A new regret analysis for Adam-type algorithms

TL;DR

This paper introduces a new theoretical framework that bridges the gap between practical usage and theoretical guarantees of Adam-type algorithms by enabling optimal regret bounds with constant momentum parameters.

Contribution

The authors develop a novel analysis method that provides regret guarantees for Adam variants with fixed momentum parameters, aligning theory with practical implementations.

Findings

Achieves data-dependent regret bounds with constant $eta_1$

Applies to a wide range of Adam-type algorithms

Removes the need for decaying $eta_1$ in theoretical analysis

Abstract

In this paper, we focus on a theory-practice gap for Adam and its variants (AMSgrad, AdamNC, etc.). In practice, these algorithms are used with a constant first-order moment parameter $\beta_{1}$ (typically between $0.9$ and $0.99$). In theory, regret guarantees for online convex optimization require a rapidly decaying $\beta_{1}\to0$ schedule. We show that this is an artifact of the standard analysis and propose a novel framework that allows us to derive optimal, data-dependent regret bounds with a constant $\beta_{1}$, without further assumptions. We also demonstrate the flexibility of our analysis on a wide range of different algorithms and settings.