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Unum Computing: An Energy Efficient and Massively Parallel Approach to Numerics

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Re: Unum Computing: An Energy Efficient and Massively Parallel Approach to Numerics

#21

Post by methodOverdrive »

simonbyrne wrote: post #14

Are there any more details on unums available online (i.e. without purchasing his book)? The presentation is quite heavy on promotion, but a bit sparse on details. I also note that he dedicates a page on his website to "Gustafson's law". http://johngustafson.net/glaw.html

Not many. I did go ahead and purchase his book. It's... kind of a weird read, honestly. Useful - it clarified some things about the proposed Unum format - but also written very casually, like "pop science" prose, which seems inappropriate - who would buy a book about a floating-point number format if they didn't want a dry, boring book full of technical details? Some of the arguments are made as though to convince a non-technical audience - maybe Gustafson wants managers (or former engineers who haven't worked as engineers for a long time) to read his book, but I think it would have been better to publish all of the details without any fluff, first.

Also worth noting that half of the book is about the "ubox" method for solving optimization problems - also cool, but may be overkill if you are just interested in the numeric format itself. Personally, I've been working on an implementation of the format that I can toy around with - I have no real interest in learning a lot about the cool algorithms I could do with it until I can show myself that it works for basic arithmetic, etc, as well as the author claims.

Gustafson also makes the code available (I think [here](https://www.crcpress.com/The-End-of-Error-Unum-Computing/Gus...). It's Mathematica code... there is a free viewer for that format (if you don't have Mathematica) which can print out a PDF with richly formatted equations.

Also, Googling for that link led me to this [Python implementation someone whipped up](https://github.com/jrmuizel/pyunum).

Re: Unum Computing: An Energy Efficient and Massively Parallel Approach to Numerics

#22

Post by methodOverdrive »

yoklov wrote: post #13
cossatot wrote: post #9

Are there architectural or other challenges to building processors for this? How compatible are unums with modern processors? Would LAPACK et al have to be re-written?

My understanding is that it's not compatible with modern processors, and new hardware would be required to efficiently execute it. It seems to be a superset of floating point (and fixed point?) though, so maybe LAPACK could still work? I may be misreading or misunderstanding this however.

New hardware would be needed. I think libraries like LAPACK basically are fast because they do a good job taking advantage of the hardware implementation of floating point math, and are written with sufficient understanding of the format to mitigate approximation errors. Therefore you would probably need to rewrite huge parts of such libraries to use the new Unum hardware or software - though doing so might be relatively straightforward - as you say, Unum can be seen as a superset of floating point and it does support all of the same operations.

Re: Unum Computing: An Energy Efficient and Massively Parallel Approach to Numerics

#23

Post by dnautics »

Lawtonfogle wrote: post #8
dnautics wrote: post #3

Earlier quoted context omitted.

All real numbers are representable, but not necessarily to arbitrary precision. You should really read the book. A lot of the atomic operations seem cumbersome, because the unum doesn't have a fixed size representation, but you just get over that when you remember that the real problem is shuttling data over buses. More compute transistors is not a problem.

If I had a number system of -100, 0, and 100, could I claim all real numbers are represented, just not to arbitrary precision? With n bits, there are 2^n possible representations max. If you can say all real numbers are represented for n around 30, or 60, or even 1,000,000, then can't you say the same for n equal to 1 or 2? Now I think we can say some representations are better than others (at least for certain appli…

Sure, you can claim that but it doesn't mean squat unless your underlying circuits correctly process operations (*,+,/etc) on the representation in a way that is faithful to your claim.

In the book he talks about a simple unum with values -inf, less than -2, -2, between -2 ... -1, -1, between -1 ... 0, 0, 0...1, 1, 1...2, 2, > 2, and inf, also intervals like 0...2, -2...0 etc 1...inf, emerge when you intentionally use less precision.

This is a mathematically closed system that represents all real numbers and is even useful!

Re: Unum Computing: An Energy Efficient and Massively Parallel Approach to Numerics

#24

Post by dnautics »

stephencanon wrote: post #10
yoklov wrote: post #7

Earlier quoted context omitted.

I got the impression that the unum explicitly modeled a range of values, so I'm not sure that applies. Not an expert here though.

