  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 $\delta$-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
$\delta$ 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.
