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Alternating Diffusion for Proximal Sampling with Zeroth Order Queries

ICLR 2026 Conference SubmissionAnonymous Authors
OpenReview Score: 6.5Download Report PDF
SamplingDiffusion-based Monte CarloZeroth-order methods
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.

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This report is AI-GENERATED using Large Language Models and WisPaper (A scholar search engine). It analyzes academic papers' tasks and contributions against retrieved prior work. While this system identifies POTENTIAL overlaps and novel directions, ITS COVERAGE IS NOT EXHAUSTIVE AND JUDGMENTS ARE APPROXIMATE. These results are intended to assist human reviewers and SHOULD NOT be relied upon as a definitive verdict on novelty.
NOTE that some papers exist in multiple, slightly different versions (e.g., with different titles or URLs). The system may retrieve several versions of the same underlying work. The current automated pipeline does not reliably align or distinguish these cases, so human reviewers will need to disambiguate them manually.
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Overview

Overall Novelty Assessment

The paper proposes a diffusion-based approximate proximal sampler that operates with zeroth-order information, treating intermediate particle distributions as Gaussian mixtures to derive Monte Carlo score estimators. It resides in the 'Diffusion and Heat Flow Proximal Sampling' leaf, which contains only two papers total: the original work and one sibling (Zeroth-Order Denoising Diffusion). This leaf sits under the broader 'Proximal Sampling and Diffusion-Based Methods' branch, indicating a relatively sparse research direction within the taxonomy of 32 papers across the field.

The taxonomy reveals that most zeroth-order proximal work concentrates on optimization methods (variance reduction, distributed algorithms, trust-region approaches) rather than sampling. The sibling paper in the same leaf explores denoising diffusion perspectives, while the neighboring 'Log-Concave and Convex Body Sampling' leaf addresses uniform distributions over convex bodies. The paper's focus on heat flow dynamics and Gaussian mixture score estimation distinguishes it from optimization-centric branches and from rejection-sampling approaches mentioned in the taxonomy's scope notes.

Among 22 candidates examined, the first contribution (diffusion-based sampler with zeroth-order queries) shows 1 refutable candidate out of 10 examined, suggesting some prior work exists but coverage is limited. The second contribution (theoretical convergence analysis) encountered 3 refutable candidates among 9 examined, indicating more substantial overlap with existing convergence theory. The third contribution (multi-particle extension) found no refutable candidates in 3 examined, appearing more novel within this limited search scope. The analysis explicitly acknowledges examining top-K semantic matches rather than exhaustive literature coverage.

Given the sparse taxonomy leaf and limited search scope, the work appears to occupy a relatively underexplored intersection of diffusion-based sampling and zeroth-order methods. The theoretical convergence analysis shows more connection to prior work than the algorithmic and multi-particle aspects. However, the 22-candidate search scale means substantial relevant literature may exist beyond the examined set, particularly in adjacent sampling or score-based modeling communities not fully captured by this taxonomy.

Taxonomy

Core-task Taxonomy Papers
32
3
Claimed Contributions
22
Contribution Candidate Papers Compared
4
Refutable Paper

Research Landscape Overview

Core task: Approximate proximal sampling with zeroth-order information. This field addresses optimization and sampling problems where gradient information is unavailable or prohibitively expensive, yet one must still handle composite objectives involving non-smooth regularizers or constraints. The taxonomy reveals four main branches. Proximal Sampling and Diffusion-Based Methods explore connections between proximal operators and diffusion processes, often leveraging heat flow or score-based techniques to sample from complex distributions. Zeroth-Order Proximal Optimization Methods focus on algorithmic designs that estimate gradients via function evaluations alone while respecting proximal structure, including works like Zeroth-Order Proximal Stochastic[3] and Lightweight Zeroth-Order Proximal[23]. Gradient-Free Optimization Without Proximal Structure encompasses classical derivative-free schemes that may not explicitly invoke proximal mappings, such as simplex methods and variance-reduced approaches. Applications and Domain-Specific Methods gather problem-driven studies in areas like Bayesian design, distributed settings, and quantum systems, illustrating how zeroth-order proximal ideas adapt to specialized constraints. A particularly active line of work examines trade-offs between sample complexity, convergence rates, and the cost of approximate proximal steps. Some methods achieve faster rates by exploiting variance reduction or preconditioning (Inexact Preconditioned Zeroth-Order[5], Double Variance Reduction[7]), while others prioritize communication efficiency in distributed scenarios (Distributed Gradient-Free Projection-Free[2], Zeroth-Order Consensus[20]). The original paper, Alternating Diffusion Proximal[0], sits within the Diffusion and Heat Flow Proximal Sampling cluster alongside Zeroth-Order Denoising Diffusion[4]. Both works integrate diffusion-based sampling with zeroth-order queries, but Alternating Diffusion Proximal[0] emphasizes an alternating scheme that interleaves diffusion steps with proximal updates, contrasting with the denoising perspective of its neighbor. This positioning highlights ongoing exploration of how continuous-time dynamics and discrete proximal operators can be harmonized when only function values are accessible.

Claimed Contributions

Diffusion-based approximate proximal sampler with zeroth-order queries

The authors propose a novel approximate proximal sampling algorithm that operates using only zeroth-order (function value) information of the potential function. Unlike existing implementations that rely on rejection sampling, this method directly simulates diffusion dynamics by treating intermediate particle distributions as Gaussian mixtures, enabling Monte Carlo score estimation without requiring gradients or learned models.

10 retrieved papers
Can Refute
Theoretical convergence analysis for diffusion-based proximal sampling

The authors establish theoretical guarantees showing that their diffusion-based approximation maintains exponential convergence to the target distribution under isoperimetric conditions, similar to the ideal proximal sampling framework. This analysis demonstrates that the method converges without requiring initialization from the Gaussian limit, unlike standard diffusion models.

9 retrieved papers
Can Refute
Multi-particle extension with empirical validation

The authors develop a multi-particle version of their algorithm that exploits parallel computation and particle interactions. Numerical experiments demonstrate faster convergence compared to existing proximal sampler implementations, with improvements in both computational efficiency and sample diversity through the multi-particle framework.

3 retrieved papers

Core Task Comparisons

Comparisons with papers in the same taxonomy category

Contribution Analysis

Detailed comparisons for each claimed contribution

Contribution

Diffusion-based approximate proximal sampler with zeroth-order queries

The authors propose a novel approximate proximal sampling algorithm that operates using only zeroth-order (function value) information of the potential function. Unlike existing implementations that rely on rejection sampling, this method directly simulates diffusion dynamics by treating intermediate particle distributions as Gaussian mixtures, enabling Monte Carlo score estimation without requiring gradients or learned models.

Contribution

Theoretical convergence analysis for diffusion-based proximal sampling

The authors establish theoretical guarantees showing that their diffusion-based approximation maintains exponential convergence to the target distribution under isoperimetric conditions, similar to the ideal proximal sampling framework. This analysis demonstrates that the method converges without requiring initialization from the Gaussian limit, unlike standard diffusion models.

Contribution

Multi-particle extension with empirical validation

The authors develop a multi-particle version of their algorithm that exploits parallel computation and particle interactions. Numerical experiments demonstrate faster convergence compared to existing proximal sampler implementations, with improvements in both computational efficiency and sample diversity through the multi-particle framework.

Table of Contents

Alternating Diffusion for Proximal Sampling with Zeroth Order Queries | Novelty Validation