# Improved Bound for Robust Causal Bandits with Linear Models

CoRR（2024）

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
𝒪̃(d^L-1/2(√(T) + C)) and lower bounded by
Ω(d^L/2-2max{√(T) , d^2C}). The proposed algorithm
achieves nearly optimal 𝒪̃(√(T)) regret when C is
o(√(T)), maintaining sub-linear regret for a broad range of C.

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Key words

Linear Model,Lower Bound,Upper Bound,Structural Equation Modeling,Causal Model,Causal Path,Causal System,Absolute Difference,Weight Matrix,Ordinary Least Squares,Column Vector,Treatment Model,Graphical Model,Graph Structure,Model Misspecification,Directed Acyclic Graph,Ordinary Least Squares Estimates,Outlier Samples,Linear Graph,Causal Structure,Upper Confidence Bound,Multi-armed Bandit,Bandit Problem,Gram Matrix,Causal Graph,Reproducing Kernel Hilbert Space,Nominal Model,Nodes In The Graph,Euclidean Norm

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