Gradient-based Stochastic Optimization of Utility-based Shortfall Risk
Provides theoretical guarantees for UBSR estimation and optimization, benefiting risk-sensitive decision-making in finance.
The paper extends utility-based shortfall risk (UBSR) to unbounded random variables, covering entropic risk, expectile risk, VaR, and quadratic risk. It derives non-asymptotic bounds for estimation and optimization, and proposes a stochastic gradient algorithm with convergence guarantees.
We consider the problems of estimation and optimization of utility-based shortfall risk (UBSR). We extend UBSR to cover possibly unbounded random variables. We cover prominent risk measures such as entropic risk, expectile risk, Value-at-Risk, and quadratic risk as special cases of the UBSR. In the context of estimation, we derive non-asymptotic bounds on the mean absolute error (MAE) and the mean-squared error (MSE) of the classical sample-average approximation (SAA) estimator for the UBSR. In the context of optimization, we derive an expression for the gradient of UBSR under a smooth parameterization. We propose a gradient estimator for the UBSR and derive non-asymptotic bounds on MAE and MSE for this estimator. We incorporate the aforementioned gradient estimator into a stochastic gradient (SG) optimization algorithm and derive non-asymptotic bounds on the convergence rate of our SG algorithm for optimizing UBSR under three objectives, namely, strongly convex, convex and non-convex. Finally, we conduct experiments on financial applications to demonstrate the performance of our proposed UBSR estimation and optimization algorithms.