Improved Bound for Robust Causal Bandits with Linear Models

TL;DR

This paper introduces a robust causal bandit algorithm for linear models that accounts for temporal fluctuations in causal structures, providing near-optimal regret bounds under model deviations.

Contribution

It proposes a new algorithm for causal bandits with time-varying models and derives tight upper and lower regret bounds considering model fluctuations.

Findings

Regret bounds depend on graph structure and model deviation.

Algorithm achieves near-optimal regret when deviations are small.

Maintains sub-linear regret for broad model fluctuation ranges.

Abstract

This paper investigates the robustness of causal bandits (CBs) in the face of temporal model fluctuations. This setting deviates from the existing literature's widely-adopted assumption of constant causal models. 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 and subject to variations over time. The goal is to design a sequence of interventions that incur the smallest cumulative regret compared to an oracle aware of the entire causal model and its fluctuations. A robust CB algorithm is proposed, and its cumulative regret is analyzed by establishing both upper and lower bounds on the regret. It is shown that in a graph with maximum in-degree $d$, length of the largest causal path $L$, and an aggregate model deviation $C$, the regret is upper bounded by $\tilde{\mathcal{O}}(d^{L-\frac{1}{2}}(\sqrt{T} + C))$ and lower bounded by $\Omega(d^{\frac{L}{2}-2}\max\{\sqrt{T}\; ,\; d^2C\})$. The proposed algorithm achieves nearly optimal $\tilde{\mathcal{O}}(\sqrt{T})$ regret when $C$ is $o(\sqrt{T})$, maintaining sub-linear regret for a broad range of $C$.