Concentration Bounds for Optimized Certainty Equivalent Risk Estimation

TL;DR

This paper develops concentration bounds and finite-sample guarantees for estimating the Optimized Certainty Equivalent (OCE) risk using sample average approximation and stochastic methods, with applications to risk-aware bandits.

Contribution

It introduces new theoretical bounds for OCE risk estimation, including concentration and finite-sample bounds for both SAA and stochastic approximation methods.

Findings

Derived mean-squared error and concentration bounds for SAA of OCE

Established finite-sample bounds for stochastic approximation-based OCE estimator

Validated theoretical bounds through numerical experiments

Abstract

We consider the problem of estimating the Optimized Certainty Equivalent (OCE) risk from independent and identically distributed (i.i.d.) samples. For the classic sample average approximation (SAA) of OCE, we derive mean-squared error as well as concentration bounds (assuming sub-Gaussianity). Further, we analyze an efficient stochastic approximation-based OCE estimator, and derive finite sample bounds for the same. To show the applicability of our bounds, we consider a risk-aware bandit problem, with OCE as the risk. For this problem, we derive bound on the probability of mis-identification. Finally, we conduct numerical experiments to validate the theoretical findings.