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The asymptotic analysis of a generic stochastic optimization algorithm mainlyrelies on the establishment of a specific descent condition. While theconvexity assumption allows for technical shortcuts and generally leads tostrict convergence, dealing with the non-convex framework, on the contrary,requires the use of specific results as those relying on theKurdyka-Lojasiewicz (KL) theory. While such tools have become popular in thefield of deterministic optimisation, they are much less widespread in thestochastic context and, in this case, the few works making use of them areessentially based on trajectory-by-trajectory approaches. In this paper, wepropose a new methodology, also based on KL theory, for deeper asymptoticinvestigations on a stochastic scheme verifying a descent condition. Thespecificity of our work is here to be of macroscopic nature insofar as ourstrategy of proof is more in-expectation-based and therefore seems more naturaltypically with respect to the noise properties, of conditional order,encountered in the stochastic literature nowadays.

A new use of the Kurdyka-Lojasiewicz property to study asymptotic  behaviours of some stochastic optimization algorithms in a non-convex  differentiable framework