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.

翻译：我们研究风险敏感在线算法的设计，其中风险度量被用于随机在线算法的竞争分析。我们引入CVaR$_\delta$竞争比（$\delta$-CR），该指标利用算法成本的条件风险价值来衡量算法在最差$(1-\delta)$部分结果上的成本期望与离线最优成本的比值，并运用该度量研究三个在线优化问题：连续时间滑雪租赁、离散时间滑雪租赁以及单峰搜索。不同问题的最优$\delta$-CR及算法结构存在显著差异：我们证明连续时间滑雪租赁的最优$\delta$-CR为$2-2^{-\Theta(\frac{1}{1-\delta})}$，该值可通过一个由延迟微分方程描述的算法获得。与之相对，在购买成本为$B$的离散时间滑雪租赁问题中，当$\delta = 1 - \Theta(\frac{1}{\log B})$时会出现急剧的相变，此后经典确定性策略即为最优。类似地，单峰搜索在$\delta = \frac{1}{2}$处呈现相变，此后经典确定性策略达到最优；我们还提出一个当$\delta \downarrow 0$时渐近最优的算法，该算法同样作为延迟微分方程的解出现。