Abstract
We give a development of the ODE method for the analysis of recursive algorithms described by a stochastic recursion. With variability modeled via an underlying Markov process, and under general assumptions, the following results are obtained: (i) Stability of an associated ODE implies that the stochastic recursion is stable in a strong sense when a gain parameter is small. (ii) The range of gain-values is quantified through a spectral analysis of an associated linear operator, providing a non-local theory, even for nonlinear systems. (iii) A second-order analysis shows precisely how variability leads to sensitivity of the algorithm with respect to the gain parameter. All results are obtained within the natural operator-theoretic framework of geometrically ergodic Markov processes.
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