Training the Classifiers - Design and Analysis of Learning Classifier Systems - page 89

LMS and NLMS algorithm is lower than for the RLS algorithm and depends on
the values of the input, and on the other hand that direct tracking of the noise
variance is more accurate than estimating it by the LMS method.

5.4.1 Experimental Setup

The following classifier setups are used:
NLMS Classifier. This classifier uses the NLMS algorithm (5.29) to estimate the
weight vector, starting with w 0 = 0 , and a constant step size of γ =0 . 2. For
one-dimensional input spaces, D X =1,with x n =1forall n> 0, the NLMS
algorithm is equivalent to the LMS algorithm (5.25), in which variable step
sizes according to the MAM update [220] are used,

⎨

1 /c N if c N ≤ 1 /γ,

γ N = (5.76)

⎩

γ otherwise ,

which is equivalent to bootstrapping the estimate by RLS (see Example 5.6).
The noise variance is estimated by the LMS algorithm (5.63), with an in-
itial τ − 0 = 0, and a step size that follows the MAM update (5.76). Thus,
the NLMS classifier uses the same techniques for weight vector and noise
variance estimation as XCS(F), with the only difference that the correct
variance rather than the mean absolute error (5.75) is estimated (see also
Sect. 5.3.7). Hence, the performance of NLMS classifiers reflects the perfor-
mance of classifiers in XCS(F).
RLSLMS Classifier. The weight vector is estimated by the RLS algorithm, using
(5.34) and (5.35), with initialisation w 0 = 0 and Λ − 0 = 1000 I . The noise
variance is estimated by the LMS algorithm, just as for the NLMS Classifier.
This setup conforms to XCSF classifiers with RLS as first introduced by
Lanzi et al. [142, 143].
RLS Classifier. As before, the weight vector is estimated by the RLS algorithm
(5.34) and (5.35), with initialisation w 0 = 0 and Λ − 0 = 1000 I . The noise
variance is estimated by tracking the sum of squared errors according to
(5.68) and evaluating (5.63) for the unbiased variance estimate.
In both experiments, all three classifiers are used for the same regression task,
with the assumption that they match all inputs, that is, m ( x n ) = 1 for all n> 0.
Their performance of estimating the weight vector is measured by the MSE of
their model evaluated with respect to the target function f over 1000 inputs
that are evenly distributed over the function's domain, using (5.11). The quality
of the estimate noise variance is evaluated by its squared error when compared
to the unbiased noise variance estimate (5.13) of a linear model trained by (5.8)
over 1000 observations that are evenly distributed over the function's domain.
For the first experiment, averaging classifiers with x n =1forall n> 0are
used to estimate weight and noise variance of the noisy target function f 1 ( x )=
5+

(0 , 1). Hence, the correct weight estimate is w = 5, with noise variance
τ − 1 = 1. As the function output is independent of its input, its domain does not

N