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：通过因果Bandit中的序贯干预设计，可有效建模因果系统中为优化奖励函数而进行的序贯实验设计。现有因果Bandit文献的关键假设是因果模型随时间保持不变，但这一假设在持续经历时变模型波动的复杂系统中未必成立。本文研究因果Bandit对此类模型波动的鲁棒性，聚焦于具有线性结构方程模型的因果系统。其中结构方程模型、时变干预前/后统计模型均未知。以累积遗憾作为设计准则，目标是设计能产生最小累积遗憾（相对于知晓完整因果模型及其波动的预言机）的干预序列。首先证明，即使仅出现少量模型偏差实例，现有方法也无法维持遗憾亚线性。具体而言，当模型偏差实例数仅达$T^\frac{1}{2L}$量级时（$T$为时间跨度，$L$为图中最长因果路径长度），现有算法将产生关于$T$的线性遗憾。其次，设计鲁棒因果Bandit算法并分析其遗憾，建立遗憾的上界与信息论下界。在含$N$个节点、最大度为$d$的图中，给定通用模型偏差度量$C$，累积遗憾上界为$\tilde{\mathcal{O}}(d^{L-\frac{1}{2}}(\sqrt{NT} + NC))$，下界为$\Omega(d^{\frac{L}{2}-2}\max\{\sqrt{T},d^2C\})$。通过对比这些界可知，当$C$为$o(\sqrt{T})$时，所提算法可实现近最优的$\tilde{\mathcal{O}}(\sqrt{T})$遗憾，并在更大范围$C$下保持亚线性遗憾。