Boţ-Nguyen Acceleration, Weighted Mean Ergodic Iteration, and the Beta-Binomial Distribution
This provides theoretical insights into a recent acceleration method for a restricted class of operators, but the results are incremental and limited to linear cases.
The paper analyzes a specific instance of the Boţ-Nguyen acceleration algorithm for linear nonexpansive operators, showing it fits into weighted mean ergodic iterations and that the weak limit is the projection onto the fixed point set. The weights relate to the beta-binomial distribution, and strong convergence is obtained for a parameter value of 4.
In 2023, Boţ and Nguyen introduced a new class of accelerated algorithms for finding a fixed point of a nonexpansive operator as the weak limit of a sequence. In this paper, we analyze a particular instance of their algorithm when the nonexpansive operator is assumed to be linear. Surprisingly, the Boţ-Nguyen acceleration then fits naturally into the framework of weighted mean ergodic iterations. This allows us to identify the weak limit as the projection of the starting point onto the fixed point set. Moreover, the weights involved are closely related to the beta-binomial distribution. Finally, when the parameter is equal to 4, then we obtain strong convergence of the iterates.