Causal discovery from interventional data is an important problem, where the task is to design an interventional strategy that learns the hidden ground truth causal graph $G(V,E)$ on $|V| = n$ nodes while minimizing the number of performed interventions. Most prior interventional strategies broadly fall into two categories: non-adaptive and adaptive. Non-adaptive strategies decide on a single fixed set of interventions to be performed while adaptive strategies can decide on which nodes to intervene on sequentially based on past interventions. While adaptive algorithms may use exponentially fewer interventions than their non-adaptive counterparts, there are practical concerns that constrain the amount of adaptivity allowed. Motivated by this trade-off, we study the problem of $r$-adaptivity, where the algorithm designer recovers the causal graph under a total of $r$ sequential rounds whilst trying to minimize the total number of interventions. For this problem, we provide a $r$-adaptive algorithm that achieves $O(\min\{r,\log n\} \cdot n^{1/\min\{r,\log n\}})$ approximation with respect to the verification number, a well-known lower bound for adaptive algorithms. Furthermore, for every $r$, we show that our approximation is tight. Our definition of $r$-adaptivity interpolates nicely between the non-adaptive ($r=1$) and fully adaptive ($r=n$) settings where our approximation simplifies to $O(n)$ and $O(\log n)$ respectively, matching the best-known approximation guarantees for both extremes. Our results also extend naturally to the bounded size interventions.

翻译：从干预数据中发现因果关系是一个重要问题，其任务是设计一种干预策略，通过学习隐藏的真实因果图$G(V,E)$（其中$|V| = n$个节点）的同时最小化执行的干预次数。以往大多数干预策略大致分为两类：非自适应和自适应。非自适应策略决定执行一组固定的干预方案，而自适应策略可根据以往干预结果顺序决定对哪些节点进行干预。尽管自适应算法可能比非自适应算法使用指数级更少的干预次数，但实际应用中对允许的自适应程度存在限制。受此权衡启发，我们研究了$r$-自适应问题，即算法设计者在总共$r$个顺序轮次中恢复因果图，同时力求最小化总干预次数。针对该问题，我们提出一种$r$-自适应算法，其关于验证数（自适应算法的一个著名下界）的近似比达到$O(\min\{r,\log n\} \cdot n^{1/\min\{r,\log n\}})$。此外，对于每个$r$，我们证明该近似比是紧的。我们对$r$-自适应的定义在非自适应（$r=1$）和完全自适应（$r=n$）设置之间实现了平滑插值，此时近似比分别简化为$O(n)$和$O(\log n)$，与两种极端情况下的已知最优近似保证相匹配。我们的结果还可自然扩展到有界规模干预场景。