Convergence and dynamical behavior of the Adam algorithm for nonconvex stochastic optimization

Adam is a popular variant of stochastic gradient descent for finding a local minimizer of a function. In the constant stepsize regime, assuming that the objective function is differentiable and nonconvex, we establish the convergence in the long run of the iterates to a stationary point under a stability condition. The key ingredient is the introduction of a continuous-time version of Adam, under the form of a nonautonomous ordinary differential equation. This continuous-time system is a relevant approximation of the Adam iterates, in the sense that the interpolated Adam process converges weakly toward the solution to the ODE. The existence and the uniqueness of the solution are established. We further show the convergence of the solution toward the critical points of the objective function and quantify its convergence rate under a Lojasiewicz \ assumption. Then, we introduce a novel decreasing stepsize version of Adam. Under mild assumptions, it is shown that the iterates are almost surely bounded and 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.