Alternating Diffusion for Proximal Sampling with Zeroth Order Queries

2026-03-20

AI Summary

This work proposes a novel proximal sampler for settings where only zeroth-order information of the potential function is accessible. The method alternates between simulating forward and backward diffusion dynamics of heat flow, modeling the intermediate particle distribution as a Gaussian mixture and constructing a Monte Carlo score estimator that admits direct sampling. In contrast to conventional rejection sampling, the proposed approach features deterministic runtime, supports flexible step sizes, and maintains exponential convergence under multi-particle interactions and parallel computation. When the target distribution satisfies an isoperimetric condition, the algorithm inherits the exponential convergence rate characteristic of proximal samplers. Numerical experiments confirm its ability to rapidly converge to the target distribution.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

This work introduces a new approximate proximal sampler that operates solely with zeroth-order information of the potential function. Prior theoretical analyses have revealed that proximal sampling corresponds to alternating forward and backward iterations of the heat flow. The backward step was originally implemented by rejection sampling, whereas we directly simulate the dynamics. Unlike diffusion-based sampling methods that estimate scores via learned models or by invoking auxiliary samplers, our method treats the intermediate particle distribution as a Gaussian mixture, thereby yielding a Monte Carlo score estimator from directly samplable distributions. Theoretically, when the score estimation error is sufficiently controlled, our method inherits the exponential convergence of proximal sampling under isoperimetric conditions on the target distribution. In practice, the algorithm avoids rejection sampling, permits flexible step sizes, and runs with a deterministic runtime budget. Numerical experiments demonstrate that our approach converges rapidly to the target distribution, driven by interactions among multiple particles and by exploiting parallel computation.

Problem

• proximal sampling
• zeroth-order queries
• diffusion-based sampling
• score estimation
• Monte Carlo methods

Innovation

• zeroth-order sampling
• proximal sampling
• alternating diffusion
• score estimation
• Gaussian mixture approximation