Concentration bounds for CVaR estimation: The cases of light-tailed and heavy-tailed distributions
This work provides theoretical guarantees for CVaR estimation, which is crucial for risk management in fields like finance, though it is incremental as it builds on existing estimators and algorithms.
The authors derived concentration bounds for Conditional Value-at-Risk (CVaR) estimation, achieving exponential decay in sample size for both light-tailed and heavy-tailed distributions, and applied these results to a multi-armed bandit problem to bound the probability of incorrect identification in best CVaR-arm selection.
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.