PAC-Bayesian Bound for the Conditional Value at Risk

Conditional Value at Risk ( \textsc{CVaR}) is a coherent risk measure'' which generalizes expectation (reduced to a boundary parameter setting). Widely used in mathematical finance, it is garnering increasing interest in machine learning as an alternate approach to regularization, and as a means for ensuring fairness. This paper presents a generalization bound for learning algorithms that minimize the \textsc{CVaR} of the empirical loss. The bound is of PAC-Bayesian type and is guaranteed to be small when the empirical \textsc{CVaR} is small. We achieve this by reducing the problem of estimating \textsc{CVaR} to that of merely estimating an expectation.