  We introduce a new class of reinforcement learning methods referred to as
{\em episodic multi-armed bandits} (eMAB). In eMAB the learner proceeds in {\em
episodes}, each composed of several {\em steps}, in which it chooses an action
and observes a feedback signal. Moreover, in each step, it can take a special
action, called the $stop$ action, that ends the current episode. After the
$stop$ action is taken, the learner collects a terminal reward, and observes
the costs and terminal rewards associated with each step of the episode. The
goal of the learner is to maximize its cumulative gain (i.e., the terminal
reward minus costs) over all episodes by learning to choose the best sequence
of actions based on the feedback. First, we define an {\em oracle} benchmark,
which sequentially selects the actions that maximize the expected immediate
gain. Then, we propose our online learning algorithm, named {\em FeedBack
Adaptive Learning} (FeedBAL), and prove that its regret with respect to the
benchmark is bounded with high probability and increases logarithmically in
expectation. Moreover, the regret only has polynomial dependence on the number
of steps, actions and states. eMAB can be used to model applications that
involve humans in the loop, ranging from personalized medical screening to
personalized web-based education, where sequences of actions are taken in each
episode, and optimal behavior requires adapting the chosen actions based on the
feedback.
