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Gradient-based Stochastic Optimization of Utility-based Shortfall Risk
[Submitted on 1 Jun 2025 (v1), last revised 17 Jun 2026 (this ve · 2026-06-18 · via stat updates on arXiv.org

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Abstract:We consider the problems of estimation and optimization of utility-based shortfall risk (UBSR). We extend UBSR to cover possibly unbounded random variables. We cover prominent risk measures such as entropic risk, expectile risk, Value-at-Risk, and quadratic risk as special cases of the UBSR. In the context of estimation, we derive non-asymptotic bounds on the mean absolute error (MAE) and the mean-squared error (MSE) of the classical sample-average approximation (SAA) estimator for the UBSR. In the context of optimization, we derive an expression for the gradient of UBSR under a smooth parameterization. We propose a gradient estimator for the UBSR and derive non-asymptotic bounds on MAE and MSE for this estimator. We incorporate the aforementioned gradient estimator into a stochastic gradient (SG) optimization algorithm and derive non-asymptotic bounds on the convergence rate of our SG algorithm for optimizing UBSR under three objectives, namely, strongly convex, convex and non-convex. Finally, we conduct experiments on financial applications to demonstrate the performance of our proposed UBSR estimation and optimization algorithms.

Submission history

From: Sumedh Gupte [view email]
[v1] Sun, 1 Jun 2025 17:53:15 UTC (1,495 KB)
[v2] Wed, 17 Jun 2026 12:29:10 UTC (573 KB)