A new regret analysis for Adam-type algorithms

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.