Why are eight bits enough fot deep neural networks

http://petewarden.com/2015/05/23/why-are-eight-bits-enough-for-deep-neural-networks/

Deep learning is a very weird technology. It evolved over decades on a very different track than the mainstream of AI, kept alive by the efforts of a handful of believers. When I started using it a few years ago, it reminded me of the first time I played with an iPhone – it felt like I’d been handed something that had been sent back to us from the future, or alien technology.
One of the consequences of that is that my engineering intuitions about it are often wrong. When I came across im2col, the memory redundancy seemed crazy, based on my experience with image processing, but it turns out it’s an efficient way to tackle the problem. While there are more complex approaches that can yield better results, they’re not the ones my graphics background would have predicted.
Another key area that seems to throw a lot of people off is how much precision you need for the calculations inside neural networks. For most of my career, precision loss has been a fairly easy thing to estimate. I almost never needed more than 32-bit floats, and if I did it was because I’d screwed up my numerical design and I had a fragile algorithm that would go wrong pretty soon even with 64 bits. 16-bit floats were good for a lot of graphics operations, as long as they weren’t chained together too deeply. I could use 8-bit values for a final output for display, or at the end of an algorithm, but they weren’t useful for much else.
It turns out that neural networks are different. You can run them with eight-bit parameters and intermediate buffers, and suffer no noticeable loss in the final results. This was astonishing to me, but it’s something that’s been re-discovered over and over again. My colleague Vincent Vanhoucke has the only paper I’ve found covering this result for deep networks, but I’ve seen with my own eyes how it holds true across every application I’ve tried it on. I’ve also had to convince almost every other engineer who I tell that I’m not crazy, and watch them prove it to themselves by running a lot of their own tests, so this post is an attempt to short-circuit some of that!

How does it work?

You can see an example of a low-precision approach in the Jetpac mobile framework, though to keep things simple I keep the intermediate calculations in float and just use eight bits to compress the weights. Nervana’s NEON library also supports fp16, though not eight-bit yet. As long as you accumulate to 32 bits when you’re doing the long dot products that are the heart of the fully-connected and convolution operations (and that take up the vast majority of the time) you don’t need float though, you can keep all your inputs and output as eight bit. I’ve even seen evidence that you can drop a bit or two below eight without too much loss! The pooling layers are fine at eight bits too, I’ve generally seen the bias addition and activation functions (other than the trivial relu) done at higher precision, but 16 bits seems fine even for those.
I’ve generally taken networks that have been trained in full float and down-converted them afterwards, since I’m focused on inference, but training can also be done at low precision. Knowing that you’re aiming at a lower-precision deployment can make life easier too, even if you train in float, since you can do things like place limits on the ranges of the activation layers.

Why does it work?

I can’t see any fundamental mathematical reason why the results should hold up so well with low precision, so I’ve come to believe that it emerges as a side-effect of a successful training process. When we are trying to teach a network, the aim is to have it understand the patterns that are useful evidence and discard the meaningless variations and irrelevant details. That means we expect the network to be able to produce good results despite a lot of noise. Dropout is a good example of synthetic grit being thrown into the machinery, so that the final network can function even with very adverse data.
The networks that emerge from this process have to be very robust numerically, with a lot of redundancy in their calculations so that small differences in input samples don’t affect the results. Compared to differences in pose, position, and orientation, the noise in images is actually a comparatively small problem to deal with. All of the layers are affected by those small input changes to some extent, so they all develop a tolerance to minor variations. That means that the differences introduced by low-precision calculations are well within the tolerances a network has learned to deal with. Intuitively, they feel like weebles that won’t fall down no matter how much you push them, thanks to an inherently stable structure.
At heart I’m an engineer, so I’ve been happy to see it works in practice without worrying too much about why, I don’t want to look a gift horse in the mouth! What I’ve laid out here is my best guess at the cause of this property, but I would love to see a more principled explanation if any researchers want to investigate more thoroughly? [Update – here’s a related paper from Matthieu Courbariaux, thanks Scott!]

What does this mean?

This is very good news for anyone trying to optimize deep neural networks. On the general CPU side, modern SIMD instruction sets are often geared towards float, and so eight bit calculations don’t offer a massive computational advantage on recent x86 or ARM chips. DRAM access takes a lot of electrical power though, and is slow too, so just reducing the bandwidth by 75% can be a very big help. Being able to squeeze more values into fast, low-power SRAM cache and registers is a win too.
GPUs were originally designed to take eight bit texture values, perform calculations on them at higher precisions, and then write them back out at eight bits again, so they’re a perfect fit for our needs. They generally have very wide pipes to DRAM, so the gains aren’t quite as straightforward to achieve, but can be exploited with a bit of work. I’ve learned to appreciate DSPs as great low-power solutions too, and their instruction sets are geared towards the sort of fixed-point operations we need. Custom vision chips like Movidius’ Myriad are good fits too.
Deep networks’ robustness means that they can be implemented efficiently across a very wide range of hardware. Combine this flexibility with their almost-magical effectiveness at a lot of AI tasks that have eluded us for decades, and you can see why I’m so excited about how they will alter our world over the next few years!