Combined Model and Task Learning, and Other Mechanisms - Computational Explorations in Cognitive Neuroscience - page 180

is clearly true of the cortex, where in the visual system
for example the original retinal array is re-represented
in a large number of different ways, many of which
build upon each other over many “hidden layers” (Des-
imone & Ungerleider, 1989; Van Essen & Maunsell,
1983; Maunsell & Newsome, 1987). We explore a
multi-hidden layer model of visual object recognition
in chapter 8.
A classic example of a problem that benefits from
multiple hidden layers is the family trees problem of
Hinton (1986). In this task, the network learns the
family relationships for two isomorphic families. This
learning is facilitated by re-representing in an inter-
mediate hidden layer the individuals in the family to
encode more specifically their functional similarities
(i.e., individuals who enter into similar relationships
are represented similarly). The importance of this re-
representation was emphasized in chapters 3 and 5 as a
central property of cognition.
However, the deep network used in this example
makes it difficult for a purely error-driven network to
learn, with a standard backpropagation network tak-
ing thousands of epochs to solve the problem (although
carefully chosen parameters can reduce this to around
100 epochs). Furthermore, the poor depth scaling of
error-driven learning is even worse in bidirectionally
connected networks, where training times are twice as
long as in the feedforward case, and seven times longer
than when using a combination of Hebbian and error-
driven learning (O'Reilly, 1996b).
This example, which we will explore in the subse-
quent section, is consistent with the general account of
the problems with remote error signals in deep networks
presented previously (section 6.2.1). We can elaborate
this example by considering the analogy of balancing a
stack of poles (figure 6.6). It is easier to balance one tall
pole than an equivalently tall stack of poles placed one
atop the other, because the corrective movements made
by moving the base of the pole back and forth have a
direct effect on a single pole, while the effects are indi-
rect on poles higher up in a stack. Thus, with a stack
of poles, the corrective movements made on the bot-
tom pole have increasingly remote effects on the poles
higher up, and the nature of these effects depends on
the position of the poles lower down. Similarly, the er-

a)

b)

Self−organizing
Learning

Gyroscopes

Flexibility
Limits

Constraints

Figure 6.6: Illustration of the analogy between error-driven
learning in deep networks and balancing a stack of poles,
which is more difficult in a) than in b) . Hebbian self-
organizing model learning is like adding a gyroscope to each
pole, as it gives each layer some guidance (stability) on how to
learn useful representations. Constraints like inhibitory com-
petition restrict the flexibility of motion of the poles.

ror signals in a deep network have increasingly indirect
and remote effects on layers further down (away from
the training signal at the output layer) in the network,
and the nature of the effects depends on the representa-
tions that have developed in the shallower layers (nearer
the output signal). With the increased nonlinearity as-
sociated with bidirectional networks, this problem only
gets worse.
One way to make the pole-balancing problem easier
is to give each pole a little internal gyroscope, so that
they each have greater self-stability and can at least par-
tially balance themselves. Model learning should pro-
vide exactly this kind of self-stabilization, because the
learning is local and produces potentially useful rep-
resentations even in the absence of error signals. At
a slightly more abstract level, a combined task and
model learning system is generally more constrained
than a purely error-driven learning algorithm, and thus
has fewer degrees of freedom to adapt through learning.
These constraints can be thought of as limiting the range
of motion for each pole, which would also make them
easier to balance. We will explore these ideas in the fol-
lowing simulation of the family trees problem, and in
many of the subsequent chapters.