Convergence and Dynamical Behavior of the ADAM Algorithm for Nonconvex Stochastic Optimization

Adam is a popular variant of stochastic gradient descent forfinding a local minimizer of a function. In the constant stepsize regime,assuming that the objective function is differentiable and nonconvex, weestablish the convergence in the long run of the iterates to a stationarypoint under a stability condition. The key ingredient is the introductionof a continuous-time version of Adam, under the form of a nonautonomousordinary differential equation. This continuous-time system is a relevantapproximation of the Adam iterates, in the sense that the interpolatedAdam process converges weakly toward the solution to the ODE. Theexistence and the uniqueness of the solution are established. We furthershow the convergence of the solution toward the critical points of theobjective function and quantify its convergence rate under a Łojasiewicz assumption.Then, we introduce a novel decreasing stepsize version of Adam.Under mild assumptions, it is shown that the iterates are almost surely boundedand converge almost surely to critical points of the objective function. Finally,we analyze the fluctuations of the algorithm by means of a conditional central limit theorem.