  The contextual bandit literature has traditionally focused on algorithms that
address the exploration-exploitation tradeoff. In particular, greedy algorithms
that exploit current estimates without any exploration may be sub-optimal in
general. However, exploration-free greedy algorithms are desirable in practical
settings where exploration may be costly or unethical (e.g., clinical trials).
Surprisingly, we find that a simple greedy algorithm can be rate optimal
(achieves asymptotically optimal regret) if there is sufficient randomness in
the observed contexts (covariates). We prove that this is always the case for a
two-armed bandit under a general class of context distributions that satisfy a
condition we term covariate diversity. Furthermore, even absent this condition,
we show that a greedy algorithm can be rate optimal with positive probability.
Thus, standard bandit algorithms may unnecessarily explore. Motivated by these
results, we introduce Greedy-First, a new algorithm that uses only observed
contexts and rewards to determine whether to follow a greedy algorithm or to
explore. We prove that this algorithm is rate optimal without any additional
assumptions on the context distribution or the number of arms. Extensive
simulations demonstrate that Greedy-First successfully reduces exploration and
outperforms existing (exploration-based) contextual bandit algorithms such as
Thompson sampling or upper confidence bound (UCB).
