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This paper analyzes the Momentum Least Mean Squares (MLMS) algorithm in nonstationary environments, deriving tracking performance and regret bounds for time-varying stochastic linear systems. The analysis addresses the challenge of analyzing MLMS stability, which involves second-order time-varying random vector difference equations, unlike the first-order equations in classical LMS. Experimental results on synthetic and real-world data validate the theoretical findings, demonstrating MLMS's rapid adaptation and robust tracking capabilities in nonstationary settings.
Momentum LMS offers surprisingly robust tracking and adaptation in non-stationary environments, despite the added complexity of analyzing its stability compared to standard LMS.
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.
