Two-query SPSA Langevin sampler for black-box neural energies

Integrate this at the transition kernel of a latent-variable or parameter sampler, rather than inside ordinary supervised backpropagation. Let $x$ be a latent vector, simulator state, or Bayesian neural-network parameter vector, and let energy(x) return a noisy scalar $\tilde f(x)$; it may execute a neural decoder, renderer, simulator, or black-box validation score. At iteration $k$, draw a fresh Rademacher vector Delta, evaluate y_plus = energy(x + c_k*Delta) and y_minus = energy(x - c_k*Delta), and form g_hat = ((y_plus-y_minus)/(2*c_k))*Delta. Draw standard Gaussian xi and update x = x - h_k*g_hat + sqrt(2*h_k)*xi. Use h_k=h0/(k+1)^alpha and c_k=c0/(k+1)^beta; initialize with alpha=0.6, beta=0.1, and tune h0 using a short pilot run. Clip updates whose norm exceeds a fixed multiple of the running median, while recording clipping frequency. Estimate oracle noise by repeated calls at one point: delta_hat^2 = 0.5*Var(energy(x)), and tune c0 so the finite-difference signal exceeds this noise. First test on a 2D or 10D Gaussian-mixture energy, then on a 32D VAE latent posterior whose decoder contains a nondifferentiable simulator. Compare against random-walk Metropolis, one-sided zeroth-order Langevin, and exact-gradient ULA when available, matching total energy evaluations. Measure MMD or $W_2$ distance to reference samples, effective sample size per oracle call, and divergence rate. Success means lower sampling error at equal two-query budgets and stable chains as $k$ increases; failure is SPSA variance overwhelming the drift in moderate dimension.