Suppose we want to send a message to a friend using as few bits as possible… And let’s also suppose that this message consists of unit-normally-distributed real numbers. Now, it turns out that it’s really really expensive to send arbitrary precision real numbers: You could find yourself sending some not-previously-known-to-man transcendental number, and just have to keep listing off digits forever. So, in reality, we allow some amount of noise tolerance in the encoding.
This is a good problem to sink some effort into: Lots of data is roughly normally distributed, and if you’re sending some machine learned representation, you can introduce some regularization to ensure that the data is close to unit-normal. (This is, for example, the whole idea behind Variational Auto Encoders.)
I’ll start with a quick outline of a standard solution, and then we’ll go into a half-baked idea involving Fibonacci codes and space filling curves.
The basic idea comes in two parts:
First, Fibonacci coding lets you encode an arbitrarily large integer, without having to send the length of the integer before hand. (There’s a bunch of similar schemes you can use here, but the Fibs are fun and interesting.)
Second, map the integers into the reals in a way that covers the normal distribution in a ‘progressive’ way, so that we can use small numbers if we don’t need high precision, and only use bigger numbers as we need more precision.
At the end we’ll run a few empirical tests and see how well the scheme works, and point to where there’s room for improvement.
I recently had the pleasure of reading James Scott’s “Seeing Like a State,” which examines a certain strain of failure in large centrally-organized projects. These failures come down to the kinds of knowledge available to administrators and governments: aggregates and statistics, as opposed to the kinds of direct experience available to the people living ‘on the ground,’ in situations where the centralized knowledge either fails to or has no chance to describe a complex reality. The book classifies these two different kinds of knowledge as techne (general knowledge) and metis (local knowledge). In my reading, the techne – in both strengths and shortcomings – bears similarity to the knowledge we obtain from traditional algorithms, while metis knowledge is just starting to become available via statistical learning algorithms.
In this (kinda long) post, I will outline some of the major points of Scott’s arguments, and look at how they relate to modern machine learning. In particular, the divides Scott observes between the knowledge of administrators and the knowledge of communities suggest an array of topics for research. Beyond simply looking at the difference between the ways that humans and machines process data, we observe areas where traditional, centralized data analysis has systematically failed. And from these failures, we glean suggestions of where we need to improve machine learning systems to be able to solve the underlying problems.
I’ve been playing around with Cities: Skylines recently, the super-popular SimCity knock-off. Dealing with traffic is a core theme of the game (as it should be). Traffic tends to accumulate at intersections, and it’s well known that one-way streets have higher traffic flow. The logical conclusion, then, is to try to build a city with a single extremely long one-way street… Unfortunately we have to compromise on this perfect vision, because people want to get back home after work and so on.
A Quick Intro to Space Filling Curves
Meanwhile, a space-filling curve is an mathematical invention of the 19th century, and one of the earlier examples of a fractal. The basic idea is to define a path that passes through every point of a square, while also being continuous. This is accomplished by defining a sequence of increasingly twisty paths (H1, H2, H3, …) in such a way that H∞ is well-defined and continuous. Of course, we don’t want a infinitely twisty road, but the model of the space filling curve will still be useful to us.
There are a few important ideas in the space filling curve. The first is a notion that by getting certain properties right in the sequence of curves H1, H2, H3, …, we’ll be able to carry those properties on to the limit curve H∞.
The second main idea is how to get continuity. Thinking of the curve as a function where you’re at the start at time 0 , and you always get to the end at time 1, we want an H∞ where small changes in time produce small changes in position. The tricky part here is that the path itself gets longer and longer as we try to fill the square, potentially making continuity hard to satisfy… When the length of the path doubles, you’re moving twice as fast.
In fact, because of continuity, you can also “go backwards:” Given a point in the square, you can approximate what time you would have passed through the point on the limit curve H∞, with arbitrary precision. This gives direct proof that the curve actually covers the whole square.
Here’s an example of a space filling curve which is not continuous. Define Bk as the curve you get from following these instructions:
Start in the lower-left corner.
Go to the top of the square, and then move right by 1/k.
Move to the bottom of the square, and move right by 1/k.
Repeat steps 2 and 3 until you get to the right side of the square.
The problem here is that a very small change in time might take us all the way from the top of the square to the bottom of the square. We need to be twistier to make sure that we don’t jump around in the square. The Moore curve, illustrated above, does his nicely: small changes in time (color) don’t move you from one side of the square to the other.
