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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | https://arxiv.org/abs/2405.09859 |
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| _version_ | 1866916259617046528 |
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| author | Christianson, Nicolas Sun, Bo Low, Steven Wierman, Adam |
| author_facet | Christianson, Nicolas Sun, Bo Low, Steven Wierman, Adam |
| contents | We study the design of risk-sensitive online algorithms, in which risk measures are used in the competitive analysis of randomized online algorithms. We introduce the CVaR$_δ$-competitive ratio ($δ$-CR) using the conditional value-at-risk of an algorithm's cost, which measures the expectation of the $(1-δ)$-fraction of worst outcomes against the offline optimal cost, and use this measure to study three online optimization problems: continuous-time ski rental, discrete-time ski rental, and one-max search. The structure of the optimal $δ$-CR and algorithm varies significantly between problems: we prove that the optimal $δ$-CR for continuous-time ski rental is $2-2^{-Θ(\frac{1}{1-δ})}$, obtained by an algorithm described by a delay differential equation. In contrast, in discrete-time ski rental with buying cost $B$, there is an abrupt phase transition at $δ= 1 - Θ(\frac{1}{\log B})$, after which the classic deterministic strategy is optimal. Similarly, one-max search exhibits a phase transition at $δ= \frac{1}{2}$, after which the classic deterministic strategy is optimal; we also obtain an algorithm that is asymptotically optimal as $δ\downarrow 0$ that arises as the solution to a delay differential equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_09859 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Risk-Sensitive Online Algorithms Christianson, Nicolas Sun, Bo Low, Steven Wierman, Adam Data Structures and Algorithms We study the design of risk-sensitive online algorithms, in which risk measures are used in the competitive analysis of randomized online algorithms. We introduce the CVaR$_δ$-competitive ratio ($δ$-CR) using the conditional value-at-risk of an algorithm's cost, which measures the expectation of the $(1-δ)$-fraction of worst outcomes against the offline optimal cost, and use this measure to study three online optimization problems: continuous-time ski rental, discrete-time ski rental, and one-max search. The structure of the optimal $δ$-CR and algorithm varies significantly between problems: we prove that the optimal $δ$-CR for continuous-time ski rental is $2-2^{-Θ(\frac{1}{1-δ})}$, obtained by an algorithm described by a delay differential equation. In contrast, in discrete-time ski rental with buying cost $B$, there is an abrupt phase transition at $δ= 1 - Θ(\frac{1}{\log B})$, after which the classic deterministic strategy is optimal. Similarly, one-max search exhibits a phase transition at $δ= \frac{1}{2}$, after which the classic deterministic strategy is optimal; we also obtain an algorithm that is asymptotically optimal as $δ\downarrow 0$ that arises as the solution to a delay differential equation. |
| title | Risk-Sensitive Online Algorithms |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2405.09859 |