Price of Correlations in Stochastic Optimization

Abstract

When decisions are made in the presence of large-scale stochastic data, it is common to pay more attention to the easy-to-see statistics (e.g., mean) instead of the underlying correlations. One reason is that it is often much easier to solve a stochastic optimization problem by assuming independence across the random data. In this paper, we study the possible loss incurred by ignoring these correlations through a distributionally-robust stochastic programming model, and propose a new concept called Price of Correlations (POC) to quantify that loss. We show that the POC has a small upper bound for a wide class of cost functions, including uncapacitated facility location, Steiner tree and submodular functions, suggesting that the intuitive approach of assuming independent distribution may actually work well for these stochastic optimization problems. On the other hand, we demonstrate that for some cost functions, POC can be particularly large, e.g., the supermodular functions. We propose alternative ways to solve the corresponding distributionally robust models for these functions. As a byproduct, our analysis yields new results on social welfare maximization and the existence of Walrasian equilibria, which may be of independent interest.

Maximizing Stochastic Monotone Submodular Functions

“…Their results show that our Theorems 3 and 4 will also hold in an adaptive submodular setting. In addition, see Agrawal et al (2012) who introduce the notion of price of correlations and study the effect of ignoring correlations in optimization problems.…”

Combinatorial Prophet Inequalities

“…For a monotone submodular function f , [14] showed that the expected value of f over any distribution of subsets is at most a constant factor larger than the expectation over subsets drawn from the product distribution with the same marginals. This bound on the correlation gap has been very useful in the past decade with applications in optimization [17], mechanism design [1,51,11,8], influence in social networks [46,7], and recommendation systems [40].…”

“…We recall some notation for to extend submodular functions from the discrete hypercube {0, 1} n to relaxations whose domain is the continuous hypercube [0,1] n . For any vector x ∈ [0, 1] n , let S ∼ x denote a random set S that contains each element i ∈ [n] independently w.p.…”

“…Roughly, on each of n days, the decision maker knows an independent prior distribution over k potential items that could appear. 1 She also has access to a combinatorial (in particular, submodular or monotone subadditive) function f that describes the value of any subset of the n · k items (see Section 2 for a formal definition and further discussion). We obtain the following combinatorial prophet inequalities: Theorem 1.1.…”

Bayesian Mechanism Design

“…(In the first-price auction an agent with value v wins with probability v and pays her bid which is v/2; in the secondprice auction an agent with value v wins with probability v and pays the expected value of the other agent conditioned on being at most v which is v/2.) Both the first-and second-price auction with bidder values drawn uniformly from [0,1] are examples of independent, single-dimensional, and linear utilities. In these auction problems, the auctioneer also has a feasibility constraint that at most one agent can win the item, i.e., i x i ≤ 1.…”