Input-Output Stability of Gradient Descent: A Discrete-Time Passivity-Based Approach
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866909403500773376 |
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| author | Moalemi, Sepehr Forbes, James Richard |
| author_facet | Moalemi, Sepehr Forbes, James Richard |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_16074 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Input-Output Stability of Gradient Descent: A Discrete-Time Passivity-Based Approach Moalemi, Sepehr Forbes, James Richard Optimization and Control Systems and Control 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. |
| title | Input-Output Stability of Gradient Descent: A Discrete-Time Passivity-Based Approach |
| topic | Optimization and Control Systems and Control |
| url | https://arxiv.org/abs/2411.16074 |