Title: Towards More Theoretically-Grounded Particle Optimization Sampling for Deep Learning. Abstract: Many deep-learning based methods such as Bayesian deep learning (DL) and deep reinforcement learning (RL) have heavily relied on the ability of a model being able to efficiently explore via Bayesian sampling. Particle-optimization sampling (POS) is a recently developed technique to generate high-quality samples from a target distribution by iteratively updating a set of interactive particles, with a representative algorithm the Stein variational gradient descent (SVGD). Though obtaining significant empirical success, the {\em non-asymptotic} convergence behavior of SVGD remains unknown. In this paper, we generalize POS to a stochasticity setting by injecting random noise in particle updates, called stochastic particle-optimization sampling (SPOS). Notably, for the first time, we develop {\em non-asymptotic convergence theory} for the SPOS framework, characterizing convergence of a sample approximation w.r.t.\! the number of particles and iterations under both convex- and noncovex-energy-function settings. Interestingly, we provide theoretical understanding of a pitfall of SVGD that can be avoided in the proposed SPOS framework, {\it i.e.}, particles tend to collapse to a local mode in SVGD under some particular conditions. Our theory is based on the analysis of nonlinear stochastic differential equations, which serves as an extension and a complementary development to the asymptotic convergence theory for SVGD such as (Liu, 2017). With such theoretical guarantees, SPOS can be safely and effectively applied on both Bayesian DL and deep RL tasks. Extensive results demonstrate the effectiveness of our proposed framework.