Stability of adaptive network algorithms in multitask environments

Online Proximal Learning Over Jointly Sparse Multitask Networks With $\ell _{\infty, 1}$ Regularization

“…Proof: By iterating the RHS of (72) from time instant n = 0, and proving the convergence of the obtained series, we arrive at condition (85) for step-size to ensure the mean stability of (19). For more details, see Appendix D. Remark 1: Equation (85) provides an upper bound for step-size µ k to ensure the mean stability of the distributed networks with the proximal multitask diffusion LMS algorithm (19). The upper bound is closely related to the second-order statistics R x,k of the input signals.…”

“…From iteration (72), we obtain the following Theorem 1 for the mean stability of proximal multitask diffusion LMS algorithm (19).…”

“…(Mean stability) Assume data model ( 1) and assumption A1 hold. Then for any initial conditions, distributed networks endowed with the proximal multitask diffusion LMS algorithm (19) are stable in the mean, if step-sizes µ k satisfy:…”

“…In [18], the authors address multitask problems over asynchronous networks and carry out a detailed theoretical analysis. In [19], the performance of multitask diffusion networks is analyzed for correlated noise and regressors. In [20], the authors propose a combination framework that aggregates several diffusion strategies.…”

“…where we use the notations E{ w n+1 2 Σ } and E{ w n+1 2 σ } interchangeably. From iteration (95), we obtain the following Theorem 2 to ensure the mean-square stability of the proximal multitask diffusion LMS algorithm (19).…”