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A variance-reduced zeroth-order Langevin sampler delivers the first non-asymptotic convergence guarantees for non-log-concave distributions without access to gradients.

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2026-06-29 08:39 UTC pith:IFUVXOJ7

load-bearing objection First non-asymptotic ZO non-log-concave sampling rates in relative Fisher info, but TV bound needs Poincaré that often fails for these targets. the 1 major comments →

arxiv 2605.30573 v1 pith:IFUVXOJ7 submitted 2026-05-28 cs.LG

Zeroth-Order Non-Log-Concave Sampling with Variance Reduction and Applications to Inverse Problems

M. Berk Sahin , Behzad Sharif , Abolfazl Hashemi This is my paper
classification cs.LG
keywords zeroth-order samplingnon-log-concave distributionsLangevin dynamicsvariance reductioninverse problemsposterior samplingscore-based generative modelsPoincaré inequality
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a gradient estimator that cuts variance in batched zeroth-order methods and removes the usual dimensional blow-up in required batch size. It proves that the resulting dynamics converge to the target distribution at explicit rates measured by relative Fisher information, and by squared total variation distance once a Poincaré inequality is assumed. The same guarantees are extended to ZO-APMC, a posterior sampler that combines the method with pre-trained score-based generative priors for black-box inverse problems. A reader cares because these are the first such finite-time bounds in the fully gradient-free, non-log-concave regime that appears in many real inverse problems.

Core claim

The central claim is that a carefully constructed variance-reduced zeroth-order estimator inside Langevin dynamics yields the first non-asymptotic convergence guarantees for sampling from non-log-concave targets, measured in ε-relative Fisher information distance, and in squared total variation distance under an additional Poincaré inequality; the same framework supplies the first such guarantees for ZO-APMC, a black-box posterior sampler that uses pre-trained score-based generative priors.

What carries the argument

A variance-reduced zeroth-order gradient estimator embedded in the Langevin update that eliminates the unfavorable dimensional dependence of classical batched estimators.

Load-bearing premise

The target distribution must satisfy a Poincaré inequality for the squared total variation distance bound to hold.

What would settle it

A concrete high-dimensional non-log-concave distribution that satisfies the paper's other assumptions yet violates the Poincaré inequality and for which the algorithm fails to converge in total variation at the claimed rate.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Black-box inverse problems become amenable to rigorous posterior sampling with pre-trained generative models.
  • The batch-size requirement for accurate zeroth-order estimation no longer grows with dimension.
  • Non-asymptotic Fisher-information bounds now exist for a broader class of gradient-free sampling algorithms.
  • The same estimator can be swapped into other Langevin-type schemes that previously lacked non-asymptotic analysis.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The technique may transfer to zeroth-order optimization or to sampling under other weak convexity conditions beyond Poincaré.
  • Practical implementations could test whether the observed empirical stability matches the derived dimensional independence.
  • If the Poincaré assumption can be relaxed or replaced by weaker functional inequalities, the total-variation guarantee might extend to a larger family of targets.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 0 minor

Summary. The paper proposes a variance-reduced zeroth-order Langevin sampling method for high-dimensional non-log-concave distributions in black-box settings where gradients are inaccessible. It claims non-asymptotic convergence guarantees in terms of ε-relative Fisher information without additional assumptions, and squared total variation distance under a Poincaré inequality assumption. It further introduces the ZO-APMC algorithm for posterior sampling in black-box inverse problems using pre-trained score-based generative priors, with analogous guarantees, and validates the approach on synthetic experiments and practical linear/nonlinear inverse problems.

Significance. If the derivations hold, this would be a meaningful contribution by supplying the first non-asymptotic guarantees for zeroth-order non-log-concave sampling and by extending the method to inverse problems with generative priors. The variance-reduced estimator that removes unfavorable dimensional dependence in batch size is a technical strength. The paper ships non-asymptotic convergence guarantees (a positive feature) rather than only asymptotic statements.

major comments (1)
  1. Abstract: the squared total variation distance guarantee is stated to hold only under a Poincaré inequality assumption. For general non-log-concave targets this inequality need not hold, so the TV claim is conditional; this assumption is load-bearing for that part of the central claim even though the ε-relative Fisher information bound is presented without it. A concrete test would be to exhibit a non-log-concave distribution violating Poincaré and verify whether the sampling method still converges in TV distance.

