Momentum LMS Theory beyond Stationarity: Stability, Tracking, and Regret

TL;DR

This paper analyzes the Momentum LMS algorithm's stability, tracking, and regret bounds in nonstationary data streams, demonstrating its effectiveness for real-time adaptive filtering in complex, time-varying environments.

Contribution

It provides the first theoretical analysis of MLMS's stability and performance bounds in nonstationary settings, extending classical LMS theory to include momentum effects.

Findings

MLMS achieves rapid adaptation in nonstationary environments

Theoretical bounds on tracking and regret are established for MLMS

Experiments confirm MLMS's robustness and effectiveness in real-world data streams

Abstract

In large-scale data processing scenarios, data often arrive in sequential streams generated by complex systems that exhibit drifting distributions and time-varying system parameters. This nonstationarity challenges theoretical analysis, as it violates classical assumptions of i.i.d. (independent and identically distributed) samples, necessitating algorithms capable of real-time updates without expensive retraining. An effective approach should process each sample in a single pass, while maintaining computational and memory complexities independent of the data stream length. Motivated by these challenges, this paper investigates the Momentum Least Mean Squares (MLMS) algorithm as an adaptive identification tool, leveraging its computational simplicity and online processing capabilities. Theoretically, we derive tracking performance and regret bounds for the MLMS in time-varying stochastic linear systems under various practical conditions. Unlike classical LMS, whose stability can be characterized by first-order random vector difference equations, MLMS introduces an additional dynamical state due to momentum, leading to second-order time-varying random vector difference equations whose stability analysis hinges on more complicated products of random matrices, which poses a substantially challenging problem to resolve. Experiments on synthetic and real-world data streams demonstrate that MLMS achieves rapid adaptation and robust tracking, in agreement with our theoretical results especially in nonstationary settings, highlighting its promise for modern streaming and online learning applications.