My first concerns, namely the usage of analog methods to solve NP-complete problems, lies along the same lines as this post: https://www.scottaaronson.com/blog/?p=2212

Moreover, from what I can see, and as you mention as your original concern, the extent of proof for the 'exponential-speedup' claim exists in the form of some benchmarks, hardly a proof that their method actually works on all instances, which would be needed to show that it is a approximation algorithm as commonly defined.

The purpose of my post was to highlight that for some problems (for example, TSP), even a really crappy approximation algorithm would imply P=NP. As highlighted by the authors of this paper, MAX-EkSAT is in a similar situation, though judging by https://www.cse.buffalo.edu/~hungngo/classes/2006/594/notes/..., unlike TSP, approximation to SOME ratio is possible, though there exists aratio >1 which cannot be beat unless P=NP.

I was simply trying to address the statement: "So it appears that they're finding approximate solutions, so already it's rather less significant than the title suggests. ", since an actual approximation algorithm for some ratio really WOULD be significant (showing P=NP).

Re: Evidence of exponential speed-up in the solution of hard optimization problems?

#18

Post by bitL »

Is this similar to what Berkeley MPC Lab is doing with quadratic programming using electricity? I.e. that nature has means to solve optimization problems in an instant and our approximations of this process in the form of differential equations allow us to do something similar, albeit not 100% accurately?

Re: Evidence of exponential speed-up in the solution of hard optimization problems?

#19

Post by spuz »

ColinWright wrote: post #11

Quoting from the abstract: > ... a non-combinatorial approach > to hard optimization problems that > ... finds better approximations than > the current state-of-the-art. So it appears that they're finding approximate solutions, so already it's rather less significant than the title suggests. Then: > We show empirical evidence that > our solver scales linearly with > the size of the problem, ... We know that for most…

> So it appears that they're finding approximate solutions, so already it's rather less significant than the title suggests

The title doesn't suggest anything about exact solutions as far as I can tell.

> We know that for most NP-Complete problems, most instances are easy. The question then is whether they are testing their algorithms on instances that are known to be hard. There's a chance they're doing something like finding factors of random integers, which we know is easy in almost every instance.

I don't know how they choose the problems they choose to solve but I assume this information is in the paper? If you are skeptical then why not read it and report your findings? Simply expressing your skepticism doesn't seem valuable.

Re: Evidence of exponential speed-up in the solution of hard optimization problems?

#20

Post by spuz »

colordrops wrote: post #9

This keeps popping up every few months for several years, and seems like BS. Here's an example: https://news.ycombinator.com/item?id=8652475 And here's Scott Aaronson's debunking: https://www.scottaaronson.com/blog/?p=2212 If these guys could really solve NP complete problems, they should have some amazing concrete results to show at this point, which they don't.

You link to research from 3 years ago. Does the latest paper (Oct 2017) not represent the concrete results you claim they don't have?
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