Gradient-based optimisation of the conditional-value-at-risk using the multi-level Monte Carlo method
In this work, we tackle the problem of minimising the Conditional-Value-at-Risk (CVaR) of output quantities of complex differential models with random input data, using gradient-based approaches in combination with the Multi-Level Monte Carlo (MLMC) method. In particular, we consider the framework o...
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| Vydáno v: | Journal of computational physics Ročník 495; s. 112523 |
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| Jazyk: | angličtina |
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15.12.2023
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| ISSN: | 0021-9991, 1090-2716 |
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| Abstract | In this work, we tackle the problem of minimising the Conditional-Value-at-Risk (CVaR) of output quantities of complex differential models with random input data, using gradient-based approaches in combination with the Multi-Level Monte Carlo (MLMC) method. In particular, we consider the framework of multi-level Monte Carlo for parametric expectations introduced in [24] and propose modifications of the MLMC estimator, error estimation procedure, and adaptive MLMC parameter selection to ensure the estimation of the CVaR and sensitivities for a given design with a prescribed accuracy. We then propose combining the MLMC framework with an alternating inexact minimisation-gradient descent algorithm, for which we prove Q-linear convergence in the optimisation iterations under the assumptions of strong convexity and Lipschitz continuity of the gradient of the objective function. We demonstrate the performance of our approach on two numerical examples of practical relevance, which evidence the same optimal asymptotic cost-tolerance behaviour as standard MLMC methods for fixed design computations of output expectations.
•Conditional-Value-at-Risk (CvaR) sensitivities using Parametric Expectations (PE).•Estimating sensitivities using Multi-Level Monte Carlo (MLMC) for PE.•Alternating minimization gradient-descent algorithm using MLMC sensitivities.•Exponential convergence predicted and obtained in the optimization iterations. |
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| AbstractList | In this work, we tackle the problem of minimising the Conditional-Value-at-Risk (CVaR) of output quantities of complex differential models with random input data, using gradient-based approaches in combination with the Multi-Level Monte Carlo (MLMC) method. In particular, we consider the framework of multi-level Monte Carlo for parametric expectations introduced in [24] and propose modifications of the MLMC estimator, error estimation procedure, and adaptive MLMC parameter selection to ensure the estimation of the CVaR and sensitivities for a given design with a prescribed accuracy. We then propose combining the MLMC framework with an alternating inexact minimisation-gradient descent algorithm, for which we prove Q-linear convergence in the optimisation iterations under the assumptions of strong convexity and Lipschitz continuity of the gradient of the objective function. We demonstrate the performance of our approach on two numerical examples of practical relevance, which evidence the same optimal asymptotic cost-tolerance behaviour as standard MLMC methods for fixed design computations of output expectations.
•Conditional-Value-at-Risk (CvaR) sensitivities using Parametric Expectations (PE).•Estimating sensitivities using Multi-Level Monte Carlo (MLMC) for PE.•Alternating minimization gradient-descent algorithm using MLMC sensitivities.•Exponential convergence predicted and obtained in the optimization iterations. |
| ArticleNumber | 112523 |
| Author | Nobile, Fabio Ganesh, Sundar |
| Author_xml | – sequence: 1 givenname: Sundar orcidid: 0000-0001-6477-3924 surname: Ganesh fullname: Ganesh, Sundar email: sundar.ganesh@epfl.ch – sequence: 2 givenname: Fabio orcidid: 0000-0002-8130-0114 surname: Nobile fullname: Nobile, Fabio email: fabio.nobile@epfl.ch |
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| Cites_doi | 10.1159/000089933 10.1113/jphysiol.1952.sp004764 10.1287/opre.1070.0496 10.1016/j.cma.2022.115371 10.1159/000089932 10.1093/biomet/66.3.605 10.1287/mnsc.1080.0901 10.1287/moor.1050.0186 10.1016/S0006-3495(61)86902-6 10.1615/Int.J.UncertaintyQuantification.2023045259 10.1137/140964552 10.1016/j.jcp.2020.109466 10.1137/17M1122992 10.1145/3242094 10.1016/j.camwa.2017.02.028 10.1137/17M1135566 10.1137/S0895479899358194 10.1093/imanum/drac083 10.1111/1467-9965.00068 10.1109/JRPROC.1962.288235 10.1137/17M1155892 10.1007/s10107-012-0572-5 10.1137/140954556 10.2514/1.C035054 10.1016/S0378-4266(02)00271-6 10.1007/s10589-008-9196-3 10.1137/17M1154679 10.1017/S096249291500001X |
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| Keywords | Uncertainty quantification Gradient descent CVaR Multilevel Monte Carlo methods VaR Optimisation under uncertainty |
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| Title | Gradient-based optimisation of the conditional-value-at-risk using the multi-level Monte Carlo method |
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