Robust stability analysis of adaptation algorithms for single perceptron

The problem of robust stability and convergence of learning parameters of adaptation algorithms in a noisy environment for the single preceptron is addressed. The case in which the same input pattern is presented in the adaptation cycle is analyzed. The algorithm proposed is of the Widrow-Hoff type. It is concluded that this algorithm is robust. However, the weight vectors do not necessarily converge in the presence of measurement noise. A modified version of this algorithm in which the reduction factors are allowed to vary with time is proposed, and it is shown that this algorithm is robust and that the weight vectors converge in the presence of bounded noise. Only deterministic-type arguments are used in the analysis. An ultimate bound on the error in terms of a convex combination of the initial error and the bound on the noise is obtained.