Correlation Robust Stochastic Optimization

TL;DR

This paper studies a robust stochastic optimization model that accounts for unknown correlations between variables, introduces the correlation gap concept, and provides approximation results for various cost functions, with implications for social welfare and equilibria.

Contribution

It introduces the correlation gap concept to compare robust and independent models, and provides approximation bounds for key cost functions in stochastic optimization.

Findings

Correlation gap is well-bounded for certain functions.

Approximation factors are derived for submodular, facility location, and Steiner tree functions.

New results in social welfare maximization and Walrasian equilibria.

Abstract

We consider a robust model proposed by Scarf, 1958, for stochastic optimization when only the marginal probabilities of (binary) random variables are given, and the correlation between the random variables is unknown. In the robust model, the objective is to minimize expected cost against worst possible joint distribution with those marginals. We introduce the concept of correlation gap to compare this model to the stochastic optimization model that ignores correlations and minimizes expected cost under independent Bernoulli distribution. We identify a class of functions, using concepts of summable cost sharing schemes from game theory, for which the correlation gap is well-bounded and the robust model can be approximated closely by the independent distribution model. As a result, we derive efficient approximation factors for many popular cost functions, like submodular functions, facility location, and Steiner tree. As a byproduct, our analysis also yields some new results in the areas of social welfare maximization and existence of Walrasian equilibria, which may be of independent interest.