# Approximation Algorithms for Distributionally Robust Stochastic Optimization with Black-Box Distributions

## TL;DR

This paper develops approximation algorithms for distributionally robust stochastic optimization problems where the underlying distribution is uncertain and accessed via black-box sampling, extending solutions to classic combinatorial problems.

## Contribution

It introduces a framework using sample average approximation and LP-rounding to solve distributionally robust problems with black-box distributions, achieving near-optimal guarantees.

## Key findings

- First approximation algorithms for distributionally robust set cover, vertex cover, edge cover, facility location, and Steiner tree.
- Guarantees within O(1) factors of deterministic problem solutions for most cases.
- Framework applicable to problems with uncertain distributions accessed via sampling.

## Abstract

Two-stage stochastic optimization is a framework for modeling uncertainty, where we have a probability distribution over possible realizations of the data, called scenarios, and decisions are taken in two stages: we make first-stage decisions knowing only the underlying distribution and before a scenario is realized, and may take additional second-stage recourse actions after a scenario is realized. The goal is typically to minimize the total expected cost. A criticism of this model is that the underlying probability distribution is itself often imprecise! To address this, a versatile approach that has been proposed is the {\em distributionally robust 2-stage model}: given a collection of probability distributions, our goal now is to minimize the maximum expected total cost with respect to a distribution in this collection. We provide a framework for designing approximation algorithms in such settings when the collection is a ball around a central distribution and the central distribution is accessed {\em only via a sampling black box}. We first show that one can utilize the {\em sample average approximation} (SAA) method to reduce the problem to the case where the central distribution has {\em polynomial-size} support. We then show how to approximately solve a fractional relaxation of the SAA (i.e., polynomial-scenario central-distribution) problem. By complementing this via LP-rounding algorithms that provide {\em local} (i.e., per-scenario) approximation guarantees, we obtain the {\em first} approximation algorithms for the distributionally robust versions of a variety of discrete-optimization problems including set cover, vertex cover, edge cover, facility location, and Steiner tree, with guarantees that are, except for set cover, within $O(1)$-factors of the guarantees known for the deterministic version of the problem.