Mixing Time of the Proximal Sampler in Relative Fisher Information via Strong Data Processing Inequality

Andre Wibisono

REVIEW 2 major objections 3 minor 1 cited by

Mixing Time of the Proximal Sampler in Relative Fisher Information via Strong Data Processing Inequality

T0 review · 2 major / 3 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read The Proximal Sampler contracts relative Fisher information exponentially fast for strongly log-concave targets, at the same rate as continuous-time Langevin dynamics.

desk verdict Direct exponential Fisher-information mixing for the Proximal Sampler, with a clean SDPI proof; watch the regularity gap between Lemma 2 and the theorems. read the letter →

arxiv 2502.05623 v2 pith:IZIYOSC2 submitted 2025-02-08 cs.IT cs.LGmath.ITmath.OCmath.STstat.TH

Andre Wibisono This is my paper · ORCID

classification cs.ITcs.LGmath.ITmath.OCmath.STstat.TH MSC 60J6094A1760J25

keywords relativeFisherinformationProximalSamplerstrongdataprocessinginequalitystronglylog-concaveLangevindynamicsmixingtimeGaussianchannelrestrictedoracle

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves that the Proximal Sampler—a Gibbs-sampling algorithm built from forward and reverse Gaussian steps—converges exponentially fast in relative Fisher information when the target distribution is strongly log-concave. Relative Fisher information measures the expected squared difference between the score functions (gradients of log-density) of the current and target distributions, so it is a stronger error notion than KL divergence. The per-iteration contraction is $FI(\rho_{k+1}^X \| \nu_X) \le FI(\rho_k^X \| \nu_X)/(1+\alpha\eta)^2$ for any step size $\eta$, exactly the rate of continuous-time Langevin dynamics evaluated at the same effective time. This fills a gap between continuous-time and discrete-time guarantees: previously the discrete Proximal Sampler had such a rate only for divergences like KL or $\Phi$-divergence, not for relative Fisher information. With rejection sampling for the internal oracle, the result yields a high-accuracy iteration complexity whose iteration count grows logarithmically in $1/\varepsilon$.

What carries the argument

The load-bearing object is a time-derivative identity for relative Fisher information along any Fokker-Planck channel (Lemma 2). It states that $$\frac{d}{dt} FI(\rho_t \| \nu_t) = -c\, E_{\rho_t}\left[\left\|\$nabla^{2}$ \log\frac{\rho_t}{\nu_t}\right\|_{\mathrm{HS}}^2\right] - 2\, E_{\rho_t}\left[\left\|\nabla \log\frac{\rho_t}{\nu_t}\right\|^2_{-c\$nabla^{2}$ \log \nu_t + (\nabla b_t)_{\mathrm{sym}}}\right],$$ where $c$ is the noise coefficient and $b_t$ the drift of the channel. Along the Gaussian channel ($b_t=0$, $c=1$), strong log-concavity of $\nu_0$ is preserved by convolution with modulus $\alpha_t = \alpha/(1+\alpha t)$, making the second term at most $-2\alpha_t FI$, which integrates to the $(1+\alpha t)^{-2}$ contraction; along the reverse Gaussian channel the drift is chosen so the second term vanishes, leaving only the non-positive second-order Fisher information. This channel-wise contraction is exactly what composes into the Proximal Sampler rate.

What would settle it

Take a strongly log-concave target whose log-density is only once differentiable (or has slow-tail decay), initialize at a Gaussian, and compute by high-precision quadrature the relative Fisher information after one Gaussian-channel or Proximal Sampler step; if the ratio ever exceeds $(1+\alpha\eta)^{-2}$, the regularity assumption in Appendix F.1 is doing real work rather than serving as bookkeeping.

Watch

Extended reading notes

Core claim

The central result, Theorem 6, states that if the target $\nu_X$ is $\alpha$-strongly log-concave, then every Proximal Sampler iteration obeys $FI(\rho_{k+1}^X \| \nu_X) \le FI(\rho_k^X \| \nu_X)/(1+\alpha\eta)^2$, and hence $FI(\rho_k^X \| \nu_X) \le FI(\rho_0^X \| \nu_X)/(1+\alpha\eta)^{2k}$. The proof splits the algorithm into its two channels: the forward Gaussian step $\rho \mapsto \rho * N(0,\eta I)$ satisfies a strong data processing inequality with contraction $(1+\alpha\eta)^{-2}$ when the target is $\alpha$-strongly log-concave, while the backward reverse-Gaussian step is only non-expansive, giving no further contraction but no increase. The exponent $\alpha\eta$ makes the discrete rate match the continuous-time Langevin bound $e^{-2\alpha t}$ in the small-step limit $\eta k \to t$. The paper also derives an eventual strong data processing inequality when the target is a log-Lipschitz perturbation of a strongly log-concave distribution, and exhibits Gaussian-channel examples where plain data-processing monotonicity of relative Fisher information fails, so some structural assumption on the second distribution is necessary.

Load-bearing premise

The load-bearing premise is the regularity hypothesis stated in Appendix F.1—that the evolving densities are smooth and decay quickly enough at infinity for every integration by parts in Lemma 2 to have zero boundary terms—while the main theorems are stated under merely $FI(\rho_0\|\nu_0)<\infty$, so the missing approximation argument is what keeps the exponential rate from being fully general.

