Smoothed $f$-Divergence Distributionally Robust Optimization
Abstract
In data-driven optimization, sample average approximation (SAA) is known to suffer from the so-called optimizer's curse that causes an over-optimistic evaluation of the solution performance. We argue that a special type of distributionallly robust optimization (DRO) formulation offers theoretical advantages in correcting for this optimizer's curse compared to simple ``margin'' adjustments to SAA and other DRO approaches: It attains a statistical bound on the out-of-sample performance, for a wide class of objective functions and distributions, that is nearly tightest in terms of exponential decay rate. This DRO uses an ambiguity set based on a Kullback Leibler (KL) divergence smoothed by the Wasserstein or L\'evy-Prokhorov (LP) distance via a suitable distance optimization. Computationally, we also show that such a DRO, and its generalized versions using smoothed -divergence, are not harder than DRO problems based on -divergence or Wasserstein distances, rendering our DRO formulations both statistically optimal and computationally viable.
Cite
@article{arxiv.2306.14041,
title = {Smoothed $f$-Divergence Distributionally Robust Optimization},
author = {Zhenyuan Liu and Bart P. G. Van Parys and Henry Lam},
journal= {arXiv preprint arXiv:2306.14041},
year = {2023}
}