Excursions of adaptive algorithms via the Poisson clumping heuristic

The authors detail the application of the Poisson clumping heuristic (PCH) to the least mean square (LMS) adaptive algorithm and its signed variants. Under certain conditions on the input and disturbance statistics, the parameter estimate errors form a Markov process. The PCH asserts that large excursions of the parameter estimates occur in clumps, and that these clumps are distributed in a Poisson manner with parameter λ_b, where b is the maximum parameter error. Expressions are derived for each of the four algorithms in the scalar case, which allows calculation of λ _b in a relatively straightforward manner. These values are compared to simulations of the algorithms. Given that the results are asymptotic in b, the close agreement between simulated and theoretical values is striking, even for very modest b. The four algorithms are then compared in terms of λ_b. Some observations are made regarding the relative performance of the four variants. No single LMS variant always outperforms the others. Suggestions are made as to how this technique might be applied in the vector case, and a crucial monotonicity property is verified