Why Are Eight Bits Enough for Deep Neural Networks?

I had wondered this myself -- it seems reasonable to see limiting the precision of activation as a form of regularization, as the author alludes to.

For me, the place we'll eventually end up is obviously custom deep learning / evaluation chips that perform analogue operations using transistors in their linear regime (like how op-amps work). These chips would be programmed merely to express the tensor operation graph, essentially analogue tensor FPGAs.

This should bring multiple order-of-magnitude reductions in power consumption and increases in evaluation speed. And when you don't have a clock there might also be interesting ways of dealing with time in which you don't discretize and unroll, like one currently does with GRUs or LSTMs.

Let the laws of physics do the recurrent math for you. Analog RNN computers would be very interesting, but they would first setting in stone the basics of the algorithms we use. We are still only beginning to explore the algorithm space, and this requires a flexibility that analog computers (or even ASICs or FPGAs) don't provide.

It's still not clear whether the future of AI will even involve neural networks at all. Intuitively, they seem so inefficient.

I agree that this kind of naive analog computing sounds very attractive with those simple linear operations (linear networks have been exhaustively studied, as you noted you essentially need only resistors and amplifiers). But it's not entirely obvious to me they ought to be better than digital electronics for comparable precision (considering their noise) and power consumption. I think you may get into trouble in the small current regime due to quantum mechanics: while you can do digital electronics with only a few electrons, you may need a large number to be able to maintain good linearity. An then there's the fact you can deal with exponentially larger numbers with roughly linearly (or polynomial) increasing memory, while if you use analog circuits you have to pay a quadratic cost on the exponential, so ~n^k vs ~exp(2n) power consumption doesn't look good from this pov. But who knows, as the article points out the nonlinearities of the network may miraculously make it work even with very poor linearity and poor precision. It remains to be tested.

Point of interest, if you do the fundamental physics on neuronal membranes, the number of levels that are actually distinguishable give the noise in the system is only about 1000. So even in a biological system there are only 4x the the number of discrete levels. I realize this isn't a good match to what is mentioned in the article but it does put some constraints on the maximum dynamic range that biological sensors have to work within.

did my thesis on this topic (at that time we were searching the lower bound of ALU needed to have them running in zero power devices)

it's interesting, NN degrade at about 6bit, and that's mostly because the transfer function become stable and the training gets stuck more often in local minimums.

we built a training methodology in two step, first you trained them in 16bit precision, finding the absolute minimum, then retrain them with 6bit precision, and the NN basically learned to cope with the precision loss on its own.

funny part is, the less bit you have, the more robust the network became, because error correcting became a normal part of its transfer function.

we couldn't make the network solution converge on 4bit however. we tried using different transfer function, but then ran out of time before getting meaningful results (Each function needs it's own back propagation adjustment and things like that take time, I'm not a mathematician :D)