  We study how to learn optimal interventions sequentially given causal
information represented as a causal graph along with associated conditional
distributions. Causal modeling is useful in real world problems like online
advertisement where complex causal mechanisms underlie the relationship between
interventions and outcomes. We propose two algorithms, causal upper confidence
bound (C-UCB) and causal Thompson Sampling (C-TS), that enjoy improved
cumulative regret bounds compared with algorithms that do not use causal
information. We thus resolve an open problem posed by
\cite{lattimore2016causal}. Further, we extend C-UCB and C-TS to the linear
bandit setting and propose causal linear UCB (CL-UCB) and causal linear TS
(CL-TS) algorithms. These algorithms enjoy a cumulative regret bound that only
scales with the feature dimension. Our experiments show the benefit of using
causal information. For example, we observe that even with a few hundreds of
iterations, the regret of causal algorithms is less than that of standard
algorithms by a factor of three. We also show that under certain causal
structures, our algorithms scale better than the standard bandit algorithms as
the number of interventions increases.
