Learning to optimize convex risk measures: The cases of utility-based shortfall risk and optimized certainty equivalent risk

Paper: arXiv 2506.01101

Authors: Sumedh Gupte, Prashanth L. A., Sanjay P. Bhat

Abstract

We consider the problems of estimation and optimization of two popular convex risk measures: utility-based shortfall risk (UBSR) and Optimized Certainty Equivalent (OCE) risk. We extend these risk measures to cover possibly unbounded random variables. We cover prominent risk measures like the entropic risk, expectile risk, monotone mean-variance risk, Value-at-Risk, and Conditional Value-at-Risk as few special cases of either the UBSR or the OCE risk. In the context of estimation, we derive non-asymptotic bounds on the mean absolute error (MAE) and mean-squared error (MSE) of the classical sample average approximation (SAA) estimators of both, the UBSR and the OCE. Next, in the context of optimization, we derive expressions for the UBSR gradient and the OCE gradient under a smooth parameterization. Utilizing these expressions, we propose gradient estimators for both, the UBSR and the OCE. We use the SAA estimator of UBSR in both these gradient estimators, and derive non-asymptotic bounds on MAE and MSE for the proposed gradient estimation schemes. We incorporate the aforementioned gradient estimators into a stochastic gradient (SG) algorithm for optimization. Finally, we derive non-asymptotic bounds that quantify the rate of convergence of our SG algorithm for the optimization of the UBSR and the OCE risk measure.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 2.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides rigorous non-asymptotic mathematical proofs for estimation and stochastic gradient optimization of UBSR and OCE risk measures for vector parameters and unbounded variables. However, it is purely theoretical with no empirical backtests or numerical experiments presented in the provided excerpt.

Research Flowchart

  flowchart TD
    A[Goal: Estimate & Optimize Unbounded Convex Risk Measures: UBSR & OCE] --> B[Inputs: Unbounded Random Variables & Smooth Parameterization]
    B --> C[Methodology: Sample Average Approximation & Gradient Formulation]
    C --> D[Computation: Derivation of Gradient Estimators using SAA]
    D --> E[Optimization: Stochastic Gradient Algorithm Execution]
    E --> F[Outcomes: Non-asymptotic MAE/MSE Bounds & Convergence Rates]