Optimal Best-Arm Identification Methods for Tail-Risk Measures
This work addresses risk assessment in finance, insurance, and safety-critical environments, providing an optimal solution for tail-risk measures, though it is incremental as it builds on existing best-arm identification frameworks.
The paper tackles the problem of identifying the best arm with the smallest tail-risk measures (CVaR, VaR, or weighted sum) in multi-armed bandits, including heavy-tailed distributions, by developing an optimal δ-correct algorithm that matches the lower bound on expected samples asymptotically as δ approaches 0.
Conditional value-at-risk (CVaR) and value-at-risk (VaR) are popular tail-risk measures in finance and insurance industries as well as in highly reliable, safety-critical uncertain environments where often the underlying probability distributions are heavy-tailed. We use the multi-armed bandit best-arm identification framework and consider the problem of identifying the arm from amongst finitely many that has the smallest CVaR, VaR, or weighted sum of CVaR and mean. The latter captures the risk-return trade-off common in finance. Our main contribution is an optimal $δ$-correct algorithm that acts on general arms, including heavy-tailed distributions, and matches the lower bound on the expected number of samples needed, asymptotically (as $δ$ approaches $0$). The algorithm requires solving a non-convex optimization problem in the space of probability measures, that requires delicate analysis. En-route, we develop new non-asymptotic empirical likelihood-based concentration inequalities for tail-risk measures which are tighter than those for popular truncation-based empirical estimators.