Stochastic Differential Equations for Modeling First Order Optimization Methods

. In this article, a family of SDEs are derived as a tool to understand the behavior of numerical optimization methods under random evaluations of the gradient. Our objective is to transpose the introduction of continuous versions through ODEs to understand the asymptotic behavior of a discrete optimization scheme to the stochastic setting. We consider a continuous version of the stochastic gradient scheme and of a stochastic inertial system. This article first studies the quality of the approximation of the discrete scheme by an SDE when the step size tends to 0. Then, it presents new asymptotic bounds on the values \ (F (Xₜ) -F^*\), where \ (Xₜ\) is a solution of the SDE and \ (F^*= F\), when \ (F\) is convex and under integrability conditions on the noise. Results are provided under two sets of hypotheses: first considering \ (C²\) and convex functions and then adding some geometrical properties of \ (F\). All of these results provide insight on the behavior of these inertial and perturbed algorithms in the setting of stochastic algorithms. KeywordsLyapunov functionsrate of convergenceSDEsoptimizationgeometrical properties of the objectiveMSC codes30D1034F0565K1090C2590C30