Stein variational gradient descent

⚠️ Curated Limitations & Negative Results

• The population derivation assumes smooth densities, required moments, Stein boundary/integration conditions and differentiable valid transforms. A local inverse-function argument alone does not establish a global invertible transport for arbitrary finite steps or domains.
• The empirical particle measure is atomic and has infinite KL relative to a continuous target. Its vector field approximates a population calculation; neither the infinitesimal argument nor the 2016 empirical results establish exact finite-particle sampling, global finite-step descent or universal convergence.
• RBF repulsion does not universally prevent collapse: initially coincident particles have equal updates and remain coincident. Kernel and bandwidth selection matter, and the median-squared/log(n) experiment heuristic is undefined or degenerate at n=1 or zero median unless an additional rule is supplied.
• The exact one-particle score-ascent reduction uses a unit diagonal kernel with zero first-argument diagonal derivative, as for RBF. A nonunit diagonal changes the step. For distinct fixed particles in the small-bandwidth RBF limit the direction includes 1/n; a claim of independent gradient ascent must respect that scale and does not cover coincident particles.
• Minibatch and subsampled-particle variants require their own sampling and scaling conditions; they need not equal the deterministic full-score/full-sum update. Direct kernel evaluation requires a quadratic number of pair interactions in n, and the source does not establish universal runtime or posterior-quality superiority. No software fallback, training or benchmark execution was inspected.
• The official supplement Lemma A.1 has a leading-sign inconsistency with its proof and the main negative KL derivative. Main Equation 9 displays log densities rather than the gradient estimator described in its prose. These source-local displays are retained rather than silently promoted into operational formulas; later arXiv v3 still contains them in the inspected passages.
• The official toy description says one particle while its Figure 1 caption says 100; no quantitative toy claim is adopted. This record covers the original scalar-kernel particle method, not later amortized samplers, Stein policy gradients, matrix-kernel methods, gradient-free variants or later convergence theorems.