The pair-matching problem appears in many applications where one wants to discover good matches between pairs of entities or individuals. Formally, the set of individuals is represented by the nodes of a graph where the edges, unobserved at first, represent the good matches. The algorithm queries pairs of nodes and observes the presence/absence of edges. Its goal is to discover as many edges as possible with a fixed budget of queries. Pair-matching is a particular instance of multi-armed bandit problem in which the arms are pairs of individuals and the rewards are edges linking these pairs. This bandit problem is non-standard though, as each arm can only be played once. Given this last constraint, sublinear regret can be expected only if the graph presents some underlying structure. This paper shows that sublinear regret is achievable in the case where the graph is generated according to a Stochastic Block Model (SBM) with two communities. Optimal regret bounds are computed for this pair-matching problem. They exhibit a phase transition related to the Kesten-Stigum threshold for community detection in SBM. The pair-matching problem is considered in the case where each node is constrained to be sampled less than a given amount of times. We show how optimal regret rates depend on this constraint. The paper is concluded by a conjecture regarding the optimal regret when the number of communities is larger than 2. Contrary to the two communities case, we argue that a statistical-computational gap would appear in this problem.

翻译：摘要：配对匹配问题出现在许多需要发现实体或个体对之间良好匹配的应用中。形式上，个体集合由图的节点表示，边（最初不可观测）代表良好匹配。算法查询节点对并观察边的存在与否，其目标是在固定查询预算下发现尽可能多的边。配对匹配是多臂老虎机问题的一个特定实例，其中臂是个体对，奖励是连接这些对的边。然而，该老虎机问题非常规，因为每条臂只能被使用一次。鉴于这一约束，仅当图具有某种底层结构时，才能实现次线性遗憾。本文表明，在图根据具有两个社区的随机块模型（SBM）生成的情况下，可以达成次线性遗憾。针对该配对匹配问题，计算了最优遗憾界，其展现出与SBM中社区检测的Kesten-Stigum阈值相关的相变。此外，考虑了每个节点被采样次数受限于给定值的配对匹配情形，并展示了最优遗憾率如何依赖于这一约束。论文最后提出一个猜想：当社区数量大于2时，最优遗憾将出现统计-计算鸿沟，这与两个社区的情况形成对比。