🤖 AI Summary
This work addresses the challenge of simultaneously achieving real-time responsiveness, stability, and tracking performance in nonstationary data streams, where conventional adaptive algorithms often fall short. Focusing on the Momentum Least Mean Squares (MLMS) algorithm, the study establishes, for the first time, rigorous stability conditions, tracking error bounds, and dynamic regret bounds under time-varying stochastic linear systems. The analysis overcomes the significant theoretical difficulty posed by second-order products of stochastic matrices induced by the momentum term. By integrating tools from stochastic vector difference equations, nonstationary time series modeling, and online learning theory, the paper demonstrates that MLMS exhibits both rapid adaptability and robust tracking capability. Empirical evaluations on synthetic and real-world data streams confirm that MLMS significantly outperforms the classical LMS algorithm.
📝 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.