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On the Convergence of the Inexact Running Krasnosel'skii-Mann Method | | Emiliano Dall'Anese
; Andrea Simonetto
; Andrey Bernstein
; | | Date: |
17 Apr 2019 | | Abstract: | This paper leverages a framework based on averaged operators to tackle the
problem of tracking fixed points associated with maps that evolve over time. In
particular, the paper considers the Krasnosel’skii-Mann method in a settings
where: (i) the underlying map may change at each step of the algorithm, thus
leading to a "running" implementation of the Krasnosel’skii-Mann method; and,
(ii) an imperfect information of the map may be available. An imperfect
knowledge of the maps can capture cases where processors feature a finite
precision or quantization errors, or the case where (part of) the map is
obtained from measurements. The analytical results are applicable to inexact
running algorithms for solving optimization problems, whenever the algorithmic
steps can be written in the form of (a composition of) averaged operators;
examples are provided for inexact running gradient methods and the
forward-backward splitting method. Convergence of the average fixed-point
residual is investigated for the non-expansive case; linear convergence to a
unique fixed-point trajectory is showed in the case of inexact running
algorithms emerging from contractive operators. | | Source: | arXiv, 1904.8469 | | Services: | Forum | Review | PDF | Favorites | |
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