Training the Classifiers - Design and Analysis of Learning Classifier Systems - page 86

Theorem 5.7 (Incremental Sum of Squared Error Update). Let the se-
quence of weight vector estimates { w 1 , w 2 ,... } satisfy the Principle of Ortho-

gonality (5.16). Then

2
M N +1

X N +1 w N +1 − y N +1 (5.68)

M N + m ( x N +1 )( w N x N +1 −

y N +1 )( w N +1 x N +1 −

2

= X N w N − y N y N +1 )

holds.

An almost equal derivation reveals that the sum of squared errors for the recency-
weighted RLS variant is given by

2
M N +1

X N +1 w N +1 − y N +1

= λ m ( x N +1 )

2 M N

X N w N − y N

+ m ( x N +1 )( w N x N +1 − y N +1 )( w N +1 x N +1 −

y N +1 ) , (5.69)

where, when compared to (5.68), the current sum of squared errors is additionally
discounted.
In summary, the unbiased noise precision estimate can be tracked by directly
solving (5.63), where the match count is updated by

c N +1 = c N + m ( x N +1 ) , (5.70)

and the sum of squared errors is updated by (5.68). As Theorem 5.7 states,
(5.68) is only valid if the Principle of Orthogonality holds. However, using the
computationally cheaper RLS implementation that involves (5.35) introduces an
initial bias and hence violates the Principle of Orthogonality. Nonetheless, if δ in
Λ − 0 = δ I is set to a very large positive scalar, this bias is negligible, and hence
(5.68) is still applicable with only minor inaccuracy.

Example 5.8 (Noise Precision Estimation for Averaging Classifiers). Consider
averaging classifiers, such that x n =1forall n> 0. Given the use of gradient-
based methods to estimate the weight vector violates the Principle of Orthogo-
nality, and hence (5.65) has to be used estimate the noise precision, resulting in

N +1 = τ − N + m ( x N +1 ) ( w N +1 −

τ − N . τ − 1 y N +1 ) 2 − (5.71)

Alternatively, we can use the RLS algorithm (5.46) for averaging classifiers, and
use (5.68) to accurately track the noise precision by

τ − 1

N +1 = τ − 1

+ m ( x N +1 )( w N − y N +1 )( w N +1 − y N +1 ) . (5.72)

N

Note that while the computational cost of both approaches is equal (in its ap-
plication to averaging classifiers), the second approach is vastly superior in its
weight vector and noise precision estimation accuracy and should therefore be
always preferred.