Stein Variational Gradient Descent (SVGD) is a popular variational inference algorithm which simulates an interacting particle system to approximately sample from a target distribution, with impressive empirical performance across various domains. Theoretically, its population (i.e, infinite-particle) limit dynamics is well studied but the behavior of SVGD in the finite-particle regime is much less understood. In this work, we design two computationally efficient variants of SVGD, namely VP-SVGD and GB-SVGD, with provably fast finite-particle convergence rates. We introduce the notion of virtual particles and develop novel stochastic approximations of population-limit SVGD dynamics in the space of probability measures, which are exactly implementable using a finite number of particles. Our algorithms can be viewed as specific random-batch approximations of SVGD, which are computationally more efficient than ordinary SVGD. We show that the $n$ particles output by VP-SVGD and GB-SVGD, run for $T$ steps with batch-size $K$, are at-least as good as i.i.d samples from a distribution whose Kernel Stein Discrepancy to the target is at most $O\left(\tfrac{d^{1/3}}{(KT)^{1/6}}\right)$ under standard assumptions. Our results also hold under a mild growth condition on the potential function, which is much weaker than the isoperimetric (e.g. Poincare Inequality) or information-transport conditions (e.g. Talagrand's Inequality $\mathsf{T}_1$) generally considered in prior works. As a corollary, we consider the convergence of the empirical measure (of the particles output by VP-SVGD and GB-SVGD) to the target distribution and demonstrate a double exponential improvement over the best known finite-particle analysis of SVGD. Beyond this, our results present the first known oracle complexities for this setting with polynomial dimension dependence.