PAC-Bayesian Bound for the Conditional Value at Risk
Zakaria Mhammedi,Benjamin Guedj,Robert C. Williamson
Conditional Value at Risk ($ extsc{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 $ extsc{CVaR}$ of the empirical loss. The bound is of PAC-Bayesian type and is guaranteed to be small when the empirical $ extsc{CVaR}$ is small. We achieve this by reducing the problem of estimating $ extsc{CVaR}$ to that of merely estimating an expectation. This then enables us, as a by-product, to obtain concentration inequalities for $ extsc{CVaR}$ even when the random variable in question is unbounded.
