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.