# Convergence Analysis of a Momentum Algorithm with Adaptive Step Size for Non Convex Optimization

arXiv: https://arxiv.org/abs/1911.07596 · HTML: https://mlsanity.com/paper/1911.07596/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Nov 2019.

## Built on by (most cited first, 3 of 3)

- [Uniform a priori bounds and error analysis for the Adam stochastic gradient descent optimization method](https://mlsanity.com/paper/2603.18899/) (2026)
- [Unified convergence analysis for gradient descent optimization methods in the training of deep neural networks](https://mlsanity.com/paper/2607.04233/) (2026)
- [Variational Stochastic Gradient Descent for Deep Neural Networks](https://mlsanity.com/paper/2404.06549/) (2024)

## Abstract (arXiv)

Although ADAM is a very popular algorithm for optimizing the weights of neural networks, it has been recently shown that it can diverge even in simple convex optimization examples. Several variants of ADAM have been proposed to circumvent this convergence issue. In this work, we study the ADAM algorithm for smooth nonconvex optimization under a boundedness assumption on the adaptive learning rate. The bound on the adaptive step size depends on the Lipschitz constant of the gradient of the objective function and provides safe theoretical adaptive step sizes. Under this boundedness assumption, we show a novel first order convergence rate result in both deterministic and stochastic contexts. Furthermore, we establish convergence rates of the function value sequence using the Kurdyka-Lojasiewicz property.