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not too large, since it would then include information from a very different
weight state; L can't be too small either otherwise the density estimation will
suffer from a lack of samples. The modification of the RTRL algorithm is on
equation (6.15) containing the weight update rule, which is rewritten as
exp
e( t
L
i ) T e( t
i ) T
e( t
i )
i ) T C
Δ w j = η
Λ j ( t
.
(6.22)
2 h 2
i =1
The minus sign on expression (6.21) cancels with the minus sign that would
have to be inserted into the Δ w j given that maximization is now performed
instead of minimization.
6.2.1.4
Experiments
This section describes experiments conducted to evaluate the performance
of the RTRL-ZED method. The obtained experimental results are compared
against those of the original RTRL method. The activation function used in
the RNNs is the standard sigmoid. The variables x and
Λ j were initialized
as follows: x(0) = 0 ,
Λ j (0) = 0 ,j =1 ,...,q .
The first experiment uses 3000 instances of the Mackey-Glass time se-
ries [143]. The first 2000 are ignored in terms of prediction evaluation (it
is considered that the network is still adapting to the signal) and the last
1000 are used for evaluation. The task consists on predicting the next point
of the series with an absolute error smaller than 0.05 (the tolerance). This
can be seen as a classification task if we consider that for each point that
the network tries to predict, it is in fact classifying the input as a member
of the Mackey-Glass series or not depending on whether it is able to predict
within the prescribed tolerance. We count the number of misclassified points
(predicted with absolute error higher than the tolerance) on the last 1000
points of the series.
Tabl e 6 . 8 Average error in percentage (with standard deviation) for 100 repeti-
tions of the experiment using the Mackey-Glass time series.
q 2 4 6
RTRL 4.37 (0.15) 4.53 (0.32) 4.56 (0.39)
RTRL-ZED 1.53 (0.09) 1.53 (0.05) 1.41 (0.09)
Table 6.8 presents the results. It contains the average errors and standard
deviations for 100 repetitions of the experiment. The parameters were varied
in the following way: for RTRL, η was varied from 1 to 20 with 0.2 steps;
for RTRL-ZED, η was varied from 0.1 to 1 with 0.05 steps, h varied from
 
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