Unified Convergence Analysis of Stochastic Momentum Methods for Convex and Non-convex Optimization

TL;DR

This paper provides a unified convergence analysis framework for stochastic momentum methods in convex and non-convex optimization, offering insights into their similarities, differences, and practical performance in deep learning.

Contribution

It introduces a unified convergence analysis framework for stochastic momentum methods, bridging theory and practice, and compares their empirical performance in deep neural network training.

Findings

Unified framework reveals similarities and differences among stochastic momentum methods.

The analysis explains the transition from gradient to Nesterov's and heavy-ball methods.

Empirical results show Nesterov's method balances convergence speed and robustness.

Abstract

Recently, {\it stochastic momentum} methods have been widely adopted in training deep neural networks. However, their convergence analysis is still underexplored at the moment, in particular for non-convex optimization. This paper fills the gap between practice and theory by developing a basic convergence analysis of two stochastic momentum methods, namely stochastic heavy-ball method and the stochastic variant of Nesterov's accelerated gradient method. We hope that the basic convergence results developed in this paper can serve the reference to the convergence of stochastic momentum methods and also serve the baselines for comparison in future development of stochastic momentum methods. The novelty of convergence analysis presented in this paper is a unified framework, revealing more insights about the similarities and differences between different stochastic momentum methods and stochastic gradient method. The unified framework exhibits a continuous change from the gradient method to Nesterov's accelerated gradient method and finally the heavy-ball method incurred by a free parameter, which can help explain a similar change observed in the testing error convergence behavior for deep learning. Furthermore, our empirical results for optimizing deep neural networks demonstrate that the stochastic variant of Nesterov's accelerated gradient method achieves a good tradeoff (between speed of convergence in training error and robustness of convergence in testing error) among the three stochastic methods.