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

2026-10-03

AI Summary

This study addresses the reliance of generative model sampling on gradient computation, which limits applicability to black-box or time-varying target distributions. To overcome this limitation, this work proposes a gradient-free sampling framework based on high-frequency cosine perturbations. By extending bounded extremum seeking to Itô processes as a substitute for the drift term, the method enables both MCMC and diffusion sampling for energy-based and score-based models. Notably, it requires no ellipticity assumptions and accommodates non-smooth energy functions. Experiments on CelebA-HQ and CIFAR-10 validate the effectiveness of the approach in latent-space tracking of time-varying images and energy-based model sampling. Overall, this research establishes a new paradigm for generative model sampling in black-box scenarios where gradient information is unavailable or unreliable.

Abstract

We introduce a sampling approach for energy- and score-based generative models that requires no gradient evaluations of the model. Replacing the drift term that would normally contain the score $\nabla_\mathbf{x} \log p_\theta(\bf{x})$ with a high-frequency dithered cosine of the model's \textit{value}, $\sqrt{\alpha\omega}\,\cos(\omega t + k \log p_\theta(\bf{x}))$, 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 It\^{o} 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 It\^{o} processes, with an explicit $O(\omega^{-1/2})$ mean-square rate under global bounds. 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; the Krylov-R\"ockner integrability class is the boundary of provability. The approach provides a hard \textit{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.

Problem

• gradient-free sampling
• generative models
• black-box model
• latent-space sampling
• time-varying targets

Innovation

• Gradient-Free Sampling
• Extremum Seeking
• Stochastic Differential Equations
• Generative Models
• Latent Space Tracking