  This paper discusses a scenario approach to robust optimization of a blackbox
function in a bandit setting. We assume that the blackbox function can be
modeled as a Gaussian Process (GP) for every realization of the uncertain
parameter. We adopt a scenario approach in which we draw fixed independent
samples of the uncertain parameter. For a given policy, i.e., a sequence of
query points and uncertain parameters in the sampled set, we introduce a notion
of regret defined with respect to additional draws of the uncertain parameter,
termed as scenario regret under re-draw. We present a scenario-based iterative
algorithm using the upper confidence bound (UCB) of the fixed independent
scenarios to compute a policy for the blackbox optimization. For this
algorithm, we characterize a high probability upper bound on the regret under
re-draw for any finite number of iterations of the algorithm. We further
characterize parameter regimes in which the regret tends to zero asymptotically
with the number of iterations with high probability. Finally, we supplement our
analysis with numerical results.
