Lanzi Pier Luca, Loiacono Daniele, Wilson Stewart W, Goldberg David E
Dipartimento di Elettronica e Informazione, Politecnico di Milano, Milano, I-20133, Italy.
Evol Comput. 2007 Summer;15(2):133-68. doi: 10.1162/evco.2007.15.2.133.
We analyze generalization in XCSF and introduce three improvements. We begin by showing that the types of generalizations evolved by XCSF can be influenced by the input range. To explain these results we present a theoretical analysis of the convergence of classifier weights in XCSF which highlights a broader issue. In XCSF, because of the mathematical properties of the Widrow-Hoff update, the convergence of classifier weights in a given subspace can be slow when the spread of the eigenvalues of the autocorrelation matrix associated with each classifier is large. As a major consequence, the system's accuracy pressure may act before classifier weights are adequately updated, so that XCSF may evolve piecewise constant approximations, instead of the intended, and more efficient, piecewise linear ones. We propose three different ways to update classifier weights in XCSF so as to increase the generalization capabilities of XCSF: one based on a condition-based normalization of the inputs, one based on linear least squares, and one based on the recursive version of linear least squares. Through a series of experiments we show that while all three approaches significantly improve XCSF, least squares approaches appear to be best performing and most robust. Finally we show how XCSF can be extended to include polynomial approximations.
我们分析了XCSF中的泛化并引入了三项改进。我们首先表明，XCSF所演化出的泛化类型会受到输入范围的影响。为了解释这些结果，我们对XCSF中分类器权重的收敛进行了理论分析，这凸显了一个更广泛的问题。在XCSF中，由于Widrow-Hoff更新的数学特性，当与每个分类器相关联的自相关矩阵的特征值分布较大时，给定子空间中分类器权重的收敛可能会很慢。作为一个主要后果，系统的准确性压力可能在分类器权重得到充分更新之前就起作用，这样XCSF可能会演化出分段常数近似，而不是预期的、更有效的分段线性近似。我们提出了三种不同的方法来更新XCSF中的分类器权重，以提高XCSF的泛化能力：一种基于输入的条件归一化，一种基于线性最小二乘法，还有一种基于线性最小二乘法的递归版本。通过一系列实验，我们表明虽然所有这三种方法都显著改进了XCSF，但最小二乘法似乎表现最佳且最稳健。最后，我们展示了如何将XCSF扩展以包括多项式近似。
