  In this work, we analyze an efficient sampling-based algorithm for
general-purpose reachability analysis, which remains a notoriously challenging
problem with applications ranging from neural network verification to safety
analysis of dynamical systems. By sampling inputs, evaluating their images in
the true reachable set, and taking their $\epsilon$-padded convex hull as a set
estimator, this algorithm applies to general problem settings and is simple to
implement. Our main contribution is the derivation of asymptotic and
finite-sample accuracy guarantees using random set theory. This analysis
informs algorithmic design to obtain an $\epsilon$-close reachable set
approximation with high probability, provides insights into which reachability
problems are most challenging, and motivates safety-critical applications of
the technique. On a neural network verification task, we show that this
approach is more accurate and significantly faster than prior work. Informed by
our analysis, we also design a robust model predictive controller that we
demonstrate in hardware experiments.
