  The design and performance analysis of bandit algorithms in the presence of
stage-wise safety or reliability constraints has recently garnered significant
interest. In this work, we consider the linear stochastic bandit problem under
additional \textit{linear safety constraints} that need to be satisfied at each
round. We provide a new safe algorithm based on linear Thompson Sampling (TS)
for this problem and show a frequentist regret of order $\mathcal{O}
(d^{3/2}\log^{1/2}d \cdot T^{1/2}\log^{3/2}T)$, which remarkably matches the
results provided by (Abeille et al., 2017) for the standard linear TS algorithm
in the absence of safety constraints. We compare the performance of our
algorithm with UCB-based safe algorithms and highlight how the inherently
randomized nature of TS leads to a superior performance in expanding the set of
safe actions the algorithm has access to at each round.
