Probability of divergence for the least-mean fourth algorithm

In this paper, it is shown that the least-mean fourth (LMF) adaptive algorithm is not mean-square stable when the regressor input is not strictly bounded (as happens, for example, if the input has a Gaussian distribution). For input distributions with infinite support, even for the Gaussian distribution, the LMF always has a nonzero probability of divergence, no matter how small the step-size is chosen. This result is proven for a slight modification of the Gaussian distribution in a one-tap filter and corroborated with several simulations. In addition, an upper bound is given for the probability of divergence of LMF as a function of the filter length, input power, step-size, and noise variance, for the case of Gaussian regressors. The results reported in this paper provide tools for designers to better understand the behavior of the LMF algorithm and decide on the convenience or not of its use for a given application.