In combinatorial causal bandits (CCB), the learning agent chooses a subset of variables in each round to intervene and collects feedback from the observed variables to minimize expected regret or sample complexity. Previous works study this problem in both general causal models and binary generalized linear models (BGLMs). However, all of them require prior knowledge of causal graph structure. This paper studies the CCB problem without the graph structure on binary general causal models and BGLMs. We first provide an exponential lower bound of cumulative regrets for the CCB problem on general causal models. To overcome the exponentially large space of parameters, we then consider the CCB problem on BGLMs. We design a regret minimization algorithm for BGLMs even without the graph skeleton and show that it still achieves $O(\sqrt{T}\ln T)$ expected regret. This asymptotic regret is the same as the state-of-art algorithms relying on the graph structure. Moreover, we sacrifice the regret to $O(T^{\frac{2}{3}}\ln T)$ to remove the weight gap covered by the asymptotic notation. At last, we give some discussions and algorithms for pure exploration of the CCB problem without the graph structure.

翻译：在组合因果强盗（CCB）问题中，学习代理在每一轮中选择变量子集进行干预，并从观测变量中收集反馈，以最小化期望遗憾或样本复杂度。先前的工作分别在一般因果模型和二元广义线性模型（BGLMs）中研究了该问题。然而，所有这些方法都需要先验的因果图结构知识。本文研究了无图结构条件下二元一般因果模型和BGLMs上的CCB问题。首先，我们给出了一般因果模型上CCB问题的累积遗憾指数下界。为克服参数空间的指数级规模，我们进而考虑BGLMs上的CCB问题。我们设计了一种无需图骨架即可用于BGLMs的遗憾最小化算法，并证明其仍能达到$O(\sqrt{T}\ln T)$的期望遗憾。该渐近遗憾与依赖图结构的最优算法相同。此外，我们将遗憾牺牲至$O(T^{\frac{2}{3}}\ln T)$以消除渐近符号所覆盖的权重差距。最后，我们给出了无图结构条件下CCB问题纯探索的一些讨论与算法。