That's fine, but then you're representing dyadic intervals, not real numbers. Simple question: what's unum[sqrt(2)]?

Rougly speaking It would be 1.414... + plus a bit that indicates the result is between two adjacent exact values. This bit would not be set if the result were exact.

It's not the same as dyadic interval math, it's kind of a hybrid.

Re: Unum Computing: An Energy Efficient and Massively Parallel Approach to Numerics

#25

Post by methodOverdrive »

Lawtonfogle wrote: post #17
yoklov wrote: post #7

Earlier quoted context omitted.

I got the impression that the unum explicitly modeled a range of values, so I'm not sure that applies. Not an expert here though.

Regardless if it is representing number, ranges, or a mix of both, there are still 2^n maximum possible representations for n bits. Perhaps you could create a mix of numbers and ranges equal to (2^n)-2 and then add one range that is from the lowest number represented to negative infinity and add another that does the same for positives. But I can do that system with -, 0, +, NAN. See, I just represented all numbers.…

I would argue that unums are a "superior replacement" for doubles in many cases, though: in the case that you support unums that are "wide" enough, you can represent doubles exactly, plus you have additional values, plus some nice rules about when approximation error occurs/is propagated and not as many bits need to be stored or moved around on buses. It'll be a while before there's an implementation anywhere near as fast as existing FPUs, but Gustafson makes a good argument for his format. Personally, I'm more interested in the correctness benefits than the space/power/time savings - even if unums are never faster than 64-bit floats, they present an interesting way to do real-number arithmetic and my brief exposure to them leaves me much more confident that I could write numerical algorithms correctly than with doubles - I do numerical/statistical algorithms with doubles in my work, and it's really a pain to reason about things that the "uncertain" (open-inverval) values of the unum format would greatly simplify.

They're also an inferior replacement in the case that you want to take advantage of highly-optimized hardware, and that getting a correct answer doesn't really matter. I don't see unums replacing floats for, say, video game graphics. But for numerical computation, it seems like the only real flaw with unums compared to doubles is the nonexistence of a hardware implementation, and the existing popularity of doubles.

Re: Unum Computing: An Energy Efficient and Massively Parallel Approach to Numerics

#26

Post by dnautics »

yoklov wrote: post #19

The big issue I see with these is that the number of bits is dependent on the stored value. It's also not a power of two. This has a lot of problematic consequences. Indexing into a list of them wouldn't be constant time, for example. You'd need to unpack them (into fixed size unums?) first. That said, information here is sparse and I'm not an expert on numerical computing (although I do graphics at work and know som…

For any unum system there's an 'archtecture spec' that puts parameters on the maximum sizes of the exponent and mantissa. You can have values that consume less bits, the format contains a tag that says how many bits you are consuming for each part of your number, a lower size indicates for exact values that less size is needed; for uncertain values it means the uncertainty is higher.

You could pad your values to achieve constant time indexing.

Re: Unum Computing: An Energy Efficient and Massively Parallel Approach to Numerics

#27

Post by methodOverdrive »

stephencanon wrote: post #6
bmh100 wrote: post #4

Earlier quoted context omitted.

I am not sure that I understand what you're saying. Given a charitable reading of the presentation, the author seems to be saying that his standard explicitly specifies range of values, as opposed to the IEEE Float. It did not seem like he was saying that he could explicitly represent an infinite amount of exact, distinct values using a finite number of bits. Low level data structures are not my area of expertise, so…

Under that reading, he's saying nothing: just represent the interval [-inf, inf] as "0" and call it a day. So assuming that he's saying anything at all, he's at least being imprecise, and the actually claim should be something like "represent any dyadic interval using a finite number of bits".

He's a bit showy about the format. Wish he would just put out a technical paper.

Anyway, I guess his motivation might be "you can represent any real number (with finite bits, and therefore finite precision)". In the book, he presents an interesting case: little 4-bit versions of the Unum that can represent:

-inf, (-inf, -2), -2, (-2, -1), (-1, -1/2), -1/2, (-1/2, -0), -0, 0, (0, 1/2), 1/2, (1/2, 1), 1, (1, 2), 2, (2, inf), inf.

