Regarding 8-bit numbers, here's a thread on why 8 bits are enough and an old product that used that to good effect:

https://news.ycombinator.com/item?id=10244398

http://www.eetimes.com/document.asp?doc_id=1140287

It's something that keeps getting rediscovered. I know embedded industry shoehorns all kinds of problems into 8- and 16-bitters. Some even use 4-bit MCU's. Might be worthwhile if someone does a survey of all the things you can handle easily or without too much work in 8-16-bit cores. This might help for people building systems out of existing parts or people trying to design heterogenous SOC's.

The lack of real 8-bit comparison data makes the whole paper a little suspect IMO...it's sort of like the early GPU papers that claimed 100x improvement over the CPU while running x87 scalar CPU instructions...the benefits are definitely still there but handicapping one architecture when it has features that are specifically capable of doing this is a bit stupid. It's not like they didn't have to do a lot of work on TF to make it output TPU instructions. When you're down to a 4x improvement...the benefits of specialized accelerators start to become somewhat questionable.

I do like that they highlighted the importance of low latency output though...that's even more critical for future non "Web" applications which have to run in real time.

The difference is the power numbers.

3.5x faster than CPU doesn't sound special, but when you're building inference capacity by the megawatt, you get a lot more of that 3.5x faster TPU inside that hard power constraint.

"The TPU only operates on 8-bit integers" The 8 bit part is fine, but integers? what the hell. that's a new one for me

Anecdotally, it seems most models can be quantized to 8 bits without much loss of accuracy, and fixed point arithmetic requires much less hardware. Training is still done with floating point though.

This, when you get right down to it a lot of models do fine with only 256 unique weights.

Agreed - however as we progress I expect a comment like this to be akin to Bill Gate's 64K comment.

The brain appears to spend about 4.7 bits per synapse (26 discernible states, given the noisy computation environment of the brain); so it seems to be plenty enough for general intelligence. This could, of course, merely be a biological limit and on silicon more fine-grained weights might be the optimum.

Here is another paper demonstrating very good results with just 6 bit gradients: https://arxiv.org/abs/1606.06160

Almost certainly, and depths would have to increase. Like all series expansions the coefficients on later terms have less and less impact so their dynamic range is less and less important to the final value. But the dynamic range on the initial terms is proportionately important. I expect the dynamic range of the weights will turn out to be logarithmic with respect to overall depth of the network.

Yeah...you could implement 16-bit multiplication/addition as two 8-bit multiplications plus carry... so worst case, if you want 16-bit multiplies, you implement it yourself

Four multiplications, not two (three if you only need the low 16b of the result).

You can get the full 32-bit result in only three multiplications using https://en.wikipedia.org/wiki/Karatsuba_algorithm , but it requires more additions.

Yeah, in practice Karatsuba isn't a performant option for small operands unless your multiplier is catastrophically slow. (And it still doesn't get you to two multiplications.)

There are hints that even less than 8 bits per weight might be usable (for certain cases and on custom hw). Not sure if it's practical but it is definitely interesting.

https://arxiv.org/pdf/1502.02551.pdf

https://arxiv.org/pdf/1610.00324.pdf

I wanted to have some basic idea about hardware so I did some "research" (googling) and ended up giving a short informal talk. My slides with some links are here:

http://ml.kjx.cz/hw4nn2017.pdf

Indeed, any 8-bit x 8-bit function with an 8-bit result is just a 64KiB look-up table.

I imagine that once they've trained the floating-point models, they'll then quantize them into integers to make inference faster. It's not something I've done, but I imagine that the limited range of the of integers may cause problems (though they say in the paper that the 16-bit product can be accumulated to something that's 32-bit). The features to do this will be coming fairly soon to regular TensorFlow too.[1]

[1] https://youtu.be/0r9w3V923rk?list=PLOU2XLYxmsIKGc_NBoIhTn2Qh...

Nvidia has shipped a couple of INT8 inference cards as well:

http://www.anandtech.com/show/10675/nvidia-announces-tesla-p...

Yes, how do they perform? Linked article cites 47 TOPS on 8 bit integer arithmetic and they are useful for training too.

Take a look at Google's gemmlowp low-precision GEMM library:

https://github.com/google/gemmlowp

(Used in tensorflow)

Note however, that on Intel it's actually slower than run off the mill float32 linear algebra library like Eigen or OpenBLAS. Its main forte seems to be ARM.

Here's a paper they published a little while ago about limited numerical precision and deep learning:

https://arxiv.org/abs/1502.02551

Those limitations sounds awfully similar to that of an FPGA..

I was going to say as well. It seems like if caches are the bane of sequential processing(CPU) then routing has to be the counterpart on the parallel(FPGA/ASIC) side of the equation.

And what's amazing is that it was built on 28nm. So TPU 2.0 could increase by another 2x in perf/W just by going to 14nm (most likely) - even more if it's built on newer processes than that.

Intel's latest chips will be even further behind compared to the next-generation TPU than Haswell was compared to TPU 1.0.

28nm was quite a cheap fabrication technology even in 2015, but it costs a lot to have a completely custom production run. My guess it that it approximately works out in savings of power and space over the lifetime of the chip. It probably doesn't make sense for them to move to something smaller (and thus more expensive) whilst the performance benefit remains so substantial. If I were Intel, I probably wouldn't lose too much sleep over it either, because you still need something to attach the highly-specialised TPU to, and that'll be a Xeon for the forseeable future.

Agreed, the market for these chips is pretty small at the moment. What other platforms would need these other than cloud? Cars? Phones .... maybe?