Re: learnable dynamics (was Goal of AI...)


Michael Olea wrote:
JGCASEY wrote:
Michael Olea wrote:
...
I tell you all this, of course, in anticipation of
ripping apart, time permitting, your latest remarks
about "The animal, not the environment". (If I get
around to it, I hope you understand it will be the
comments I am ripping, not the commentator).

Hope you get around to it then :)



Hehe.

First I want to finish a little "just for fun" code
to investigate the performance of some unsupervised
learning algorithms on a time series generated by a
chaotic dynamical system. In this case the time series
itself is "complex" in the sense that it "looks random",
but there are simple rules generating the dynamics.
Do the algorithms learn a compact set of rules that
are a "good" approximation of the actual rules?


The simplest chaotic dynamical system I can think of
is the "logistic map":

x[i+1] = k*x[i]*(1-x[i])

This is a "discrete dynamical system", meaning we
sample values at discrete times, say every day, every
month, whatever. So x could be the size of a population
of insects, which we sample every generation.

The mathematical biologist Robert May investigated the
logistic map as a simplified model of population dynamics.
In this case x is not actually the number of individuals
in the population, but a fraction of some maximum possible
population size (under the assumption that the population,
because of factors like competition for finite resources
and so on, can never exceed some maximum). So in this case
x can only take values ranging from 0 to 1. So x[i] is the
value of x at generation i. The equation determines how the
size of the population in one generation depends on the
size of the population in the previous generation. Here k
is a constant, a "fecundity factor" approximating the
average number of surviving offspring per individual.

1) the population reaches some stable size and stays
there - this is a "fixed point attractor".


2) the population oscillates between some finite number n
values of x - a "periodic attractor of period n".


3) the population is aperiodic, ("chaotic") never repeating
any value it has assumed in the past.


The main interest for population dynamics is not that
this is a realistic model, but that (seemingly random)
fluctuations in population need not be due to random
outside influences, but simply reflect the inherent
dynamics of the system.


My interest is how unsupervised learning algorithms perform
in the chaotic regime, given only the stream of numbers x[i].

The "rate of growth of excess entropy (aka predictive
information" of the time series is a measure of the
"complexity" of the dynamics, which should indicate there
is an underlying simple model generating the series even
though the series itself "looks random".


Another twist is to turn the stream of values x[i] into a
stream of bits b[i] using a rule like: if (x <= 0.5) output 0
else output 1. This bit stream has many of the properties of
a sequence of tosses of a fair coin.


Anyhow, a simple C++ "functor" for the logistic map:


template<typename TREAL = float>
class LogisticMap
{
public:
//
// Constructor
//
LogisticMap(TREAL l, TREAL x_0 = 0.5)
: lambda(l), x(x_0) {}


//
// Accessors
//
TREAL Lambda() const { return lambda; }
TREAL X() const { return x; }


//
// Functor
//
TREAL operator()()
{
return x = lambda * x * (1-x);
}


private:
TREAL lambda;
TREAL x;

};


And a snippet of code that calls it:

LogisticMap<> map(4.0, 0.4);


...


for (int i = 0; i < nIterations; ++i)
{
std::cout << map() << '\n';
}


Unfortunately I am just a humble BASIC programmer who has
done some c programming but only dabbled in c++.

I had to look up lambda which is some kind of c++ library
and TREAL which is a general purpose real number. I never
got to use or understand c++ templates.

I recognize x[i+1] = k*x[i]*(1-x[i]) from reading I have
done on chaos but have never actually looked much into
the subject.

The variable x is a line segment running from 0 to 1.

Thus in BASIC I guess you would do something like this,

k = 3
for x = 0 to 1 step 0.1
PRINT k * x * ( 1 - x)
next i

So with your unsupervised learning algorithms do
they need a critic or do they just work by themselves
to learn a compact set of rules that are a "good"
approximation of the actual rules?

And do you think this is related to how the brain might
learn about its input? Is our input chaotic in any way?


--
JC

.