  Sequential learning problems are common in several fields of research and
practical applications. Examples include dynamic pricing and assortment, design
of auctions and incentives and permeate a large number of sequential treatment
experiments. In this paper, we extend one of the most popular learning
solutions, the $\epsilon_t$-greedy heuristics, to high-dimensional contexts
considering a conservative directive. We do this by allocating part of the time
the original rule uses to adopt completely new actions to a more focused search
in a restrictive set of promising actions. The resulting rule might be useful
for practical applications that still values surprises, although at a
decreasing rate, while also has restrictions on the adoption of unusual
actions. With high probability, we find reasonable bounds for the cumulative
regret of a conservative high-dimensional decaying $\epsilon_t$-greedy rule.
Also, we provide a lower bound for the cardinality of the set of viable actions
that implies in an improved regret bound for the conservative version when
compared to its non-conservative counterpart. Additionally, we show that
end-users have sufficient flexibility when establishing how much safety they
want, since it can be tuned without impacting theoretical properties. We
illustrate our proposal both in a simulation exercise and using a real dataset.
