Optimizing Shortfall Risk Metric for Learning Regression Models
This work addresses a specific risk optimization problem in regression for applications requiring robust loss metrics, but it appears incremental as it builds on existing UBSR concepts with algorithmic improvements.
The paper tackles the challenge of estimating and optimizing utility-based shortfall risk (UBSR) for regression models, where UBSR is a non-linear function of the loss distribution. It presents a bisection-type algorithm with convergence guarantees to find the UBSR-optimal solution.
We consider the problem of estimating and optimizing utility-based shortfall risk (UBSR) of a loss, say $(Y - \hat Y)^2$, 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 $\mathcal D$ of the loss $(Y- \hat Y)^2$. We construct a gradient oracle for the UBSR objective and a linear minimization oracle (LMO) for the set $\mathcal D$. Using these oracles, we devise a bisection-type algorithm, and establish convergence to the UBSR-optimal solution.