  In Reinforcement Learning (RL), an agent acts in an unknown environment to
maximize the expected cumulative discounted sum of an external reward signal,
i.e., the expected return. In practice, in many tasks of interest, such as
policy optimization, the agent usually spends its interaction budget by
collecting episodes of fixed length within a simulator (i.e., Monte Carlo
simulation). However, given the discounted nature of the RL objective, this
data collection strategy might not be the best option. Indeed, the rewards
taken in early simulation steps weigh exponentially more than future rewards.
Taking a cue from this intuition, in this paper, we design an a-priori budget
allocation strategy that leads to the collection of trajectories of different
lengths, i.e., truncated. The proposed approach provably minimizes the width of
the confidence intervals around the empirical estimates of the expected return
of a policy. After discussing the theoretical properties of our method, we make
use of our trajectory truncation mechanism to extend Policy Optimization via
Importance Sampling (POIS, Metelli et al., 2018) algorithm. Finally, we conduct
a numerical comparison between our algorithm and POIS: the results are
consistent with our theory and show that an appropriate truncation of the
trajectories can succeed in improving performance.
