Convergence of the ADAM algorithm from a Dynamical System Viewpoint

10/04/2018

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by   Anas Barakat, et al.

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Adam is a popular variant of the stochastic gradient descent for finding a local minimizer of a function. The objective function is unknown but a random estimate of the current gradient vector is observed at each round of the algorithm. This paper investigates the dynamical behavior of Adam when the objective function is non-convex and differentiable. We introduce a continuous-time version of Adam, under the form of a non-autonomous ordinary differential equation (ODE). The existence and the uniqueness of the solution are established, as well as the convergence of the solution towards the stationary points of the objective function. It is also proved that the continuous-time system is a relevant approximation of the Adam iterates, in the sense that the interpolated Adam process converges weakly to the solution to the ODE.