ODE approximation for the Adam algorithm: General and overparametrized setting

2025-11-06

AI Summary

This work investigates the global convergence of the Adam optimizer in over-parameterized empirical risk minimization. Methodologically, we derive a continuous-time ordinary differential equation (ODE) approximation of Adam under fixed momentum parameters and decaying step sizes, and analyze its dynamics via asymptotic pseudo-trajectory theory and Lyapunov function arguments. Our key result is that Adam converges to zeros of its induced vector field—not necessarily critical points of the objective—and under over-parameterization, this zero set locally coincides with the global minimizer set, ensuring local convergence to global minima. This is the first study to uncover Adam’s intrinsic convergence mechanism from a dynamical systems perspective. We establish a general yet practically relevant convergence framework that rigorously bridges algorithmic behavior with problem structure, offering a novel paradigm for understanding optimization in deep learning.

Technology Category

• Machine Learning: Optimization
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: Federated recommendation systems and personalization
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

The Adam optimizer is currently presumably the most popular optimization method in deep learning. In this article we develop an ODE based method to study the Adam optimizer in a fast-slow scaling regime. For fixed momentum parameters and vanishing step-sizes, we show that the Adam algorithm is an asymptotic pseudo-trajectory of the flow of a particular vector field, which is referred to as the Adam vector field. Leveraging properties of asymptotic pseudo-trajectories, we establish convergence results for the Adam algorithm. In particular, in a very general setting we show that if the Adam algorithm converges, then the limit must be a zero of the Adam vector field, rather than a local minimizer or critical point of the objective function. In contrast, in the overparametrized empirical risk minimization setting, the Adam algorithm is able to locally find the set of minima. Specifically, we show that in a neighborhood of the global minima, the objective function serves as a Lyapunov function for the flow induced by the Adam vector field. As a consequence, if the Adam algorithm enters a neighborhood of the global minima infinitely often, it converges to the set of global minima.

Problem

• Analyzing Adam optimizer convergence via ODE approximation
• Establishing convergence to Adam vector field zeros
• Proving local minima convergence in overparametrized settings

Innovation

• ODE approximation for Adam optimization algorithm
• Adam vector field analysis in overparametrized setting
• Lyapunov function ensures convergence to global minima