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arXiv · 2411.16074

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

Sepehr Moalemi·James Richard Forbes
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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.

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Sepehr Moalemi, James Richard Forbes. 2024-11-25. Input-Output Stability of Gradient Descent: A Discrete-Time Passivity-Based Approach. https://arxiv.org/abs/2411.16074

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