Typically, a Reinforcement Learning learning problem is formalized as finding an optimal policy for a Markov Decision Process (MDP). In many real-life situations, however, an agent can only get partial information from the environment. For example, Partially Observable MDPs are used to model the case where the agent does not fully observe the state.
I was wondering whether there is any well-established formalism for the case where the agent does not fully observe the reward signal.
In particular, I am thinking about the case where for every state-action pair $(s, a)$ the agent receives the reward $R(s, a)$ with probability $1 - \varepsilon$ and does not receive anything with probability $\varepsilon$. Of course, in principle, this setting can be thought as a regular MDP with a stochastic reward, but here I would like the agent to behave optimally w.r.t. to $R$.
I would really appreciate if you could point some relevant literature to me!