Sensitivity analysis of Wasserstein distributionally robust optimization problems

TL;DR

This paper analyzes how Wasserstein distributionally robust optimization problems respond to model uncertainty, providing explicit formulas for sensitivity and applications across statistics, finance, and machine learning.

Contribution

It introduces explicit first-order sensitivity formulas for Wasserstein DRO problems, extending to linear constraints and diverse applications.

Findings

Explicit first-order correction formulas for value and optimizer.

Application to LASSO regression showing coefficient shrinkage.

New sensitivity measure for Black-Scholes option pricing.

Abstract

We consider sensitivity of a generic stochastic optimization problem to model uncertainty. We take a non-parametric approach and capture model uncertainty using Wasserstein balls around the postulated model. We provide explicit formulae for the first order correction to both the value function and the optimizer and further extend our results to optimization under linear constraints. We present applications to statistics, machine learning, mathematical finance and uncertainty quantification. In particular, we provide explicit first-order approximation for square-root LASSO regression coefficients and deduce coefficient shrinkage compared to the ordinary least squares regression. We consider robustness of call option pricing and deduce a new Black-Scholes sensitivity, a non-parametric version of the so-called Vega. We also compute sensitivities of optimized certainty equivalents in finance and propose measures to quantify robustness of neural networks to adversarial examples.