Modelling

Question 1

What is a braitenberg vehicle

Answer 1

• a simple mapping between two sensors.
• different mappings can generate different behaviours
• the behaviour depends on its environment.

Question 2

Provide mathmatical basis for gradient descent

Answer 2

m-s1-s2 in the environment s(X)= x2.
Because of the silt s1= p(point on the vehicle)- d (distance from point to sensor) and s2= p+d
if s1 and s2 are renamed x.
p2+d2-2pd. =s1
p2 +d2 + spd = s2
M= s1- s2 so m= -4pd. This shows how models can be used to generate predictions.
This vehicle can now be plotted in its environment, and this shows that the behaviour of the vehicle will move towards to source, slowing down the closer it gets until it stops at the center= when the motor = 0 (therefore when there is no difference in sensor readings between 1 and 2) . - Basis for gradient descent.

Question 3

How can modelling using braitenberg vehicles help us on the way to gradient descent.

Answer 3

show complex looking behaviour as simple.

Provides predictions

Question 4

Describe gradient descent

Answer 4

Gradient descent is an important technique for neural networks as it is the basis for deriving models of homeostasis and the supervised learning model (The delta rule/ the rescorla- wagner model). Both of these models involve following the gradient downhill in order to reduce the error, until the gradient = 0.
To perform gradient descent the error surface must be written as a function of x (i.e. all the possible states a vehicle could be in) Then the derivative must be found of the error surface and a minus sign should be added to it. The result is a function which can be applied again and again by adding values in to reduce the error until it =0.
This process has been modelled by a braitenberg vehicle, for instance the vehicle m=s1-s2 in the environemnt f(x) = x2 can generate predictions of gradient descent. If s1 and s2 are rewritten as x1 and x2 retrospectively where they are equal distances from a point on the vehicle they will be x1= p+d and x2= p-d then they can be squared in the function. The result will look something like x1= p2 +d2 +2pd and x2= p2 +d2 -2pd. This can now be plugged back into the original equation to form m= -4pd and if pd is replaced with x then the result is m=-4x which is gradient descent on a vehicle, the derivative was found and a minus sign was added and the resulting function provides a prediction that the vehicle will travel in its environment with until it = 0 (when there is no difference between s1 and s2.
Gradient descent is not only used in neural networking, it can provide useful insights into other areas of science for instance in evolutionary biology there is a fitness landscape which is about reducing the error of our genes in order to increase our potential fitness. Furthermore, in physics there is the idea of an energy landscape which is about turning high energy states into low energy states. Furthermore, in developmental psychology the idea of an ontological landscape can provide us with detail about how the small changes at the starting point of development (genomes) could make large changes to the final overall phenotype.