Risk-Sensitive Online Algorithms

TL;DR

We introduce the CVaR$_{\delta}$-competitive ratio ($\delta$-CR) to study risk-sensitive online algorithms and apply it to CSR, DSR, and OMS, revealing problem-dependent optimal strategies and phase transitions.The continuous-time ski rental optimal strategy is characterized by a delay differential equation for the inverse CDF, yielding $\alpha_\delta^* = 2 - 2^{-{\Theta}(1/(1-\delta))}$, while discrete-time ski rental exhibits a phase transition near $\delta = 1 - \Theta(1/\log B)$ with analytic optimality for small $\delta$; one-max search shows a phase transition at $\delta = 1/2$ with an asymptotically optimal small-$\delta$ algorithm.Our analytic framework expresses CVaR in terms of inverse-CDFs and, for CSR and OMS, reduces to delay differential equations that determine optimal strategies; these techniques provide structural insights into tail-risk optimization under online uncertainty.These results illustrate sharp limits to the benefits of randomization in certain risk-sensitive online problems and highlight qualitative differences between continuous and discrete settings, with practical implications for tail-safe decision-making.

Abstract

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.

Figures (2)

• Figure 1: $\mathop{\mathrm{\mathrm{CVaR}_\delta}}\nolimits$-competitive ratios from Theorems \ref{['theorem:cvar_skirental_firstalg']} (Suboptimal) and \ref{['theorem:optimal_cont_time_ski_rental']} (Optimal) and lower bound from Theorem \ref{['theorem:cont_time_ski_lower_bound']} for continuous-time ski rental.
• Figure 2: $\mathop{\mathrm{\mathrm{CVaR}_\delta}}\nolimits$-competitive ratio of the algorithm in Theorem \ref{['theorem:one_max_search_upper_bound']} along with the upper bound \ref{['theorem:one_max_dcr_ub_root']} and lower bound \ref{['eq:one_max_dcr_root_lb']} for one-max search.

Theorems & Definitions (27)

• Definition 1: Conditional Value-at-Risk
• Definition 2: Competitive ratio
• Definition 3: $\mathop{\mathrm{\mathrm{CVaR}_\delta}}\nolimits$-Competitive Ratio
• Lemma 4
• Lemma 5
• Theorem 6
• Theorem 7
• Theorem 8
• Theorem 9
• Theorem 10