Asymptotic Analysis of the LMS Algorithm with Momentum

A widely studied filtering algorithm in signal processing is the least mean square (LMS) method, due to B. Widrow and T. Hoff, 1960. A popular extension of the LMS algorithm, which is also important in deep learning, is the LMS method with momentum, originated by S. Roy and J.J. Shynk back in 1988. This is a fixed gain (or constant step-size) version of the LMS method modified by an additional momentum term that is proportional to the last correction term. Recently, a certain equivalence of the two methods has been rigorously established by K. Yuan, B. Ying and A.H. Sayed, assuming martingale difference gradient noise. The purpose of this paper is to present the outline of a significantly simpler and more transparent asymptotic analysis of the LMS algorithm with momentum under the assumption of stationary, ergodic and mixing signals.