  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.
