Local Convergence of Adaptive Gradient Descent Optimizers

TL;DR

This paper provides a local convergence analysis for ADAM and other adaptive gradient descent algorithms, establishing necessary hyperparameter conditions for convergence in deterministic settings.

Contribution

It introduces a method for local convergence analysis of ADAM and similar optimizers, deriving hyperparameter bounds for convergence in non-convex, twice differentiable functions.

Findings

ADAM converges locally under specific hyperparameter conditions

The analysis applies to other adaptive gradient methods

Most examined algorithms show local convergence with proper hyperparameters

Abstract

Adaptive Moment Estimation (ADAM) is a very popular training algorithm for deep neural networks and belongs to the family of adaptive gradient descent optimizers. However to the best of the authors knowledge no complete convergence analysis exists for ADAM. The contribution of this paper is a method for the local convergence analysis in batch mode for a deterministic fixed training set, which gives necessary conditions for the hyperparameters of the ADAM algorithm. Due to the local nature of the arguments the objective function can be non-convex but must be at least twice continuously differentiable. Then we apply this procedure to other adaptive gradient descent algorithms and show for most of them local convergence with hyperparameter bounds.