Analytical Convergence Regions of Accelerated Gradient Descent in Nonconvex Optimization under Regularity Condition

TL;DR

This paper analytically characterizes the convergence regions of accelerated gradient descent algorithms in nonconvex optimization problems satisfying the Regularity Condition, using robust control theory to better understand their practical effectiveness.

Contribution

It provides the first analytical convergence region characterization for AGD in nonconvex problems under RC, bridging robust control and optimization analysis.

Findings

Convergence regions are explicitly characterized using control theory.

AGD algorithms often work well in practice under proper initialization.

Results apply to applications like phase retrieval and neural network training.

Abstract

There is a growing interest in using robust control theory to analyze and design optimization and machine learning algorithms. This paper studies a class of nonconvex optimization problems whose cost functions satisfy the so-called Regularity Condition (RC). Empirical studies show that accelerated gradient descent (AGD) algorithms (e.g. Nesterov's acceleration and Heavy-ball) with proper initializations often work well in practice. However, the convergence of such AGD algorithms is largely unknown in the literature. The main contribution of this paper is the analytical characterization of the convergence regions of AGD under RC via robust control tools. Since such optimization problems arise frequently in many applications such as phase retrieval, training of neural networks and matrix sensing, our result shows promise of robust control theory in these areas.