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Sensitivity analysis of Wasserstein distributionally robust optimization problems.

Daniel Bartl1, Samuel Drapeau2, Jan Obłój3

  • 1Department of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria.

Proceedings. Mathematical, Physical, and Engineering Sciences
|February 14, 2022
PubMed
Summary

This study quantifies the impact of model uncertainty on stochastic optimization using Wasserstein balls. Results offer insights into robust decision-making across statistics, machine learning, and finance.

Keywords:
Wasserstein metricnon-parametric uncertaintyrobust stochastic optimizationsensitivity analysisuncertainty quantification

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Area of Science:

  • Optimization Theory
  • Robust Statistics
  • Machine Learning

Background:

  • Stochastic optimization problems are sensitive to uncertainties in the underlying models.
  • Quantifying this sensitivity is crucial for reliable decision-making in various fields.
  • Existing methods often rely on parametric assumptions about uncertainty.

Purpose of the Study:

  • To develop a non-parametric framework for analyzing the sensitivity of stochastic optimization to model uncertainty.
  • To derive explicit formulas for first-order corrections to value functions and optimizers.
  • To extend the analysis to constrained optimization problems and demonstrate practical applications.

Main Methods:

  • Utilizing Wasserstein balls to define and quantify model uncertainty non-parametrically.
  • Deriving analytical expressions for first-order sensitivity.
  • Applying the framework to statistical regression, option pricing, and neural network robustness.

Main Results:

  • Explicit formulas for first-order corrections to value functions and optimizers under model uncertainty.
  • Demonstrated coefficient shrinkage in square-root LASSO regression compared to ordinary least-squares.
  • Introduced a non-parametric analog of Vega for option pricing and measures for neural network robustness.

Conclusions:

  • The developed non-parametric approach provides a robust method for assessing model uncertainty in optimization.
  • The findings have broad applicability in statistics, machine learning, mathematical finance, and uncertainty quantification.
  • The study offers practical tools for enhancing the reliability of models in the presence of uncertainty.