  Particle-optimization-based sampling (POS) is a recently developed effective
sampling technique that interactively updates a set of particles. A
representative algorithm is the Stein variational gradient descent (SVGD). We
prove, under certain conditions, SVGD experiences a theoretical pitfall, {\it
i.e.}, particles tend to collapse. As a remedy, we generalize POS to a
stochastic setting by injecting random noise into particle updates, thus
yielding particle-optimization sampling (SPOS). Notably, for the first time, we
develop {\em non-asymptotic convergence theory} for the SPOS framework (related
to SVGD), characterizing algorithm convergence in terms of the 1-Wasserstein
distance w.r.t.\! the numbers of particles and iterations. Somewhat
surprisingly, with the same number of updates (not too large) for each
particle, our theory suggests adopting more particles does not necessarily lead
to a better approximation of a target distribution, due to limited
computational budget and numerical errors. This phenomenon is also observed in
SVGD and verified via an experiment on synthetic data. Extensive experimental
results verify our theory and demonstrate the effectiveness of our proposed
framework.
