Sequential design of experiments for optimizing a reward function in causal systems can be effectively modeled by the sequential design of interventions in causal bandits (CBs). In the existing literature on CBs, a critical assumption is that the causal models remain constant over time. However, this assumption does not necessarily hold in complex systems, which constantly undergo temporal model fluctuations. This paper addresses the robustness of CBs to such model fluctuations. The focus is on causal systems with linear structural equation models (SEMs). The SEMs and the time-varying pre- and post-interventional statistical models are all unknown. Cumulative regret is adopted as the design criteria, based on which the objective is to design a sequence of interventions that incur the smallest cumulative regret with respect to an oracle aware of the entire causal model and its fluctuations. First, it is established that the existing approaches fail to maintain regret sub-linearity with even a few instances of model deviation. Specifically, when the number of instances with model deviation is as few as $T^\frac{1}{2L}$, where $T$ is the time horizon and $L$ is the longest causal path in the graph, the existing algorithms will have linear regret in $T$. Next, a robust CB algorithm is designed, and its regret is analyzed, where upper and information-theoretic lower bounds on the regret are established. Specifically, in a graph with $N$ nodes and maximum degree $d$, under a general measure of model deviation $C$, the cumulative regret is upper bounded by $\tilde{\mathcal{O}}(d^{L-\frac{1}{2}}(\sqrt{NT} + NC))$ and lower bounded by $\Omega(d^{\frac{L}{2}-2}\max\{\sqrt{T},d^2C\})$. Comparing these bounds establishes that the proposed algorithm achieves nearly optimal $\tilde{\mathcal{O}}(\sqrt{T})$ regret when $C$ is $o(\sqrt{T})$ and maintains sub-linear regret for a broader range of $C$.

翻译：在因果系统中优化奖励函数的序贯实验设计可通过因果赌博机中的序贯干预设计有效建模。现有因果赌博机文献的一个关键假设是因果模型随时间保持不变。然而，该假设在经历持续性时间模型波动的复杂系统中并不必然成立。本文研究了因果赌博机对此类模型波动的鲁棒性。重点关注采用线性结构方程模型的因果系统。结构方程模型以及时变的干预前与干预后统计模型均未知。采用累积遗憾作为设计准则，目标是设计一系列干预序列，使其相对于知晓完整因果模型及其波动的理想基准产生最小的累积遗憾。首先证明：即使只出现少数模型偏离实例，现有方法也无法维持遗憾的次线性性。具体而言，当模型偏离发生次数仅为$T^\frac{1}{2L}$时（其中$T$为时间范围，$L$为图中最长因果路径），现有算法将产生关于$T$的线性遗憾。接下来，设计了一种鲁棒因果赌博机算法，并分析其遗憾，建立了关于遗憾的上界与信息论下界。具体而言，在包含$N$个节点、最大度为$d$的图中，在模型偏离的一般度量$C$下，累积遗憾的上界为$\tilde{\mathcal{O}}(d^{L-\frac{1}{2}}(\sqrt{NT} + NC))$，下界为$\Omega(d^{\frac{L}{2}-2}\max\{\sqrt{T},d^2C\})$。对比这些界限表明：当$C$为$o(\sqrt{T})$时，所提算法实现了近乎最优的$\tilde{\mathcal{O}}(\sqrt{T})$遗憾，并在更广泛的$C$范围内保持了亚线性遗憾。