High Probability Convergence of Adam Under Unbounded Gradients and Affine Variance Noise

TL;DR

This paper proves that Adam algorithm converges with high probability in non-convex stochastic optimization under affine variance noise without requiring bounded gradients, providing theoretical guarantees for practical scenarios.

Contribution

It establishes high probability convergence of Adam under affine variance noise without strong assumptions, expanding its theoretical understanding in real-world applications.

Findings

Adam converges to stationary points with high probability at a rate of O(poly(log T)/√T).

Adam's gradients are confined within an order of O(poly(log T)).

A simplified Adam variant also achieves adaptive convergence based on noise level.

Abstract

In this paper, we study the convergence of the Adaptive Moment Estimation (Adam) algorithm under unconstrained non-convex smooth stochastic optimizations. Despite the widespread usage in machine learning areas, its theoretical properties remain limited. Prior researches primarily investigated Adam's convergence from an expectation view, often necessitating strong assumptions like uniformly stochastic bounded gradients or problem-dependent knowledge in prior. As a result, the applicability of these findings in practical real-world scenarios has been constrained. To overcome these limitations, we provide a deep analysis and show that Adam could converge to the stationary point in high probability with a rate of $\mathcal{O}\left({\rm poly}(\log T)/\sqrt{T}\right)$ under coordinate-wise "affine" variance noise, not requiring any bounded gradient assumption and any problem-dependent knowledge in prior to tune hyper-parameters. Additionally, it is revealed that Adam confines its gradients' magnitudes within an order of $\mathcal{O}\left({\rm poly}(\log T)\right)$. Finally, we also investigate a simplified version of Adam without one of the corrective terms and obtain a convergence rate that is adaptive to the noise level.