  Multi-armed bandits (MAB) are extensively studied in various settings where
the objective is to \textit{maximize} the actions' outcomes (i.e., rewards)
over time. Since safety is crucial in many real-world problems, safe versions
of MAB algorithms have also garnered considerable interest. In this work, we
tackle a different critical task through the lens of \textit{linear stochastic
bandits}, where the aim is to keep the actions' outcomes close to a target
level while respecting a \textit{two-sided} safety constraint, which we call
\textit{leveling}. Such a task is prevalent in numerous domains. Many
healthcare problems, for instance, require keeping a physiological variable in
a range and preferably close to a target level. The radical change in our
objective necessitates a new acquisition strategy, which is at the heart of a
MAB algorithm. We propose SALE-LTS: Safe Leveling via Linear Thompson Sampling
algorithm, with a novel acquisition strategy to accommodate our task and show
that it achieves sublinear regret with the same time and dimension dependence
as previous works on the classical reward maximization problem absent any
safety constraint. We demonstrate and discuss our algorithm's empirical
performance in detail via thorough experiments.
