Optimized Certainty Equivalent Risk Minimization Using Samples: Algorithms, Convergence Rates, and Applications
This work provides a unified framework and theoretical guarantees for optimizing a broad class of risk measures, which is valuable for researchers and practitioners in finance and machine learning who need to minimize risk-sensitive objectives.
The paper develops estimators and stochastic gradient algorithms for optimizing the Optimized Certainty Equivalent (OCE) risk, covering popular special cases like entropic risk, mean-variance, and smooth CVaR. They derive non-asymptotic bounds on the MSE of their estimators and convergence rates for the algorithm, and demonstrate its application in portfolio optimization and uncertainty quantification.
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.