  We study a variant of the stochastic linear bandit problem wherein we
optimize a linear objective function but rewards are accrued only orthogonal to
an unknown subspace (which we interpret as a \textit{protected space}) given
only zero-order stochastic oracle access to both the objective itself and
protected subspace. In particular, at each round, the learner must choose
whether to query the objective or the protected subspace alongside choosing an
action. Our algorithm, derived from the OFUL principle, uses some of the
queries to get an estimate of the protected space, and (in almost all rounds)
plays optimistically with respect to a confidence set for this space. We
provide a $\tilde{O}(sd\sqrt{T})$ regret upper bound in the case where the
action space is the complete unit ball in $\mathbb{R}^d$, $s < d$ is the
dimension of the protected subspace, and $T$ is the time horizon. Moreover, we
demonstrate that a discrete action space can lead to linear regret with an
optimistic algorithm, reinforcing the sub-optimality of optimism in certain
settings. We also show that protection constraints imply that for certain
settings, no consistent algorithm can have a regret smaller than
$\Omega(T^{3/4}).$ We finally empirically validate our results with synthetic
and real datasets.
