Optimization of utility-based shortfall risk: A non-asymptotic viewpoint
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866929297074159616 |
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| author | Gupte, Sumedh A., Prashanth L. Bhat, Sanjay P. |
| author_facet | Gupte, Sumedh A., Prashanth L. Bhat, Sanjay P. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_18743 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Optimization of utility-based shortfall risk: A non-asymptotic viewpoint Gupte, Sumedh A., Prashanth L. Bhat, Sanjay P. Machine Learning 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. |
| title | Optimization of utility-based shortfall risk: A non-asymptotic viewpoint |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2310.18743 |