Optimized Certainty Equivalent Risk Minimization Using Samples: Algorithms, Convergence Rates, and Applications

2026-08-07

AI Summary

This study addresses the optimization of optimized certainty equivalents (OCE), a risk measure widely employed in portfolio optimization and uncertainty quantification in machine learning. Focusing on unbounded random variables, it establishes the first OCE optimization framework by leveraging the duality between OCE and utility-based shortfall risk (UBSR). The work introduces a sample average approximation (SAA)-based OCE estimator and a corresponding stochastic gradient algorithm. Theoretical contributions include the design of an OCE gradient estimator, non-asymptotic mean squared error bounds, and convergence rates for the proposed algorithm. Empirical evaluations on portfolio optimization and uncertainty quantification tasks demonstrate the method’s effectiveness, offering both strong theoretical guarantees and practical performance.

Abstract

We consider the optimization of the Optimized Certainty Equivalent (OCE) risk, with applications including portfolio optimization in finance, and uncertainty quantification, classification, and regression in machine learning. Our contributions cover popular special cases of OCE, such as entropic risk, mean-variance risk, and smooth variants of Conditional Value-at-Risk. Our treatment sets out the conditions that facilitate the extension of OCE to unbounded r.v.s.. We provide a useful characterization of OCE that links OCE to utility-based shortfall risk (UBSR). Our characterization enables us to form an OCE estimator from the classic sample-average approximation (SAA) of UBSR. We derive mean-squared error (MSE) bounds for our proposed OCE estimator. For OCE optimization, we first derive an expression for the OCE gradient using the characterization linking OCE to UBSR. This expression serves as the basis for a gradient estimator for the OCE. We derive non-asymptotic bounds on the MSE for the proposed OCE gradient estimator. We incorporate the aforementioned gradient estimator into a stochastic gradient (SG) algorithm to optimize OCE and quantify its convergence rate using non-asymptotic bounds that we derive. Finally, we present three experiments that use our OCE optimization algorithm to solve portfolio optimization and uncertainty quantification problems.

Problem

• Optimized Certainty Equivalent
• risk minimization
• sample-based estimation
• stochastic optimization
• uncertainty quantification

Innovation

• Optimized Certainty Equivalent
• Utility-Based Shortfall Risk
• Non-asymptotic Convergence
• Stochastic Gradient Optimization
• Sample Average Approximation