Entropy-dissipating Stein-Langevin particle optimizer

Integrate the method at the optimizer or sampler level by maintaining \(N\) complete neural-network parameter vectors \(\theta_1,\ldots,\theta_N\). Let \(V(\theta)\) be the negative log posterior, or a scaled training loss plus a prior penalty. At every step, use the same minibatch for all particles so the Stein interaction sees comparable stochastic gradients. Compute \(g_j=\nabla_\theta V(\theta_j)\), pairwise kernels \(K_{ij}=k(\theta_i,\theta_j)\), and kernel derivatives. For an RBF kernel, use \(K_{ij}=\exp(-\|\theta_i-\theta_j\|^2/(2h^2))\) and \(\nabla_{\theta_j}K_{ij}=K_{ij}(\theta_i-\theta_j)/h^2\). Pseudocode: compute all \(g_j\); for each \(i\), initialize \(v_i=0\); for each \(j\), add \((-K_{ij}g_j+\nabla_{\theta_j}K_{ij})/N\); update \(\theta_i\leftarrow\theta_i+\eta v_i+\sqrt{2\eta T}\xi_i\). Here \(T\) is the temperature and \(\xi_i\) is standard Gaussian noise. Estimate bandwidth \(h\) from the median pairwise distance, optionally using a random parameter projection. Compute empirical KSD from the RKHS velocity norm and track particle diversity; these are diagnostics, while the entropy identity is the mathematical motivation rather than a directly observed quantity. First test on a two-layer MLP on MNIST with \(N=8\) or \(16\), comparing independent Adam, pure SVGD, SGLD, and the hybrid at equal gradient-evaluation budgets. Measure test NLL, accuracy, calibration error, empirical KSD, pairwise output diversity, and particle-collapse rate. Success means lower test NLL or calibration error at matched compute, stable last-iterate performance, and larger useful diversity without uncontrolled loss variance.