Concentration bounds for CVaR estimation: The cases of light-tailed and heavy-tailed distributions

TL;DR

This paper establishes exponential concentration bounds for CVaR estimation in both light-tailed and heavy-tailed distributions, and applies these bounds to improve CVaR-based decision-making in multi-armed bandit problems.

Contribution

It derives novel exponential concentration bounds for CVaR estimators under different tail conditions and adapts bandit algorithms for CVaR optimization using these bounds.

Findings

Exponential decay of concentration bounds for light-tailed CVaR estimates.

Exponential bounds for heavy-tailed CVaR estimators under mild moment conditions.

Modified bandit algorithm with proven bounds on incorrect identification probability.

Abstract

Conditional Value-at-Risk (CVaR) is a widely used risk metric in applications such as finance. We derive concentration bounds for CVaR estimates, considering separately the cases of light-tailed and heavy-tailed distributions. In the light-tailed case, we use a classical CVaR estimator based on the empirical distribution constructed from the samples. For heavy-tailed random variables, we assume a mild `bounded moment' condition, and derive a concentration bound for a truncation-based estimator. Notably, our concentration bounds enjoy an exponential decay in the sample size, for heavy-tailed as well as light-tailed distributions. To demonstrate the applicability of our concentration results, we consider a CVaR optimization problem in a multi-armed bandit setting. Specifically, we address the best CVaR-arm identification problem under a fixed budget. We modify the well-known successive rejects algorithm to incorporate a CVaR-based criterion. Using the CVaR concentration result, we derive an upper-bound on the probability of incorrect identification by the proposed algorithm.