Proximal Gradient Dynamics: Monotonicity, Exponential Convergence, and Applications
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| Format: | Preprint |
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2024
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| _version_ | 1866910706656346112 |
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| author | Gokhale, Anand Davydov, Alexander Bullo, Francesco |
| author_facet | Gokhale, Anand Davydov, Alexander Bullo, Francesco |
| contents | In this letter we study the proximal gradient dynamics. This recently-proposed continuous-time dynamics solves optimization problems whose cost functions are separable into a nonsmooth convex and a smooth component. First, we show that the cost function decreases monotonically along the trajectories of the proximal gradient dynamics. We then introduce a new condition that guarantees exponential convergence of the cost function to its optimal value, and show that this condition implies the proximal Polyak-Łojasiewicz condition. We also show that the proximal Polyak-Łojasiewicz condition guarantees exponential convergence of the cost function. Moreover, we extend these results to time-varying optimization problems, providing bounds for equilibrium tracking. Finally, we discuss applications of these findings, including the LASSO problem, certain matrix based problems and a numerical experiment on a feed-forward neural network. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_10664 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Proximal Gradient Dynamics: Monotonicity, Exponential Convergence, and Applications Gokhale, Anand Davydov, Alexander Bullo, Francesco Optimization and Control Systems and Control Signal Processing In this letter we study the proximal gradient dynamics. This recently-proposed continuous-time dynamics solves optimization problems whose cost functions are separable into a nonsmooth convex and a smooth component. First, we show that the cost function decreases monotonically along the trajectories of the proximal gradient dynamics. We then introduce a new condition that guarantees exponential convergence of the cost function to its optimal value, and show that this condition implies the proximal Polyak-Łojasiewicz condition. We also show that the proximal Polyak-Łojasiewicz condition guarantees exponential convergence of the cost function. Moreover, we extend these results to time-varying optimization problems, providing bounds for equilibrium tracking. Finally, we discuss applications of these findings, including the LASSO problem, certain matrix based problems and a numerical experiment on a feed-forward neural network. |
| title | Proximal Gradient Dynamics: Monotonicity, Exponential Convergence, and Applications |
| topic | Optimization and Control Systems and Control Signal Processing |
| url | https://arxiv.org/abs/2409.10664 |