expected return (the objective)

The discounted return for any single run is a random number — the environment may roll dice, and the agent itself may act randomly. So we cannot optimize one lucky episode; instead reinforcement learning aims to maximize the expected return, the average return you would get if you played the same policy over and over forever. This single number is the north star of the whole field: every algorithm, however it dresses it up, is ultimately trying to push this average higher.

Formally the objective is the expectation of the return under the distribution of trajectories that the policy and environment together produce. Because it is an average over all the ways an episode could unfold, improving it means doing better on the whole, not just getting lucky once. This framing matters: an agent that occasionally scores big but usually fails has a low expected return, and RL will correctly prefer a steadier policy that wins more on average.