Risk-Sensitive Online Algorithms

TL;DR

This paper introduces a risk-sensitive competitive ratio using CVaR for online algorithms, analyzing its structure and optimal strategies across different online optimization problems with phase transitions at certain risk levels.

Contribution

It defines the CVaR$_ ext{\delta}$-competitive ratio for risk-sensitive analysis and characterizes the optimal algorithms and phase transitions in continuous-time ski rental, discrete-time ski rental, and one-max search.

Findings

Optimal CVaR-competitive ratio for continuous-time ski rental is $2-2^{- heta}$.

Phase transition in discrete-time ski rental at $ ext{\delta} = 1 - \Theta(1/\log B)$.

Phase transition in one-max search at $ ext{\delta} = 1/2$.

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$_\delta$-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.