Optimality of Correlated Sampling Strategies

In the correlated sampling problem, two players are given probability distributions P and Q, respectively, over the same finite set, with access to shared randomness. Without any communication, the two players are each required to output an element sampled according to their respective distributions, while trying to minimize the probability that their outputs disagree. A well known strategy due to Kleinberg-Tardos and Holenstein, with a close variant (for a similar problem) due to Broder, solves this task with disagreement probability at most 2 delta/(1 + delta), where delta is the total variation distance between P and Q. This strategy has been used in several different contexts, including sketching algorithms, approximation algorithms based on rounding linear programming relaxations, the study of parallel repetition and cryptography. In this paper, we give a surprisingly simple proof that this strategy is essentially optimal. Specifically, for every delta is an element of (0, 1), we show that any correlated sampling strategy incurs a disagreement probability of essentially 2 delta/(1 + delta) on some inputs P and Q with total variation distance at most delta. This partially answers a recent question of Rivest. Our proof is based on studying a new problem that we call constrained agreement. Here, the two players are given subsets A subset of [n] and B subset of [n] , respectively, and their goal is to output an element i is an element of A and j is an element of B, respectively, while minimizing the probability that i not equal j. We prove tight bounds for this question, which in turn imply tight bounds for correlated sampling. Though we settle basic questions about the two problems, our formulation leads to more fine-grained questions that remain open.