Fastest Global Convergence Analysis of NLMS: A Matrix-Algebraic Approach

The normalized least-mean-square (NLMS) algorithm is widely used in adaptive filtering due to its simplicity and robustness. Conventional convergence analyses of NLMS focus on one-step update behavior, and characterize step-size optimality only in a local sense. However, a locally optimal step-size does not necessarily yield the best global mean-square error (MSE) trajectory, which practically governs the overall convergence performance. This letter derives a closed-form characterization of the global mean-square error trajectory and establishes a comparison framework using the Loewner partial order. Under a commutativity condition on the relevant matrices, this letter proves that among all variable step-size LMS algorithms with step sizes $0 < \mu _{k} < 2\Vert \mathbf {x}_{k}\Vert ^{2}$, NLMS with $\mu _{k}^\star = 1\Vert \mathbf {x}_{k}\Vert ^{2}$ provides the fastest global convergence speed, which theoretically justifies the empirical numerical observations of fast convergence behavior for NLMS.