CONVERGENCE OF STOCHASTIC ALGORITHMS - FROM THE KUSHNER-CLARK THEOREMTO THE LYAPOUNOV FUNCTIONAL METHOD

In the first part of this paper a global Kushner-Clark theorem about t he convergence of stochastic algorithms is proved: we show that, under some natural assumptions, one can 'read' from the trajectories of its ODE whether or not an algorithm converges. The classical stochastic o ptimization results are included in this theorem. In the second part, the above smoothness assumption on the mean vector field of the algori thm is relaxed using a new approach based on a path-dependent Lyapouno v functional. Several applications, for non-smooth mean vector fields and/or bounded Lyapounov function settings, are derived. Examples and simulations are provided that illustrate and enlighten the field of ap plication of the theoretical results.