Chapter 2. Prehension

Appendix C - Computational Neural Modelling 391
Another common rule which uses the presynaptic signal and
postsynaptic error is called the delta rule4 (or Widrow-Hoff rule), and
it uses a target value as follows (right side of Figure C.5):
Instead of using the output of neuron i to determine by how much to
change the weights, the actual output Oi is first subtracted from the
desired output ti. To do this, supervised learning is used. A small
training set is provided to the network for learning some part of the
task space. The training set is a collection of input/output pairs (or
patterns) that identifies to the network what the output should be for a
given set of inputs. For linear units, this rule minimizes the squares of
the differences between the actual and desired output values summed
over the output neuons and all pairs of input/output vectors
(Rumelhart, Hinton & Williams, 1986a).
The delta rule can be derived from a technique called gradient
descent. ‘Gradient descent’ means ‘go down hill’ in a mathematical
sense. For a traveller in a hilly area, define z to be the person’s
altitude, x to be longitude (i.e., how far east she is located) and y to be
latitude (how far north she is). Then z is a function of x and y: the
altitude of the ground varies with its x-y location. The variables x and
y are independent ones and the traveler can shift these at will by
walking. Since the traveler’s altitude changes as x and y do, z is the
dependent variable. The gradient, or change, of z is the vector of
partial derivatives of z with respect to its independent variables:
(az/ax, az/ay). In this case the gradient represents a horizontal vector
which points in the direction of travel for which z increases most
rapidly. The gradient descent rule simply says “move in the direction
opposite to which the gradient vector points”. Mathematically, it is
written as:
4An early model was a percepuon, which consisted of linear threshold units using
the delta rule for synaptic weight modification mosenblatt, 1962).