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

The asymptotic analysis of a generic stochastic optimization algorithm mainly relies on the establishment of a specific descent condition. While the convexity assumption allows for technical shortcuts and generally leads to strict convergence, dealing with the non-convex framework, on the contrary, requires the use of specific results as those relying on the Kurdyka-Lojasiewicz (KL) theory. While such tools have become popular in the field of deterministic optimisation, they are much less widespread in the stochastic context and, in this case, the few works making use of them are essentially based on trajectory-by-trajectory approaches. In this paper, we propose a new methodology, also based on KL theory, for deeper asymptotic investigations on a stochastic scheme verifying a descent condition. The specificity of our work is here to be of macroscopic nature insofar as our strategy of proof is more in-expectation-based and therefore seems more natural typically with respect to the noise properties, of conditional order, encountered in the stochastic literature nowadays.