# Optimization of utility-based shortfall risk: A non-asymptotic viewpoint

arXiv: https://arxiv.org/abs/2310.18743 · HTML: https://mlsanity.com/paper/2310.18743/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 27 references that are in the map.
- First posted on arXiv 28 Oct 2023.

## Builds on (most cited first, 5 of 5)

- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [A Wasserstein distance approach for concentration of empirical risk estimates](https://mlsanity.com/paper/1902.10709/) (2019)
- [Non-asymptotic bounds for stochastic optimization with biased noisy gradient oracles](https://mlsanity.com/paper/2002.11440/) (2020)
- [Smoothed functional-based gradient algorithms for off-policy reinforcement learning: A non-asymptotic viewpoint](https://mlsanity.com/paper/2101.02137/) (2021)
- [Online Estimation and Optimization of Utility-Based Shortfall Risk](https://mlsanity.com/paper/2111.08805/) (2021)

## Built on by (most cited first, 2 of 2)

- [Concentration Bounds for Optimized Certainty Equivalent Risk Estimation](https://mlsanity.com/paper/2405.20933/) (2024)
- [Optimized Certainty Equivalent Risk Minimization Using Samples: Algorithms, Convergence Rates, and Applications](https://mlsanity.com/paper/2608.07113/) (2026)

## Abstract (arXiv)

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.