Chapter 2. Prehension

392 A pp e n dic e s
where q is a small positive number defining the traveler’s speed.
Gradient descent is a very general technique: first, define some
quantity to minimize (a dependent variable) in terms of controllable
quantities (the independent variables). Then, take the gradient and use
it, as shown above, to ‘tweak’ the independent variables, moving
‘downhill’. The process is repeated until a minimum value is reached,
at which time the gradient wil be zero.
In a neural network, the independent variables are the synaptic
weights. The dependent value to be minimized is taken as some
measure of the difference between the actual and desired perfomance
of the network. Thus, gradient descent is used to decrease the
performance error. For example, imagine a network with a single
output neuron (neuron i), which is being trained with a pattern p
which is a collection of input/output pairs. Let the desired output
value be i, and the actual value of.. Define the error to be E = 1/2($,i
- Opi)’. ken for all weights wij w ich feed into unit i,
dE/dwij = - 1 (tpi - opi>dopi/dwij
Recalling the neuron definition equations,
then, by the chain rule for differentiation,
Now, by the gradient descent definition,
For a linear neuron f (ai) = 1 for all ai, so
(8)
Awij = q (tpi - Opiloj (12)
which is the delta rule. (The complete derivation is shown in
Rumelhart et al., 1986a).
For networks with hidden layers, the generalized delta rule is used
(see Figure C.6), which states: