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

TL;DR

This paper analyzes the convergence and dynamical behavior of the Adam optimization algorithm in non-convex stochastic settings, introducing a continuous-time approximation and a decreasing stepsize variant, with theoretical guarantees and fluctuation analysis.

Contribution

It provides a novel continuous-time ODE framework for Adam, establishes convergence results, and introduces a decreasing stepsize version with almost sure convergence.

Findings

Convergence of Adam iterates to stationary points under stability conditions.

Weak convergence of the interpolated Adam process to the ODE solution.

Almost sure convergence of the decreasing stepsize Adam to critical points.

Abstract

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.