Convergence of Adam Under Relaxed Assumptions

TL;DR

This paper proves that Adam converges to stationary points under realistic conditions and introduces a variance-reduced variant with faster convergence, enhancing theoretical understanding of its optimization guarantees.

Contribution

The paper provides the first convergence proof of Adam under relaxed assumptions and proposes a variance-reduced version with improved gradient complexity.

Findings

Adam converges to ε-stationary points with O(ε^{-4}) complexity.

Gradient boundedness along Adam's trajectory is established under generalized smoothness.

A variance-reduced Adam variant achieves O(ε^{-3}) complexity.

Abstract

In this paper, we provide a rigorous proof of convergence of the Adaptive Moment Estimate (Adam) algorithm for a wide class of optimization objectives. Despite the popularity and efficiency of the Adam algorithm in training deep neural networks, its theoretical properties are not yet fully understood, and existing convergence proofs require unrealistically strong assumptions, such as globally bounded gradients, to show the convergence to stationary points. In this paper, we show that Adam provably converges to $\epsilon$-stationary points with ${O}(\epsilon^{-4})$ gradient complexity under far more realistic conditions. The key to our analysis is a new proof of boundedness of gradients along the optimization trajectory of Adam, under a generalized smoothness assumption according to which the local smoothness (i.e., Hessian norm when it exists) is bounded by a sub-quadratic function of the gradient norm. Moreover, we propose a variance-reduced version of Adam with an accelerated gradient complexity of ${O}(\epsilon^{-3})$.