Arithmetic Vs. Expected Mean of Probabilistic Asynchronous Affine Inference

Distributed execution of algorithms over various terminals is a topic that regains increasing popularity; when tolerance to failures is also required, asynchronous operation is brought to the light, while probabilistic asynchronous operation can model the probability of failure for each terminal. This work focuses on the probabilistic asynchronous affine update model, applicable in a wide range of inference algorithms, possibly executed over distributed terminals. The existing literature focuses on the asymptotic properties of the expected mean. Instead, this work offers the asymptotic analysis for the arithmetic mean, utilized for discovering fixed points, as it is the only quantity that can be practically offered experimentally. It is shown that the asymptotic behavior of the arithmetic mean is different than the expected mean's and a sufficient condition is provided for convergence of the arithmetic mean to a fixed point. The lack of necessity for this condition is explained and the subcases, where the arithmetic mean converges, diverges or has an unpredictable behavior, are distinguished. Additionally, cases where the individual iterations never converge (e.g., oscillate infinitely) but their arithmetic mean does and offers fixed point, are also highlighted. This is another concrete example of the arithmetic mean utility. Applications of the affine model are also briefly discussed. Finally, simulations corroborate theoretical findings for various affine model setups.