/Machine Learning, Kolmogorov Complexity, and Squishy Bunnies (2019)

Machine Learning, Kolmogorov Complexity, and Squishy Bunnies (2019)

Created on Aug. 27, 2019, 9:48 a.m.

Machine Learning

We know that Machine Learning is an extremely powerful tool for tackling complex
problems which we don’t know how to solve by conventional means. Problems like
image classification can be solved effectively by Machine Learning because at
the end of the day, gathering data for that kind of task is much easier than
coming up with hand-written rules for such a complex and difficult problem.

But what about problems we already know how to solve? Is there any reason to
apply Machine Learning to problems we already have working solutions for? Tasks
such as physics simulation, where the rules and equations governing the task are
already well known and explored? Well it turns out in many cases there are good
reasons to do this – reasons related to many interesting concepts
in computer science such as the trade-off between memorization and computation,
and a concept called Kolmogorov complexity.

The way to start thinking about it is this: although it might not be
obvious, for any phenomenon, problem, or mathematical function we
are interested in (provided there is a way to find an answer in the first place) there is always a way to perform a trade-off between how
much computation we perform and how much memory we use.

For example, let’s consider a very simple Python program which computes sin(x):

def f(x):
    return np.sin(x)

This is what we would call a “direct” computation of the sin function, but let
us imagine for a moment than the sin function was a very expensive operation
to compute. In this case we might want to take a different approach. We could,
for example, pre-compute values of sin(x) for many different values of x
and store them in a big lookup table like follows:

def f(x):
    return {
        0.00: 0.00,
        0.34: 0.34,
        0.69: 0.64,
        1.04: 0.86,
        1.39: 0.98,
        ... :  ...,
        1.74: 0.98,
        2.09: 0.86,
        2.44: 0.64,
        2.79: 0.34,
        3.14: 0.00,

Now, instead of computing the value of sin(x) “directly” by some expensive operation, we
can instead “compute” the value of sin(x) almost instantly by simply looking up the
value of our input in the lookup table. The downside is this – now we need to pre-compute the
sin function hundreds of thousands of times and keep all of those pre-computed values of
sin(x) in memory. We have traded more memory usage (and pre-computation) in exchange for
less computation. And, although the lookup table may become
unfathomably huge, and the pre-computation time insanely large for complicated
functions, in theory this very same trick is applicable to any function we are
interested in – not just very simple functions like sin – functions even
as complicated as a physics simulations.

We can think about these two programs like two extreme data points – two different ways to compute the same function with vastly different uses of computation and memory. If we were to measure their respective memory usage and computation time and plot them on a graph it would probably look something like this:

graph programs

The natural question to ask next is if there are any generic programs
which lie somewhere in-between these two extremes. Programs which trade off memory
and computation in a way that is agnostic to the underlying function they are computing. In fact
there are! We can, for example, write a generic program which caches computations
to avoid re-computation. This saves some performance if we get an input we’ve
already computed the output for at the cost of some additional memory usage:

cache = {}

def f(x):
    if x in cache:
        return cache[x]
        y = np.sin(x)
        cache[x] = y
        return y

Like before, we’ve found a “generic” way to trade-off memory for computation which
can work (in theory at least) no matter what function we are interested in

There are also programs that simply approximate the function we are
interested in computing by using a bit more memory in exchange for faster and
less accurate computation. Here is one such approximation for the sin function:

def f(x):
    x = x % (2*np.pi)
    return (
        (16*x*(np.pi - x)) / 
        (5*np.pi*np.pi - 4*x*(np.pi - x)))

It might not be clear at first where the additional memory usage in these
functions is, but in this case the constants such as 2, 16, 5, 4, and
pi are all additional “special” values, just like those in the lookup table –
these are the “memory”. And while this specific approximation is limited to sin, it isn’t that
difficult to find a generic version of this program which looks similar but
can approximate any function simply by finding different appropriate constant values.

graph programs full

Given these two additions, another question to ask is this: Is there a
generic kind of program like the ones we’ve presented so far which lies in the
sweet spot at the bottom left of the graph – not using too much memory – not
using too much computation – providing the best of both worlds?

One direction we can look towards is Machine Learning – because if we can
evaluate the function we are interested in approximating offline we can gather
training data and then train a generic Machine Learning algorithm to
approximate the function. In Machine Learning we have a whole class of
different generic algorithms we could add to our graph to see their performance.

As an example, let’s take a look at Neural Networks – if we examine the
computational properties of Neural Networks, we can probably guess where they
might lie on this graph without even having to try them.

First of all, let’s look at the computation time. Now, the computation time of
a Neural Network is basically proportional to the number of weights it has, and
the number of weights it has also dictates the memory usage. So computation
time and memory usage are coupled in a standard Neural Network – visually, Neural Networks
always lie somewhere on the diagonal of the graph.

