Input-Output Stability of Gradient Descent: A Discrete-Time Passivity-Based Approach

TL;DR

This paper introduces a passivity-based framework for analyzing the stability and convergence of gradient descent in discrete time, leveraging control theory concepts to provide new insights and methods.

Contribution

It develops a passivity-based analysis approach for gradient descent, including a new variable step size method using gain-scheduling.

Findings

Guarantees input-output stability and global convergence of gradient descent.

Extends stability guarantees to larger step sizes with weak passivity.

Proposes a novel variable step size gradient descent method.

Abstract

This paper presents a discrete-time passivity-based analysis of the gradient descent method for a class of functions with sector-bounded gradients. Using a loop transformation, it is shown that the gradient descent method can be interpreted as a passive controller in negative feedback with a very strictly passive system. The passivity theorem is then used to guarantee input-output stability, as well as the global convergence, of the gradient descent method. Furthermore, provided that the lower and upper sector bounds are not equal, the input-output stability of the gradient descent method is guaranteed using the weak passivity theorem for a larger choice of step size. Finally, to demonstrate the utility of this passivity-based analysis, a new variation of the gradient descent method with variable step size is proposed by gain-scheduling the input and output of the gradient.