Unbiased Path-Rejection Langevin Corrector

(1) Integration point: modify the inference-time sampler for a neural energy model E_theta(x), where V(x)=E_theta(x) or V is a known data-consistency energy plus a neural prior. Maintain position x and momentum p. Use a cheap proposal drift ilde b_p, for example -gamma p minus a stale, cached, low-rank, or smaller-network gradient, while the target drift uses the current full gradient nabla E_theta(x). Do not alter network weights during the first experiment. (2) Pseudocode: initialize x,p; for each proposal draw Brownian increments and integrate dX=p dt, dP= ilde b_p(x,p)dt+sqrt(2 gamma)dB with a fine internal step; query the full gradient at randomized Poisson times U_i and at the endpoints; compute Delta=(-gamma p-nabla E_theta(x))- ilde b_p; accumulate the stochastic integral Delta^T dB/sqrt(2 gamma) and the quadratic term ||Delta||^2/(4 gamma); form the nonnegative unbiased estimator \hat R_T using the paper's randomized path-ratio estimator, or initially use the Poisson product estimator for the quadratic exponential and a discretized stochastic-integral factor; accept with probability min(1,\hat R_T/M), otherwise restart from the previous state. (3) Compute from the mathematics: the underdamped proposal, Girsanov correction, Poisson randomization, and rejection decision. Estimate empirically: the envelope M, the maximal discrepancy bound lambda, the acceptance rate, gradient queries per accepted sample, and bias from time discretization. Start with bounded-gradient synthetic potentials so that lambda and M are known; then test a neural energy with gradient clipping to enforce a practical bound. (4) Cheap experiment: use 2D and 10D strongly convex quartic potentials and a small MLP energy, comparing corrected proposals against MALA, ULA, and ordinary underdamped Euler using the same number of full energy-gradient calls. Measure target mean and covariance error, KL or Renyi-divergence estimates, acceptance rate, and effective samples per gradient query. Success is lower discretization bias at equal compute, or at least a 2x larger stable integration step than ULA while preserving the same distributional error. First validate unbiasedness by fixing a path and averaging thousands of ratio-estimator repetitions.