Linear Causal Bandits: Unknown Graph and Soft Interventions

TL;DR

This paper develops new algorithms and theoretical bounds for causal bandit problems with unknown graphs and stochastic interventions, advancing understanding of regret minimization in complex causal settings.

Contribution

It introduces the first regret bounds for causal bandits with unknown graphs and stochastic interventions, along with a computationally efficient algorithm for this setting.

Findings

Regret scales as ((cd)^{L-rac{1}{2}}\u007d ext{T} + d + RN)

Established a minimax lower bound of (d^{L-rac{3}{2}} ext{T})

Graph size N has diminishing impact on regret as T increases

Abstract

Designing causal bandit algorithms depends on two central categories of assumptions: (i) the extent of information about the underlying causal graphs and (ii) the extent of information about interventional statistical models. There have been extensive recent advances in dispensing with assumptions on either category. These include assuming known graphs but unknown interventional distributions, and the converse setting of assuming unknown graphs but access to restrictive hard/$\operatorname{do}$ interventions, which removes the stochasticity and ancestral dependencies. Nevertheless, the problem in its general form, i.e., unknown graph and unknown stochastic intervention models, remains open. This paper addresses this problem and establishes that in a graph with $N$ nodes, maximum in-degree $d$ and maximum causal path length $L$, after $T$ interaction rounds the regret upper bound scales as $\tilde{\mathcal{O}}((cd)^{L-\frac{1}{2}}\sqrt{T} + d + RN)$ where $c>1$ is a constant and $R$ is a measure of intervention power. A universal minimax lower bound is also established, which scales as $\Omega(d^{L-\frac{3}{2}}\sqrt{T})$. Importantly, the graph size $N$ has a diminishing effect on the regret as $T$ grows. These bounds have matching behavior in $T$, exponential dependence on $L$, and polynomial dependence on $d$ (with the gap $d\ $). On the algorithmic aspect, the paper presents a novel way of designing a computationally efficient CB algorithm, addressing a challenge that the existing CB algorithms using soft interventions face.