Particle-based inference algorithm is a promising method to efficiently generate samples for an intractable target distribution by iteratively updating a set of particles. As a noticeable example, Stein variational gradient descent (SVGD) provides a deterministic and computationally efficient update, but it is known to underestimate the variance in high dimensions, the mechanism of which is poorly understood. In this work we explore a connection between SVGD and MMD-based inference algorithm via Stein's lemma. By comparing the two update rules, we identify the source of bias in SVGD as a combination of high variance and deterministic bias, and empirically demonstrate that the removal of either factors leads to accurate estimation of the variance. In addition, for learning high-dimensional Gaussian target, we analytically derive the converged variance for both algorithms, and confirm that only SVGD suffers from the "curse of dimensionality".
Analyze the underlying mechanisms of variance collapse of SVGD in high dimensions.