Optimization of utility-based shortfall risk: A non-asymptotic viewpoint
Sumedh Gupte, Prashanth L. A., Sanjay P. Bhat
TL;DR
The paper extends utility-based shortfall risk (UBSR) to unbounded random variables and provides non-asymptotic guarantees for both UBSR estimation and UBSR-based optimization. It derives a gradient representation for UBSR with a multivariate parameter and develops a biased gradient estimator that becomes asymptotically unbiased, enabling a stochastic gradient algorithm with non-asymptotic convergence rates under strong convexity. Non-asymptotic error bounds are established for both the SAA UBSR estimator and the UBSR gradient estimator, including analyses under Lipschitz and smooth loss functions and leveraging Wasserstein-distance bounds. The framework is demonstrated via simulation experiments, including UBSR estimation and a portfolio optimization example, highlighting practical applicability to risk-sensitive decision-making in finance and reinforcement learning.
Abstract
We consider the problems of estimation and optimization of utility-based shortfall risk (UBSR), which is a popular risk measure in finance. In the context of UBSR estimation, we derive a non-asymptotic bound on the mean-squared error of the classical sample average approximation (SAA) of UBSR. Next, in the context of UBSR optimization, we derive an expression for the UBSR gradient under a smooth parameterization. This expression is a ratio of expectations, both of which involve the UBSR. We use SAA for the numerator as well as denominator in the UBSR gradient expression to arrive at a biased gradient estimator. We derive non-asymptotic bounds on the estimation error, which show that our gradient estimator is asymptotically unbiased. We incorporate the aforementioned gradient estimator into a stochastic gradient (SG) algorithm for UBSR optimization. Finally, we derive non-asymptotic bounds that quantify the rate of convergence of our SG algorithm for UBSR optimization.