Actual Simulated Cities
What happens if we try to use space filling curves to build a city in Cities: Skylines?
My first attempt at building ‘Hilbertville’ was to make large blocks, with a single, winding one-way road for access, using the design of a (second-order) Hilbert Curve. In addition to the roads, though, I placed a number of pedestrian walkways, which allow people on foot to get in and out of these neighborhoods directly. I like to think that this strongly encourages pedestrian transit, though it’s hard to tell what people’s actual overall commuting choices are from the in-game statistics.
Skylines only allows building directly facing a road; corners tend to lead to empty space. You can see a large empty square in the middle of the two blocks pictured above. There are also two smaller rectangles and two small empty squares inside of each of these two blocks. Making the top ‘loop’ a little bit longer removed most of the internal empty space. This internal space is bad from the game perspective; ideally we would still be able to put a park in the empty spaces to allow people to extra space, but even parks require road access.
Intersections with the main connecting roads end up as ‘sinks’ for all of the traffic congestion. So we should try to reduce the number of such intersections… The Moore curve is a slight variation on the Hilbert curve which puts the ‘start’ and ‘finish’ of the path next to one another. If we merge the start and finish into a wide two-way road, we get this:
We still get the wasted square between neighborhoods, but somewhat reduce the amount of empty interior space. Potentially, we could develop a slightly different pattern and alternate between blocks to eliminate the lost space between blocks. Also, because the entrance and exit to the block coincide, we get to halve the number of intersections with the main road, which is a big win for traffic congestion.
Here’s a view of the full city; it’s still not super big, with a population of about 25k. We still get pretty heavy congestion on the main ring road, though the congestion is much less bad than in some earlier cities I built. In particular, the industrial areas (with lots of truck traffic) do much better with these long and twisty one-way streets.
The empty space is actually caused by all of the turns in the road; fewer corners implies fewer wasted patches of land. The easiest way to deal with this is to just use a ‘back-and-forth’ one-way road, without all of the fancy twists.
The other major issue with this style of road design is access to services. Fire trucks in particular have a long way to go to get to the end of a block; the ‘fire danger’ indicators seem to think this is a bad idea. I’m not sure if it’s actually a problem, though, as the amount of traffic within a block is next to none, allowing pretty quick emergency response in spite of the distance.
Overall, I would say it’s a mixed success. There’s not a strong reason to favor the twisty space-filling curves over simpler back-and-forth one-way streets, and in either case the access for fire and trash trucks seems to be an issue. The twistiness of the space-filling curve is mainly used for getting the right amount of locality to ensure continuity in the limit curve; this doesn’t serve a clear purpose in the design of cities, though, and the many turns end up creating difficult-to-access corner spaces. On the bright side, though, traffic is reduced and pedestrian transit is strongly encouraged by the design of the city.
Once upon a time in the late nineties, the internet was a crypto-anarchist’s dream. It was a new trans-national cyberspace, mostly free of the meddling of any kind of government, where information could be exchanged with freedom, anonymity, and (with a bit of work) security. For a certain strain of crypto-anarchist, Temporary Autonomous Zone was a guiding document, advocating small anarchist societies in the blank spaces of existing society temporarily beyond the reach of government surveillance or regulation. This was a great idea with some obvious drawbacks: On the one hand, TAZ served as a direct inspiration for Burning Man. On the other hand, it eventually came out that Peter Lamborn Wilson (who authored TAZ under the pseudonym Hakim Bey) was an advocate of pedophilia, which had clear implications as to why he wanted freedom from regulation. It’s a document whose history highlights the simultaneous boundless possibilities and severe drawbacks of anarchism.
Against this background, Lawrence Lessig’s Code made the case that the internet TAZ was in fact temporary. Lessig argued that the internet’s behaviour is determined by a combination of computer code and legal code, and that while the legal code hadn’t been written yet, it would be soon. His prediction (which has largely been realized) was that the internet would lose its anarchic character through government regulation mixed with a need for security and convenience in commercial transactions. (In addition to these forces, social media also came along, in which people largely sacrificed their anonymity willingly for the convenience of being able to easily communicate with their meatspace social networks.)
In thinking about Bitcoin, it’s useful to see how the regulation came to change the internet. The prediction (again pretty much correct) was that regulations would target large companies instead of individual users. Companies are compelled to follow the law under the ultimate threat of not being allowed to operate at all. Because of the tendency for people to glom onto just a few instances of working solutions, it becomes easy to target a few large entities to enact regulation on a broad base of users.