And what dictates how many weights we need in a Neural Network? Well roughly,
this is dictated by two things – how accurate we want the Neural Network to be,
and how complex the function we are fitting is. If we look at all these
properties together we see something like this

            Computation Time     is proportional to     Memory Usage
            Memory Usage         is proportional to     Accuracy
            Accuracy   is inversely proportional to     Complexity

So, how far up the diagonal our Neural Network will be is governed ultimately by the
accuracy we want, and the complexity of the function we are interested in fitting. This is both good and bad news – good
because if the function we want to approximate isn’t complex then we can be
confident we will hit the sweet spot – low computation time and low memory
usage – and bad because if the function we want to approximate is complex we
get the worst of both worlds – expensive evaluation and lots of memory usage.

graph programs full

But what exactly do we mean by the complexity of a function? If we could get a
good intuitive understanding of this we would be able to predict more accurately
when Neural Networks might hit the sweet spot. This is where
Kolmogorov Complexity comes in – an intuitive measure for deciding how complex
a function is.

Kolmogorov Complexity

Put simply, the Kolmogorov Complexity of some function is the length of the
shortest possible program which can produce exactly the same outputs as the
function for all given inputs. In addition, according to the rules of
Kolmogorov Complexity, programs are not allowed to open any files or communicate
with the outside world in any way – all their data must be stored in the source code.

For example, the function which takes no inputs and outputs the string:


can be produced pretty simply by the small program:

def f():
    print "ab" * 16

Which implies this function which outputs ab repeated is not a particularly complex
function. On the other hand, the function which takes no inputs and produces the string:


is comparatively more complex – it basically requires a program to print out
the string verbatim:

def f():
    print "4c1j5b2p0cv4w1x8rx2y39umgw5q85s7"

Storing this string makes the program much longer, which means the function which produces it
must be complex.

There is already something interesting to note here – the first program computes
something (using the * operator), while the second simply memorizes
the string. This immediately gives us a good intuition: functions which need to do memorization are often
more complex.

And Kolmogorov Complexity is not limited to functions which output strings. We can use
exactly the same analysis for functions which produce images, physics
simulations, or anything else. Consider the following images, do we think
either of them are produced from complex functions?

simple 0simple 1

Well not really – probably there is a simple program which can produce either
image – it looks like they took simple rules to generate. What about these images?

complex 0complex 1

Now these are much more complex. It seems difficult to think of a program which
could produce either of these images without storing lots of raw data inside the
program. Here we gain another intuition: that natural data is often complex and
requires memorization.

What about these two images?


Aha – this time it’s a trick question. The first image is of course of a fractal –
an image which appears complex but actually as we well know has a simple program
that can be used to generate it.

The second is random noise – and can have two answers. If this noise is from a
pseudo random number generator or we just want to produce noise but not this exact noise,
it is technically not complex as we know that pseudo
random number generators are relatively simple programs – however – if this is
true random noise and we want to reproduce it exactly, then it is maximally complex – there is no possible program
which could generate it without simply storing it verbatim and writing it out. This
gives us another intuition: knowing the complexity of a function simply by
observing its input and output is extremely difficult.

So what about something like a physics simulation? Forget about the edge cases
and trick questions for now and imagine you had to write a program to produce
the movement of the cloth in this video given the movement of the ball.

simple swing

Is this complex?

Well…relative to our other examples certainly, but perhaps it isn’t as complex as you might think at first – I think it would probably be possible to guess
the motion of the cloth fairly accurately simply by knowing the phase of the movement of the
ball. My gut feeling is that if you do something clever there may well be a simple program which could
take just that one parameter and get a good guess at the state of the cloth in

What about this one?

fine folds

Okay, now this is complex – there are all kind of tiny folds and chaotic
movements going on and it would seem like you would need to write a massive
complex program to produce this exact behavior.

But is there a way we can actually compute the complexity directly rather than
just feeling around with our intuition? It has been shown that this is impossible in the general case –
but we can compute an approximation of it with an algorithm called
Principal Component Analysis (PCA). When you apply PCA to some data, what you get back is a guess at how many numbers might be required to express it for a given error threshold – a guess at how compressible the data is – and a simple algorithm for how to decompress it.

Let’s try applying this to some data gathered from a physics simulation.
Interestingly, when applied to physical simulation data, PCA has a special
behavior – in addition to telling us the “complexity” of the simulation it also
extracts the major axes of deformation for the physically simulated object we are interested in:


We can measure the complexity by looking at how many of these axes of
deformation are required to reconstruct the original motion. For a simple motion
like our sheet of cloth swinging back and forth we might only need one or two
axes to almost entirely reconstruct the motion, while for our cloth with
complex fine folds we could expect to need hundreds or even thousands.

