PAC-Bayesian Bound for the Conditional Value at Risk

Zakaria Mhammedi, Benjamin Guedj, Robert Williamson

Spotlight presentation: Orals & Spotlights Track 11: Learning Theory
on 2020-12-08T08:00:00-08:00 - 2020-12-08T08:10:00-08:00
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on 2020-12-08T09:00:00-08:00 - 2020-12-08T11:00:00-08:00
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Abstract: 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. This then enables us, as a by-product, to obtain concentration inequalities for $\textsc{CVaR}$ even when the random variable in question is unbounded.

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