Simulated Author's Rebuttal

1 responses · 1 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comment on the abstract. We respond to the major comment below.

read point-by-point responses

  1. Referee: [—] Abstract: the squared total variation distance guarantee is stated to hold only under a Poincaré inequality assumption. For general non-log-concave targets this inequality need not hold, so the TV claim is conditional; this assumption is load-bearing for that part of the central claim even though the ε-relative Fisher information bound is presented without it. A concrete test would be to exhibit a non-log-concave distribution violating Poincaré and verify whether the sampling method still converges in TV distance.

    Authors: We appreciate the referee's observation. The abstract already states that the squared total variation distance guarantee holds under the Poincaré inequality assumption while the ε-relative Fisher information bound does not require it. This distinction is deliberate: our analysis derives the Fisher information convergence for general non-log-concave targets without additional functional inequalities, whereas total variation convergence for Langevin-type methods typically relies on such assumptions (e.g., Poincaré or log-Sobolev) even when gradients are available. The presentation therefore accurately reflects the scope of the results. Regarding the proposed concrete test, identifying a specific non-log-concave distribution that violates Poincaré and then verifying non-convergence in total variation distance would require substantial new theoretical and empirical work outside the manuscript's scope. revision: no

standing simulated objections not resolved
  • Constructing and verifying an explicit non-log-concave distribution violating the Poincaré inequality to test total variation convergence of the proposed sampler.

Circularity Check

0 steps flagged

No circularity: guarantees rest on external assumptions

full rationale

The provided abstract and description contain no equations, fitted parameters, or self-citations that reduce the claimed convergence bounds to inputs by construction. The ε-relative Fisher information bound and the conditional TV bound (under Poincaré inequality) are presented as derived results relying on standard assumptions rather than self-definitional or renamed quantities. The ZO-APMC application inherits the same structure without evidence of circular reduction in the given text.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on a variance-reduced zeroth-order gradient estimator whose construction is not detailed in the abstract and on the Poincaré inequality assumption needed for the total-variation bound. No free parameters or invented entities are mentioned.

axioms (1)
  • domain assumption Poincaré inequality assumption on the target distribution
    Required to convert the relative Fisher information bound into a squared total variation distance guarantee.

pith-pipeline@v0.9.1-grok · 5750 in / 1295 out tokens · 19347 ms · 2026-06-29T08:39:26.440034+00:00 · methodology

0 comments
read the original abstract

Sampling from high-dimensional, non-log-concave distributions with unnormalized densities remains a fundamental challenge in machine learning, particularly in black-box settings where gradient information is inaccessible or computationally prohibitive. While Langevin dynamics provides a principled framework for sampling when gradients are accessible, its extension to the black-box settings suffers from high variance and lacks non-asymptotic convergence guarantees for non-log-concave sampling. To address these limitations, we propose a variance-reduced zeroth-order Langevin sampling method. Our method employs a gradient estimator that substantially reduces the variance of the classical batched zeroth-order estimator and eliminates the unfavorable dimensional dependence of the batch size required for accurate estimation, enabling practical and stable sampling. We establish the first non-asymptotic convergence guarantees for zeroth-order non-log-concave sampling in terms of $\varepsilon$-relative Fisher information, and, under a Poincar\'e inequality assumption, squared total variation distance. We further propose ZO-APMC, a posterior sampling algorithm for black-box inverse problems with pre-trained score-based generative priors, establishing the first non-asymptotic convergence guarantees for such methods. We validate our theory through synthetic experiments and demonstrate strong empirical performance on practical linear and nonlinear inverse problems.

