2023/10/28 by Gupte, Sumedh, A., Prashanth L., Bhat, Sanjay P. · 2 citations
#FOS: Computer and information sciences #Machine Learning (cs.LG)
paper · doi:10.48550/arxiv.2310.18743
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.