Improved Bound for Robust Causal Bandits with Linear Models

This paper investigates the robustness of causal bandits (CBs) in the face oftemporal model fluctuations. This setting deviates from the existingliterature's widely-adopted assumption of constant causal models. The focus ison causal systems with linear structural equation models (SEMs). The SEMs andthe time-varying pre- and post-interventional statistical models are allunknown and subject to variations over time. The goal is to design a sequenceof interventions that incur the smallest cumulative regret compared to anoracle aware of the entire causal model and its fluctuations. A robust CBalgorithm is proposed, and its cumulative regret is analyzed by establishingboth upper and lower bounds on the regret. It is shown that in a graph withmaximum in-degree d, length of the largest causal path L, and an aggregatemodel deviation C, the regret is upper bounded by𝒪̃(d^L-1/2(√(T) + C)) and lower bounded byΩ(d^L/2-2max{√(T) , d^2C}). The proposed algorithmachieves nearly optimal 𝒪̃(√(T)) regret when C iso(√(T)), maintaining sub-linear regret for a broad range of C.