  We provide finite-sample performance guarantees for control policies executed
on stochastic robotic systems. Given an open- or closed-loop policy and a
finite set of trajectory rollouts under the policy, we bound the expected
value, value-at-risk, and conditional-value-at-risk of the trajectory cost, and
the probability of failure in a sparse cost setting. The bounds hold, with
user-specified probability, for any policy synthesis technique and can be seen
as a post-design safety certification. Generating the bounds only requires
sampling simulation rollouts, without assumptions on the distribution or
complexity of the underlying stochastic system. We adapt these bounds to also
give a constraint satisfaction test to verify safety of the robot system. We
provide a thorough analysis of the bound sensitivity to sim-to-real
distribution shifts and provide results for constructing robust bounds that can
tolerate some specified amount of distribution shift. Furthermore, we extend
our method to apply when selecting the best policy from a set of candidates,
requiring a multi-hypothesis correction. We show the statistical validity of
our bounds in the Ant, Half-cheetah, and Swimmer MuJoCo environments and
demonstrate our constraint satisfaction test with the Ant. Finally, using the
20 degree-of-freedom MuJoCo Shadow Hand, we show the necessity of the
multi-hypothesis correction.
