Asymptotic study of stochastic adaptive algorithm in non-convex landscape

This paper studies some asymptotic properties of adaptive algorithms widely\nused in optimization and machine learning, and among them Adagrad and Rmsprop,\nwhich are involved in most of the blackbox deep learning algorithms. Our setup\nis the non-convex landscape optimization point of view, we consider a one time\nscale parametrization and we consider the situation where these algorithms may\nbe used or not with mini-batches. We adopt the point of view of stochastic\nalgorithms and establish the almost sure convergence of these methods when\nusing a decreasing step-size point of view towards the set of critical points\nof the target function. With a mild extra assumption on the noise, we also\nobtain the convergence towards the set of minimizer of the function. Along our\nstudy, we also obtain a "convergence rate" of the methods, in the vein of the\nworks of \\cite{GhadimiLan}.\n