Gradient-Free Sampling from Generative Models via Stochastic Bounded Extremum Seeking

We introduce a sampling approach for energy-based and score-based generative models that requires no gradient evaluations of the model. Replacing the drift term that would normally contain the score ${\nabla }_{\mathbf{x}}\log p_{\theta }(\mathbf{x})$ with a high-frequency dithered cosine of the model's $\text{value}$, $\sqrt{\alpha \omega }\cos (\omega t+k\log p_{\theta }(\mathbf{x}))$, produces, in the high-frequency averaging limit, Langevin Markov chain Monte Carlo for energy-based models and the reverse-time SDE of score-based diffusion. We prove that trajectories of the dithered Itô SDE converge to those of the target SDE, driven by the same Brownian motion, uniformly on compact time intervals in probability, by an averaging argument that extends bounded extremum seeking (ES) to Itô processes, with an explicit $O({\omega }^{-1/2})$ mean-square rate under global bounds. The algorithm does not require derivatives: only forward evaluations are used, one per time step regardless of dimension. The approach is not confined to smooth targets: it extends to $C^{1,1}$ energies with discontinuous curvature (without ellipticity requirement) and to Sobolev energies whose Hessians exist only off measure zero sets; for Lipschitz energies with gradient kinks the averaged limit remains well posed, and the convergence proof uses an $\omega$-uniform occupation estimate for the dithered process, which we prove by a near-identity change of variables that removes the dither pathwise before Girsanov's theorem is applied; the Krylov-Röckner integrability class is the boundary of provability. The approach provides a hard $\text{a priori}$ bound on the per-step update rate and applies to explicitly time-varying targets on finite horizons. Gradient-free pixel-space sampling is not competitive with well-tuned backpropagation-based samplers at practical evaluation budgets; the regime where the approach offers an advantage is latent-space sampling when the model is a black box and the target drifts in time. We demonstrate latent-space tracking for time-varying images on CelebA-HQ ($256\times 256$) from limited 1D projection measurements and latent-space EBM sampling on CIFAR-10.