Convergence and Dynamical Behavior of the Adam Algorithm for Non Convex Stochastic Optimization

Abstract : 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. 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. The key ingredient is the introduction of a continuous-time version of Adam, under the form of a non-autonomous ordinary differential equation. 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. 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.
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Contributor : Anas Barakat <>
Submitted on : Friday, November 15, 2019 - 6:05:47 PM
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Anas Barakat, Pascal Bianchi. Convergence and Dynamical Behavior of the Adam Algorithm for Non Convex Stochastic Optimization. 2019. ⟨hal-02366280⟩

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