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Two-stage stochastic optimization is a widely used 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 common criticism levied at this model is that the underlying probability distribution is itself often imprecise! To address this, an approach that is quite versatile and has gained popularity in the stochastic-optimization literature is the distributionally robust 2-stage model: given a collection D of probability distributions, our goal now is to minimize the maximum expected total cost with respect to a distribution in D. We provide a framework for designing approximation algorithms in such settings when the collection D is a ball around a central distribution and the central distribution is accessed only via a sampling black box. We first show that one can utilize the sample average approximation (SAA) method—solve the distributionally robust problem with an empirical estimate of the central distribution—to reduce the problem to the case where the central distribution has polynomial-size support. Complementing this, we show how to approximately solve a fractional relaxation of the SAA (i.e., polynomial-scenario central-distribution) problem. Unlike in 2-stage stochastic- or robust- optimization, this turns out to be quite challenging. We utilize the ellipsoid method in conjunction with several new ideas to show that this problem can be approximately solved provided that we have an (approximation) algorithm for a certain max-min problem that is akin to, and generalizes, the k-max-min problem—find the worst-case scenario consisting of at most k elements—encountered in 2-stage robust optimization. We obtain such a procedure for various discrete-optimization problems; by complementing this via LP-rounding algorithms that provide local (i.e., per-scenario) approximation guarantees, we obtain the 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.

Approximation algorithms for distributionally-robust stochastic optimization with black-box distributions