The central problem in network learning is that of determining, for
a given input, the set of elements whose outputs can be altered so
as to correct the final element, and which will do the least amount
of damage to previous adaptations to other inputs. Once this set has
been determined, the incrementing rule given for a single element will
apply in this case as well (subject to the restriction of leaving
interconnecting weights positive), since the desired final output
coincides with that desired for each of the elements to be changed
(because of positive interconnecting weights).
In the process of arriving at such a decision three factors need to be
considered. Elements selected for change should tend to be those whose
output would thereby be affected for a minimum number of other possible
inputs. At the same time it should be ascertained that a change in
each of the elements in question does indeed contribute significantly
towards correcting the final output. Finally, a minimum number of such
elements should be used.
It would appear at first that this kind of decision is impossible to
achieve if the complexity of the decision apparatus is kept comparable
to that of the basic input-output network as mentioned earlier.
However, in the method to be described it is felt that a reasonable
approximation to these requirements will be achieved without an undue
increase in complexity.
It is assumed that in addition to its normal inputs, each element
receives a variable input bias which we can call b. The output of every
element should then be determined by the sign of the usual weighted
sum of its inputs plus this bias quantity. This bias is to be the same
for each element of the network. If b = 0 the network will behave
as before. However, if b is increased gradually, various elements
throughout the network will commence changing from -1 to +1, with one
or a few changing at any one time as a rule. If b is decreased, the
opposite will occur.
Now suppose that for a given input the final output ought to be +1 but
actually is -1. Assume that b is then raised so high that this final
output is corrected. Then commence a gradual decline in b. Various
elements may revert to -1, but until the final output does, no weights
are changed. When the final output does revert to -1, it is due to an
element’s having a sum (weighted sum plus bias) which just passed down
through zero. This then caused a chain effect of changing elements
up to the final element, but presumably this element is the only one
possessing a zero sum. This can then be the signal for the weights
on an element to change—a change of final output from right to wrong
accompanied simultaneously by a zero sum in the element itself.
After such a weight change, the final output will be correct once more
and the bias can again proceed to fall. Before it reaches zero, this
process may occur a number of times throughout the network. When the
bias finally stands at zero with the final output correct, the network
is ready for the next input. Of course if -1 is desired, the bias will
change in the opposite direction.
It is possible that extending the weight change process a little past
the zero bias level may have beneficial results. This might increase
the life expectancy of each learned input-output combination and
thereby reduce the total number of errors. This is because the method
used above can stop the weight correction process so that even though
the final output is correct, some elements whose output are essential
to the final output have sums close to zero, which are easily changed
by subsequent weight changes.
It will be noted that this method conforms to all three considerations
mentioned previously. First, by furnishing each element the same bias,
and by not changing weights until the final output becomes incorrect
with dropping bias, there is a strong tendency to select elements
which, with b = 0, would have sums close to zero. But the size of the
sum in an element is a good measure of the amount of damage done to
an element for other inputs if its current output is to be changed.
Second, it is obvious that each element changed has had a demonstrable
effect on the final output. Finally, there will be a clear tendency to
change only a minimum of elements because changes never occur until the
output clearly requires a change.
On the other hand this method requires little more added complexity to
the network than it already has. Each element requires a bias, an error
signal, and the desired final output, these things being uniform for
all elements in a network. Some external device must manipulate the
bias properly, but this is a simple behavior depending only on an error
signal and the desired final output—not on the state of individual
elements in the network. What one has, then, is a network consisting
of elements which are nearly autonomous as regards their decisions
to change weights. Such a scheme appears to be the only way to avoid
constructing a central weight-change decision apparatus of great
complexity. This rather sophisticated decision is made possible by
utilizing the computational capabilities the network already possesses
in producing outputs from inputs.
It should be noted here that this varying bias method requires that
the variable bias be furnished to just those elements which have
variable weights and to no others. Any fixed portion of the network,
such as preliminary layers or final majority function for example,
must operate independently of the variable bias. Otherwise, the final
output may go from right to wrong as the bias moves towards zero and no
variable-weight element be to blame. In such a case the network would
be hung up.