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

TL;DR

This paper develops methods for estimating and optimizing convex risk measures like UBSR and OCE, providing theoretical bounds and gradient estimators, applicable to unbounded variables, with convergence guarantees for stochastic gradient algorithms.

Contribution

It introduces new estimation and optimization techniques for convex risk measures, including gradient estimators and convergence bounds, extending to unbounded random variables.

Findings

Derived non-asymptotic error bounds for estimators

Proposed gradient estimators for UBSR and OCE

Established convergence rates for stochastic gradient algorithms

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.