Temporal Variabilities Limit Convergence Rates in Gradient-Based Online Optimization
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866912646857490432 |
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| author | Van Scoy, Bryan Bianchin, Gianluca |
| author_facet | Van Scoy, Bryan Bianchin, Gianluca |
| contents | This paper investigates the fundamental performance limits of gradient-based algorithms for time-varying optimization. Leveraging the internal model principle and root locus techniques, we show that temporal variabilities impose intrinsic limits on the achievable rate of convergence. For a problem with condition ratio $κ$ and time variation whose model has degree $n$, we show that the worst-case convergence rate of any minimal-order gradient-based algorithm is $ρ_\text{TV} = (\frac{κ-1}{κ+1})^{1/n}$. This bound reveals a fundamental tradeoff between problem conditioning, temporal complexity, and rate of convergence. We further construct explicit controllers that attain the bound for low-degree models of time variation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_12512 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Temporal Variabilities Limit Convergence Rates in Gradient-Based Online Optimization Van Scoy, Bryan Bianchin, Gianluca Optimization and Control Systems and Control This paper investigates the fundamental performance limits of gradient-based algorithms for time-varying optimization. Leveraging the internal model principle and root locus techniques, we show that temporal variabilities impose intrinsic limits on the achievable rate of convergence. For a problem with condition ratio $κ$ and time variation whose model has degree $n$, we show that the worst-case convergence rate of any minimal-order gradient-based algorithm is $ρ_\text{TV} = (\frac{κ-1}{κ+1})^{1/n}$. This bound reveals a fundamental tradeoff between problem conditioning, temporal complexity, and rate of convergence. We further construct explicit controllers that attain the bound for low-degree models of time variation. |
| title | Temporal Variabilities Limit Convergence Rates in Gradient-Based Online Optimization |
| topic | Optimization and Control Systems and Control |
| url | https://arxiv.org/abs/2510.12512 |