  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 non-convex, 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 non-autonomous 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 towards the solution to the ODE. The existence and the uniqueness of the
solution are established. We further show the convergence of the solution
towards 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.
