  The standard feedback model of reinforcement learning requires revealing the
reward of every visited state-action pair. However, in practice, it is often
the case that such frequent feedback is not available. In this work, we take a
first step towards relaxing this assumption and require a weaker form of
feedback, which we refer to as \emph{trajectory feedback}. Instead of observing
the reward obtained after every action, we assume we only receive a score that
represents the quality of the whole trajectory observed by the agent, namely,
the sum of all rewards obtained over this trajectory. We extend reinforcement
learning algorithms to this setting, based on least-squares estimation of the
unknown reward, for both the known and unknown transition model cases, and
study the performance of these algorithms by analyzing their regret. For cases
where the transition model is unknown, we offer a hybrid optimistic-Thompson
Sampling approach that results in a tractable algorithm.
