Uniform a priori bounds and error analysis for the Adam stochastic gradient descent optimization method

2026-03-19

AI Summary

This work addresses a critical gap in the theoretical understanding of the Adam optimizer in strongly convex stochastic optimization, where existing convergence analyses rely on the restrictive assumption that the solution sequence remains bounded and lack unconditional error bounds. For the first time, this study establishes a uniform priori boundedness of the Adam iterates, thereby removing the need for such an assumption. By leveraging tools from stochastic optimization theory, recursive inequalities, and uniform estimation techniques, the authors rigorously prove the unconditional convergence of Adam for a broad class of strongly convex stochastic problems and derive explicit upper bounds on the optimization error. These results provide a solid theoretical foundation that significantly strengthens the justification for Adam’s widespread practical use.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• User Modeling, Personalization and Recommendation: User modeling for targeted and personalized online advertising

Abstract

The adaptive moment estimation (Adam) optimizer proposed by Kingma & Ba (2014) is presumably the most popular stochastic gradient descent (SGD) optimization method for the training of deep neural networks (DNNs) in artificial intelligence (AI) systems. Despite its groundbreaking success in the training of AI systems, it still remains an open research problem to provide a complete error analysis of Adam, not only for optimizing DNNs but even when applied to strongly convex stochastic optimization problems (SOPs). Previous error analysis results for strongly convex SOPs in the literature provide conditional convergence analyses that rely on the assumption that Adam does not diverge to infinity but remains uniformly bounded. It is the key contribution of this work to establish uniform a priori bounds for Adam and, thereby, to provide -- for the first time -- an unconditional error analysis for Adam for a large class of strongly convex SOPs.

Problem

• Adam optimizer
• error analysis
• uniform a priori bounds
• strongly convex stochastic optimization
• stochastic gradient descent

Innovation

• Adam optimizer
• uniform a priori bounds
• error analysis
• strongly convex stochastic optimization
• unconditional convergence