Adam-HNAG: A Convergent Reformulation of Adam with Accelerated Rate

TL;DR

This paper introduces Adam-HNAG, a new convergent reformulation of Adam that combines variable splitting and gradient correction, providing the first convergence proof for Adam-type methods in convex optimization.

Contribution

It develops a unified Lyapunov analysis framework for Adam-HNAG, establishing convergence guarantees and accelerated rates in convex smooth settings.

Findings

Adam-HNAG achieves accelerated convergence in convex optimization.

Numerical experiments validate the theoretical convergence and empirical differences.

First convergence proof for Adam-type methods in convex optimization.

Abstract

Adam has achieved strong empirical success, but its theory remains incomplete even in the deterministic full-batch setting, largely because adaptive preconditioning and momentum are tightly coupled. In this work, a convergent reformulation of full-batch Adam is developed by combining variable and operator splitting with a curvature-aware gradient correction. This leads to a continuous-time Adam-HNAG flow with an exponentially decaying Lyapunov function, as well as two discrete methods: Adam-HNAG, and Adam-HNAG-s, a synchronous variant closer in form to Adam. Within a unified Lyapunov analysis framework, convergence guarantees are established for both methods in the convex smooth setting, including accelerated convergence. Numerical experiments support the theory and illustrate the different empirical behavior of the two discretizations. To the best of our knowledge, this provides the first convergence proof for Adam-type methods in convex optimization.