Convergence of Adam for Non-convex Objectives: Relaxed Hyperparameters and Non-ergodic Case

TL;DR

This paper provides a comprehensive theoretical analysis of Adam's convergence in non-convex stochastic optimization, introducing relaxed hyperparameter conditions and establishing non-ergodic convergence results, including last-iterate convergence.

Contribution

It introduces weaker conditions for Adam's convergence, proves last-iterate convergence in non-convex settings, and establishes non-ergodic convergence rates under the PL condition.

Findings

Almost sure ergodic convergence rate close to o(1/√K)

First proof of last-iterate convergence to stationary points

Non-ergodic convergence rate of O(1/K) under PL condition

Abstract

Adam is a commonly used stochastic optimization algorithm in machine learning. However, its convergence is still not fully understood, especially in the non-convex setting. This paper focuses on exploring hyperparameter settings for the convergence of vanilla Adam and tackling the challenges of non-ergodic convergence related to practical application. The primary contributions are summarized as follows: firstly, we introduce precise definitions of ergodic and non-ergodic convergence, which cover nearly all forms of convergence for stochastic optimization algorithms. Meanwhile, we emphasize the superiority of non-ergodic convergence over ergodic convergence. Secondly, we establish a weaker sufficient condition for the ergodic convergence guarantee of Adam, allowing a more relaxed choice of hyperparameters. On this basis, we achieve the almost sure ergodic convergence rate of Adam, which is arbitrarily close to $o(1/\sqrt{K})$. More importantly, we prove, for the first time, that the last iterate of Adam converges to a stationary point for non-convex objectives. Finally, we obtain the non-ergodic convergence rate of $O(1/K)$ for function values under the Polyak-Lojasiewicz (PL) condition. These findings build a solid theoretical foundation for Adam to solve non-convex stochastic optimization problems.