Accelerated randomized Hamiltonian posterior sampler

Integrate this at the sampling component for a Bayesian neural-network head, not at ordinary deterministic SGD. First train a feature extractor and freeze it. Let z_i be its features and let w be the final linear or small MLP-head parameters. Use a Gaussian prior and a convex likelihood such as binary logistic regression or squared-error regression, giving f(w)=-log p(D|w)+(lambda/2)||w||^2. For logistic regression, estimate a conservative strong-convexity lower bound alpha>=lambda and set tau=c/sqrt(alpha), with c swept over 0.25, 0.5, 1, and 2. At every sampler iteration: (1) draw y from N(0,I); (2) draw T from the triangular density above, or from an exponential distribution with mean tau; (3) choose leapfrog step size h and integer L=ceil(T/h); (4) perform L leapfrog updates, using y <- y-(h/2) gradient f(w), w <- w+h y, and y <- y-(h/2) gradient f(w); (5) accept with probability min(1, exp(-H(w',y')+H(w,y))), or monitor energy error if Metropolis correction is too expensive; (6) store w' and discard y'. The paper supplies the alpha^{-1/2} trajectory-scale rule and the mixing target; estimate alpha, gradient noise, acceptance rate, and effective sample size empirically. Start with MNIST or CIFAR-10 frozen ResNet features and a Bayesian logistic head. Compare against unadjusted Langevin, preconditioned Langevin, and fixed-length HMC at equal gradient evaluations. Success means higher effective samples per gradient evaluation, faster decay of predictive posterior error, and stable acceptance above 0.7; test whether the advantage grows as alpha decreases.