Combining the Perceptron Algorithm with Logarithmic Simulated Annealing

摘要

We present results of computational experiments with an extension of the Perceptron algorithm by a special type of simulated annealing. The simulated annealing procedure employs a logarithmic cooling schedule \(c\left( k \right) = \Gamma /\ln \left( {k + 2} \right)\), where \(\Gamma\) is a parameter that depends on the underlying configuration space. For sample sets S of n-dimensional vectors generated by randomly chosen polynomials \(w_1 \cdot x_1^{a_1 } + \cdot \cdot \cdot + w_n \cdot x_n^{a_n } \geqslant \vartheta\), we try to approximate the positive and negative examples by linear threshold functions. The approximations are computed by both the classical Perceptron algorithm and our extension with logarithmic cooling schedules. For \(n{\text{ = }}{\kern 1pt} {\kern 1pt} {\text{256,}}...{\text{,1024}}\) and \(a_i = 3,...,7\), the extension outperforms the classical Perceptron algorithm by about 15% when the sample size is sufficiently large. The parameter Γ was chosen according to estimations of the maximum escape depth from local minima of the associated energy landscape.\(\Gamma \)