Page 65 - Rapid Learning in Robotics

4.3 The Best-Match Search 51

beforehand, one can construct a separate mapping for each. However, it
is clearly more desirable to work with a single network that can complete
different sets of missing variable values from different sets of known val-
ues.

4.3 The Best-Match Search

Returning to Eq. 4.4, we see that the discrete best-match search in the stan-
dard SOM is now replaced by solving the continuous minimization prob-
lem for the cost function

d
X
E s dist x w s p k x k w k s (4.10)

k

A very straightforward solution is to start with a good guess s t
,
a taken as the node location of the best-matching reference vector w a
analog to the SOM best-match step in Eq. 3.9. Then an iterative gradient
descent is performed

d

X
k ws t
s t x k w k s t (4.11)
s t p k

s
k
with as a step size parameter.
However the simple first order gradient descent method given is very
sensitive to the proper choice of the step size parameter . Too small values
lead to unnecessarily many iteration steps, too large values can lead to
divergence.
We tested various algorithms for automatically adjusting t on
the basis of observing the sequence in cost reduction and gradient direc-
tion. Some can indeed increase robustness and the rate of convergence
but bear the disadvantage of introducing other critical parameters to the
algorithm. Other techniques were considered, which are often used for im-
proving the back-propagation learning algorithm: Eq. 4.11 is augmented
by a “momentum term”

s t s t s t (4.12)

with the intention to suppress oscillations. This technique is very helpful if
the parameter is well-chosen. One method which determines is known