Optimizing Shortfall Risk Metric for Learning Regression Models

Abstract

We consider the problem of estimating and optimizing utility-based shortfall risk (UBSR) of a loss, say (Y - Y)^2, in the context of a regression problem. Empirical risk minimization with a UBSR objective is challenging since UBSR is a non-linear function of the underlying distribution. We first derive a concentration bound for UBSR estimation using independent and identically distributed (i.i.d.) samples. We then frame the UBSR optimization problem as minimization of a pseudo-linear function in the space of achievable distributions D of the loss (Y- Y)^2. We construct a gradient oracle for the UBSR objective and a linear minimization oracle (LMO) for the set D. Using these oracles, we devise a bisection-type algorithm, and establish convergence to the UBSR-optimal solution.

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Reproductions