  Many physical systems have underlying safety considerations that require that
the strategy deployed ensures the satisfaction of a set of constraints.
Further, often we have only partial information on the state of the system. We
study the problem of safe real-time decision making under uncertainty. In this
paper, we formulate a conservative stochastic contextual bandit formulation for
real-time decision making when an adversary chooses a distribution on the set
of possible contexts and the learner is subject to certain safety/performance
constraints. The learner observes only the context distribution and the exact
context is unknown, and the goal is to develop an algorithm that selects a
sequence of optimal actions to maximize the cumulative reward without violating
the safety constraints at any time step. By leveraging the UCB algorithm for
this setting, we propose a conservative linear UCB algorithm for stochastic
bandits with context distribution. We prove an upper bound on the regret of the
algorithm and show that it can be decomposed into three terms: (i) an upper
bound for the regret of the standard linear UCB algorithm, (ii) a constant term
(independent of time horizon) that accounts for the loss of being conservative
in order to satisfy the safety constraint, and (ii) a constant term
(independent of time horizon) that accounts for the loss for the contexts being
unknown and only the distribution being known. To validate the performance of
our approach we perform extensive simulations on synthetic data and on
real-world maize data collected through the Genomes to Fields (G2F) initiative.
