  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 $C$.