Putting together a pair of them, the book outlines simple interval arithmetic (where pairs of numbers can represent any interval between numbers on the line above, and single numbers can represent some of the closed intervals as above). The reason these are kind of neat is that using the standard Unum algorithms (without any fudging), you can get "correct" (albeit terribly imprecise) results for many real number computations. Questions like "is there a number satisfying a numerical predicate in some range?" or the value of a trigonmetric or exponential expression will come out "correct" (but you might get an answer like (-inf, inf)). If things work out as well as he claims (and demonstrates for some cases), then you can basically do the math to figure out how precise you want to be and choose an appropriate specialization of the format - or take advantage of the format's flexibility and do computations starting at a low precision and increasing precision until you are satisfied. In particular, it's kind of cool that you can do computations with little 8-bit intervals, and possibly circumvent doing more expensive computations (e.g. if you test if a property will hold anywhere in the Unum range and it won't hold anywhere, assuming you (and Gustafson) have done the math right, you can avoid doing more expensive checks with increased precision).

Anyway, point is, the presentations are kind of flashy and misleading - and you're right, you can't represent any real number (just finitely representable dyadic intervals)... but the format itself _does_ seem promising...

Re: Unum Computing: An Energy Efficient and Massively Parallel Approach to Numerics

#28

Post by dnautics »

methodOverdrive wrote: post #25
Lawtonfogle wrote: post #17

Earlier quoted context omitted.

Regardless if it is representing number, ranges, or a mix of both, there are still 2^n maximum possible representations for n bits. Perhaps you could create a mix of numbers and ranges equal to (2^n)-2 and then add one range that is from the lowest number represented to negative infinity and add another that does the same for positives. But I can do that system with -, 0, +, NAN. See, I just represented all numbers.…

I would argue that unums are a "superior replacement" for doubles in many cases, though: in the case that you support unums that are "wide" enough, you can represent doubles exactly, plus you have additional values, plus some nice rules about when approximation error occurs/is propagated and not as many bits need to be stored or moved around on buses. It'll be a while before there's an implementation anywhere near as…

Agreed. For neural networks. I would argue the opposite is true, you should just have a 16bit float that casts really large values to infinity silently without throwing errors, with a logistic lookup that maps "inf" to +/- 1... A mathematically incorrect float is operationally superior to the correct one.

Re: Unum Computing: An Energy Efficient and Massively Parallel Approach to Numerics

#29

Post by semi-extrinsic »

trsohmers wrote: post #18
cossatot wrote: post #9

Are there architectural or other challenges to building processors for this? How compatible are unums with modern processors? Would LAPACK et al have to be re-written?

It requires additional complexity than what is in modern FPUs, but it arguably more efficient when actually operating due to being able to have the same accuracy while using fewer bits. Most people don't realize that it is the data movement that is most expensive thing in a processor... It takes 100 picojoules to do a double precision (64 bit) floating point operation, but a humongous 4200 picojoules to actually move…

There's an even worse disparity in the amount of time required for those two processes. How is that ratio affected by your technology?

Re: Unum Computing: An Energy Efficient and Massively Parallel Approach to Numerics

#30

Post by trsohmers »

semi-extrinsic wrote: post #29
trsohmers wrote: post #18

Earlier quoted context omitted.

It requires additional complexity than what is in modern FPUs, but it arguably more efficient when actually operating due to being able to have the same accuracy while using fewer bits. Most people don't realize that it is the data movement that is most expensive thing in a processor... It takes 100 picojoules to do a double precision (64 bit) floating point operation, but a humongous 4200 picojoules to actually move…

There's an even worse disparity in the amount of time required for those two processes. How is that ratio affected by your technology?

We bring the ratio down as well... A load/store to a cores local scratchpad (Our software managed and power efficient version of a traditional L1 cache) is 1 cycle, compared to 4 cycles for an Intel processor. Add in the fact that we have 128KB of memory per scratchpad (compared to 16 to 32KB L1 D$ for Intel), you don't need to go to DRAM as much, greatly increasing performance/throughout on top of the 10x+ efficiency gain.

Even in the case of a core access accessing another cores local scratchpad when they are on opposite corners of the chip, it takes only one cycle per hop on the Network on Chip... meaning for our 256 core chip, you can go all away across the chip (and access a total of 32MB of memory) in 32 cycles... Less than the ~40 cycles it takes to access L3 cache on an Intel chip.

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