Figures

Figures reproduced from arXiv: 2605.30573 by Abolfazl Hashemi, Behzad Sharif, M. Berk Sahin.

Figure 1
Figure 1. Figure 1: Illustration of weighted annealing in PMC through weighted posteriors π (αk) σk N−1 k=0 . Solid lines and shaded regions denote the distribution mean and density, respectively, while un￾shaded regions correspond to ∇ log π(x) = 0. By gradually reducing the prior smoothing parameter σk and its weight αk rela￾tive to the likelihood ℓ, weighted annealing enables PMC to escape plateaus in ∇ log π(x). where u∼… view at source ↗
Figure 2
Figure 2. Figure 2: (a) Convergence of ZO-APMC with b = 10, b ′ = 5, and εk∗ = 2.5 for various p, alongside APMC convergence with gradient access. (b) Convergence results for fixed per-iteration cost. Each × indicate parameter settings (p, b) for which the FI drops below 0.01 after 2000 iterations. (c) Comparison of sample statistics generated by ZO-APMC and APMC with those of the analytical ground truth posterior. Theorem 4.… view at source ↗
Figure 3
Figure 3. Figure 3: Visualization of averaged reconstructions generated by APMC in the setting with gradient access and ZO-APMC in the black-box setting. Despite relying solely on function evaluations, ZO-APMC achieves reconstruction quality comparable to APMC [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Visualization of mean reconstructions for the black-hole imaging inverse problem. Two representative test cases are shown (top and bottom rows), with ground truths in the left column and black-box method reconstructions in the remaining columns. PSNR (dB) and closure-phase error (χ 2 cph, lower is better) are reported below each reconstruction. 4.1. Toy Experiments Numerical Validation. To validate the con… view at source ↗
Figure 5
Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Effect of p on the convergence of ZO-APMC to the true posterior distribution in terms of relative Fisher information. The solid lines show the mean values and shaded areas show the minimum and maximum ranges [PITH_FULL_IMAGE:figures/full_fig_p038_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Comparison of naive ZO-APMC (p = 1) and ZO-APMC with p < 1 for Fisher information convergence under Gaussian (left) and Laplace (right) measurement noise types in an inverse problem. Each plot performs the same number of forward model evaluations per iteration on average. used in practice. Accordingly, we conduct experiments under both noise modeling assumptions. We present the results in [PITH_FULL_IMAGE… view at source ↗
Figure 8
Figure 8. Figure 8: Convergence of APMC and ZO-APMC to the target posterior in KL divergence for varying initial values p0. The probability parameter p is decreased throughout sampling, and the reported values denote its initialization. ZO-APMC converges for all initial values of p0 [PITH_FULL_IMAGE:figures/full_fig_p039_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Comparison of naive ZO-APMC (p = 1) and ZO-APMC with p < 1 for estimating the ground truth posterior mean and variance statistics under Gaussian (left) and Laplace (right) measurement noise types. First row shows the posterior means and the second row shows the posterior variance. The colorbar for each statistic is shown at the end of its corresponding row. Each column shows the statistics estimated by ZO-… view at source ↗
Figure 10
Figure 10. Figure 10: Comparison of the ground-truth brain MRI with APMC and ZO-APMC reconstructions for various probabilities p ∈ {0.1, 0.2, 0.4, 0.5}, using a large batch size of b = 104 and a small batch size of b ′ = 103 . PSNR values for each reconstruction are displayed in the lower-left corner of the corresponding image [PITH_FULL_IMAGE:figures/full_fig_p041_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Comparison of the ground-truth brain MRI with reconstructions from ZO-APMC and the gradient-based approaches DPS and APMC. Each method generates 20 samples from the same measurements; the first row shows the mean reconstructions and the second row shows the corresponding variance maps. Owing to its variance-reduction mechanism, ZO-APMC produces variance maps comparable to those of the gradient-based algor… view at source ↗

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Reference graph

Works this paper leans on

2 extracted references · 1 canonical work pages

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