Error-Driven Task Learning - Computational Explorations in Cognitive Neuroscience - page 164

GeneRec, CHL and other Algorithms

5.8 Biological Considerations for GeneRec

The CHL algorithm traces its roots to the mean field
(Peterson & Anderson, 1987) or deterministic Boltz-
mann machine (DBM) (Hinton, 1989b) learning al-
gorithms, which also use locally available activation
variables to perform error-driven learning in recurrently
connected networks. The DBM algorithm was derived
originally for networks called Boltzmann machines
that have noisy units whose activation states can be de-
scribed by a probability distribution known as the Boltz-
mann distribution (Ackley et al., 1985). In this prob-
abilistic framework, learning amounts to reducing the
distance between the two probability distributions that
arise in the minus and plus phases of settling in the net-
work.
The CHL/DBM algorithm has been derived from the
Boltzmann machine learning algorithm through the use
of approximations or restricted cases of the probabilistic
network (Hinton, 1989b; Peterson & Anderson, 1987),
and derived without the use of the Boltzmann distri-
bution by using the continuous Hopfield energy func-
tion (Movellan, 1990). However, all of these deriva-
tions require problematic assumptions or approxima-
tions, which led some to conclude that CHL was funda-
mentally flawed for deterministic (non-noisy) networks
(Galland, 1993; Galland & Hinton, 1990). Furthermore,
the use of the original (noisy) Boltzmann machine has
been limited by the extreme amounts of computation
required, requiring many runs of many cycles to obtain
the averaged probability estimates needed for learning.
Thus, the derivation of CHL directly from the
backpropagation algorithm for completely determinis-
tic (non-noisy) units (via GeneRec) restores some basis
for optimism in its ability to learn difficult problems.
Further, although the generic form of CHL/GeneRec
does have some remaining performance limitations,
these are largely eliminated by the use of this learning
rule in the context of a kWTA inhibition function and
in conjunction with the CPCA Hebbian learning rule.
Most of the problems with plain CHL/GeneRec can be
traced to the consequences of using purely error-driven
learning in a unconstrained bidirectionally connected
network (O'Reilly, 1996b, in press). We will explore
some of these issues in chapter 6.

We have seen that GeneRec can implement error back-
propagation using locally available activation variables,
making it more plausible that such a learning rule
could be employed by real neurons. Also, the use of
activation-based signals (as opposed to error or other
variables) increases plausibility because it is relatively
straightforward to map unit activation onto neural vari-
ables such as time-averaged membrane potential or
spiking rate (chapter 2). However, three main features
of the GeneRec algorithm could potentially be problem-
atic from a biological perspective: 1) weight symmetry,
2) the origin of plus and minus phase activation states,
and 3) the ability of these activation states to influence
synaptic modification according to the learning rule.

5.8.1 Weight Symmetry in the Cortex

Recall that the mathematical derivation of GeneRec de-
pends on symmetric weights for units to compute their
sending error contribution based on what they receive
back from other units. Three points address the biolog-
ical plausibility of the weight symmetry requirement in
GeneRec:

As mentioned above, a symmetry preserving learning
algorithm like the CHL version of GeneRec, when
combined with either soft weight bounding or small
amounts of weight decay, will automatically lead to
symmetric weights even if they did not start out that
way. Thus, if the brain is using something like CHL,
then as long as there is bidirectional connectivity ,
the weight values on these connections will naturally
take on symmetric values. The next two points ad-
dress this much weaker constraint of bidirectional
connectivity.

The biological evidence strongly suggests that the
cortex is bidirectionally connected at the level of
cortical areas (e.g., Felleman & Van Essen, 1991;
White, 1989a). The existence of this larger-scale
bidirectional connectivity suggests that the cortex
may have come under some kind of evolutionary
pressure to produce reciprocal bidirectional connec-
tivity — the use of such connectivity to perform