Adaptive projected subgradient method - a unified view of projection based adaptive filtering algorithms

摘要

This paper presents an efficient numerical algorithm named adaptive projected subgradient method for minimizing asymptotically a certain class of sequences of nonnegative convex functions. The proposed algorithm is a natural extension of the Polyak's subgradient algorithm with a fixed target value, for unsmooth convex optimization problem, to the case where the convex objective itself keeps changing in the whole process. A main theorem on the proposed algorithm can serve as a useful mathematical foundation of a wide range of Projection based adaptive filtering algorithms. Indeed, by designing certain sequences of convex objectives, a variety of adaptive filtering algorithms are derived in a unified manner as simple examples of the adaptive projected subgradient method. These include not only the existing adaptive filtering techniques e.g., NLMS, Projected NLMS Constrained NLMS, APA, and Adaptive parallel outer projection algorithm etc, but also new techniques e.g., Adaptive parallel min-max projection algorithm, and their embedded constraint versions. These new techniques are well-suited for nowadays applications to robust acoustic signal processing as well as to adaptive array signal processing.