Risk-Sensitive Online Algorithms
<|reference_start|>Risk-Sensitive Online 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$_\delta$-competitive ratio ($\delta$-CR) using the conditional value-at-risk of an algorithm's cost, which measures the expectation of the $(1-\delta)$-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 $\delta$-CR and algorithm varies significantly between problems: we prove that the optimal $\delta$-CR for continuous-time ski rental is $2-2^{-\Theta(\frac{1}{1-\delta})}$, 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 $\delta = 1 - \Theta(\frac{1}{\log B})$, after which the classic deterministic strategy is optimal. Similarly, one-max search exhibits a phase transition at $\delta = \frac{1}{2}$, after which the classic deterministic strategy is optimal; we also obtain an algorithm that is asymptotically optimal as $\delta \downarrow 0$ that arises as the solution to a delay differential equation.<|reference_end|>
@article{christianson2024risk-sensitive,
    title={Risk-Sensitive Online Algorithms},
    author={Nicolas Christianson, Bo Sun, Steven Low, Adam Wierman},
    journal={arXiv preprint arXiv:2405.09859},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.09859},
    primaryClass={cs.DS}
}