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Stochastic differentiable approximation schemes are widely used for solvinghigh dimensional problems. Most of existing methods satisfy some desirableproperties, including conditional descent inequalities, and almost sure (a.s.)convergence guarantees on the objective function, or on the involved gradient.However, for non-convex objective functions, a.s. convergence of the iterates,i.e., the stochastic process, to a critical point is usually not guaranteed,and remains an important challenge. In this article, we develop a framework tobridge the gap between descent-type inequalities and a.s. convergence of theassociated stochastic process. Leveraging a novel Kurdyka-Lojasiewicz property,we show convergence guarantees of stochastic processes under mild assumptionson the objective function. We also provide examples of stochastic algorithmsbenefiting from the proposed framework and derive a.s. convergence guaranteeson the iterates.

A stochastic use of the Kurdyka-Lojasiewicz property: Investigation of optimization algorithms behaviours in a non-convex differentiable framework