The P-Norn Generalization of the LMS Algorithm for Adaptive Filtering

Abstract Recently much work has been done analyzing online machine learning algorithms in a worst case setting, where no probabilistic assumptions are made about the data. This is analogous to the H ∞ setting used in adaptive linear filtering. Bregman divergences have become a standard tool for analyzing online Machine Learning algorithms. Using these divergences, we motivate a generalization of the the Least Mean Squared (LMS) algorithm. The loss bounds for these so-called p-norm algorithms involve other norms than the standard 2-norm. The bounds can be significantly better if a large proportion of the input variables are irrelevant, i.e., if the weight vector we are trying to learn is sparse. We also prove result for nonstationary targets. We only know how to apply kernel methods to the standard LMS algorithm (i.e., p = 2). However even in the general p-norm case we can handle generalized linear models where the output of the system is a linear function combined with a nonlinear transfer function (e.g., the logistic sigmoid).