Below you can see a comparison of what happens if we choose different numbers of axes to
reconstruct the movement of a cape attached to a character. With fewer axes you
get less detail – the complexity is effectively reduced. In this case, although
the original cloth has around 3000 vertices we need just 256 axes (sometimes also called Basis) to represent
the state of the cloth without too much loss in quality, detail, and

PCA comparison

This tells us something interesting – that physics simulations are almost
always less complex than they may first appear (there is a good reason for this too, based on the theory of adding constraints). It tells us that
if we try to approximate a fairly simple physics simulation with a Neural
Network, we have a good chance of hitting the sweet spot between computation and

Squishy Bunnies

So let’s try it – lets set up a bunch of simulations and run them for a long
time to extract lots of different simulation data for some different situations
we are interested in.

Training Data

As you’d expect, extracting this kind of data can take a long time – up to
several days – but once complete we have a massive database of physics
simulation data we can learn from.

The next step is to apply PCA. At this point we can decide exactly how many
axes we want to use by examining how well we can reproduce the original
data. Fewer axes means fewer details, but also makes it much more likely we
can hit the sweet spot in regards to performance.

Once we’ve applied PCA, for each frame we have N numbers, where N is the number
of axes, each number representing the deformation on that axis. Using
this representation, we want to train a Neural Network to predict the PCA
values for the next frame, given the frame before, and the positions of all
the different external objects such as the ball or whatever else.

In fact, we can ask the network to predict a correction to an
extrapolation of the current deformation using the rate of change of the current deformation –
in this case we get a more accurate prediction because most of the time objects
change very consistently between frames.

Once trained, we can drop in our Neural Network as a replacement to the simulation function. And, while the normal physics simulation takes as input
the full state of the cloth including all of the thousands of vertex positions –
our network only takes as input just the N numbers representing the deformation
on the PCA axes – and outputs the same. In this way it does vastly less computation and produces a much more efficient simulation.

And since the deformations computed from the PCA have a specific mathematical
property (they are orthonormal), this exact formulation has a nice physical interpretation too –
that under some basic assumptions we can say the Neural Network is actually being used to predict the
forces applied in the simulation in a highly efficient way. Forces such as those
introduced by collisions and internal tension. You can see the paper for more details.

Here you can see it applied to two small examples – a simple ball and sheet and a cloth pinned at four corners. At face value you would
never know that behind the scenes no physics simulation is actually being
performed – all of it is being approximated by a neural network!

Sheet and Trampoline

We can also include as input to the network other things – such as the wind
direction and speed. Here we can use it to control a flag:


What about a more complicated example? In this case the network gets as input
the joint positions of the blue character and learns to predict the movement
of the cape. It learns to do everything itself, including all of the collisions and other
interaction dynamics. Here we can compare it to the ground truth simulation – we can see
that while some details are lost, overall it does a pretty good job of
approximating the result.


Here we plug it into an interactive system to see how it behaves in a more
realistic environment.


It isn’t just cloth, we can also approximate the deformation of soft bodies
like this deformable bunny or this dragon. In these cases we get even more
massive performance gains simply because simulating deformable objects is
even more expensive.


If we adjust the number of PCA axes we use, we can also trade runtime
performance and memory in exchange for quality (or we could say complexity in this case).


Naturally, our network only performs well on the kind of situations it’s
trained on. If we move the objects faster or further than what we had in the
training data we don’t get realistic behavior from the simulation:


Similarly, if we try to train it on situations which are too complex it simply
takes too long to even get all the training data we need to cover every possible
different situation we might be interested in covering. Additionally, if the complexity gets too high we hit the worst of
both worlds – a massive neural network which requires huge amounts of memory
and takes a long time to compute.

But as long as we remain in the range of the training data and the complexity
is relatively low, we really can hit the sweet spot.
Performance goes between 35us per frame, and 350us per frame – which is roughly
within the budget for a character or other special entity in a AAA game
production – and about 300 to 5000 times faster than the original simulation we
used to get the training data. Having such fast performance also allows us to
simulate a lot of things at once! Including things that would be totally impossible with normal simulation:



So Neural Networks are not just good for things we don’t know how to solve,
they can provide massive performance gains on problems we already know how
to solve. In fact, we can use the concept of Kolmogorov Complexity
to get a kind of intuition (and even use PCA for a simple kind of measure) for how well we expect Neural Networks to
perform on a task.

But, thinking toward the future, there is still something awkward about all of this – which is that we
didn’t actually do what we originally wanted to do – which was to find a way to easily trade computation for memory usage and vice versa – in standard Neural Networks the two are coupled and computation time is always proportional to the number of
weights. Yes, we can hit the sweet spot if the complexity is low, but what if it is high? Wouldn’t it be nice to (for example) allow for a bit more memory usage to get computation time within a certain budget?

Wouldn’t it be great in game development if we could adapt Neural Networks in a
way which allowed us to set our budgets for computation and memory and have the
result presented to us in whatever accuracy was possible. Recent research has shown some promising results, but it looks like we will have to wait a bit longer before that dream becomes a reality…

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