Editorial extensions

If this is right

For any $\alpha$-strongly log-concave target, the relative Fisher information of the last iterate decays as $(1+\alpha\eta)^{-2k}$, making the iteration count to reach precision $\varepsilon$ logarithmic in $1/\varepsilon$.
With $\eta=1/(dL)$ and rejection sampling for the Restricted Gaussian Oracle, each iteration uses $O(1)$ expected queries to the target log-density, so the total cost is $O((dL/\alpha)\log(dL/\varepsilon))$ up to constants.
The rate matches continuous-time Langevin dynamics and, through the sampling-as-optimization analogy, the proximal gradient method for strongly convex objectives.
When the initial distribution is symmetric and satisfies a Poincaré inequality, the forward step contracts by $(1+\beta\eta)^{-1}(1+\alpha\eta)^{-2}$, yielding a faster overall mixing rate.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same channel-composition argument would extend to any sampler built from one strongly contractive forward channel and one non-expansive reverse channel, so other proximal or diffusion-based samplers may inherit a Fisher-information mixing bound under strong log-concavity; the paper develops this only for the Proximal Sampler.
Because the Gaussian-channel SDPI uses pointwise curvature of the second distribution, weakening the target assumption from strong log-concavity to a log-Sobolev inequality would require a genuinely new argument rather than a tighter constant.
A Fisher-information guarantee controls convergence of score functions, so it may sharpen downstream analyses that need good score estimates—for example mean-field Langevin systems—but this application is not explored in the paper.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

You should know about this one: it gives the first direct exponential convergence guarantee for the Proximal Sampler in relative Fisher information under strong log-concavity, matching the continuous-time Langevin rate. The route is a new strong data processing inequality for relative Fisher information along the Gaussian channel, plus DPI along the reverse Gaussian channel. The proof structure is clean: Lemma 2 gives a time-derivative formula, Theorem 3 extracts contraction, and Theorem 6 composes the channels. I checked the algebra in the appendices; it is detailed and the key steps are straightforward to verify. The examples showing that DPI fails in general are a nice touch.

What is genuinely new: prior work on relative Fisher information was either low-accuracy and biased (ULA in Balasubramanian et al.) or indirect (Chewi et al. Appendix A.2 via heat flow after chi-square). This paper proves the rate directly and matches the proximal-gradient bound from optimization. No fitting, no circularity; the SDPI constants come from the strong log-concavity parameter.

The soft spot, as you might expect from the stress test, is regularity. Lemma 2 is proved under smoothness and sufficient decay so that boundary terms vanish. Theorems 3, 4, and 6 are stated under only FI(ρ0||ν0)<∞. The paper does not provide an approximation argument to bridge this. The stress-test example of a 1-SLC ν0 with a cusp in its log-density (exp(-x²/2-|x|)) is valid: such a target is not C², so the time-derivative formula is not directly applicable. This does not sink the paper—the result is certainly true for smooth targets, and it is very plausible that a standard smoothing argument extends it. But as written, the stated generality is not fully proven. That is a genuine referee-level issue, not a fatal flaw. There is also a harmless sign typo in Appendix F.4.

Bottom line: this is a strong, useful paper for the sampling theory community. It deserves a serious referee, and the referee should be asked to fill the regularity gap or restrict the statements to smooth densities. I would take it to a reading group and would cite it in future work on log-concave sampling.

Recommendation: accept with the regularity issue as a required revision.

Referee Report

2 major / 3 minor

Summary. The paper studies the mixing time of the Proximal Sampler when the error is measured in relative Fisher information. Its main algorithmic result, Theorem 6, states that for an alpha-strongly log-concave target, each Proximal Sampler iteration contracts relative Fisher information by the factor (1 + alpha*eta)^-2, giving an exponential convergence rate that matches the continuous-time Langevin rate from Eq. (14). The proof is built on a time-derivative formula for relative Fisher information along simultaneous Fokker-Planck channels (Lemma 2), a strong data processing inequality along the Gaussian channel under strong log-concavity (Theorem 3), and a data processing inequality along the reverse Gaussian channel. The paper also gives an eventual SDPI along the Ornstein-Uhlenbeck channel, counterexamples showing that DPI can fail in general, and a high-accuracy iteration complexity corollary for log-smooth targets under rejection-sampling implementation of the restricted Gaussian oracle.

Significance. If the main result is fully established, it closes a natural gap: the Proximal Sampler is shown to inherit the continuous-time Langevin exponential convergence in relative Fisher information, a metric that is stronger than KL divergence under a log-Sobolev inequality. The proof is self-contained, the contraction constants are explicit and depend only on the strong log-concavity parameter, and the paper provides tight Gaussian examples and counterexamples. These are substantial strengths. The main caveat is a regularity gap between the proof of Lemma 2 and the stated assumptions of the theorems that use it.

major comments (2)

[Appendix F.1 and Theorem 6]
[Appendix F.2, Theorem 3(iv)]

minor comments (3)

[Section 3.3.1, Example 4]
[Section 4.3.1, Example 5]
[Appendix F.1, proof of Lemma 2]

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the SDPI rates are derived from alpha-strong log-concavity via an explicit time-derivative computation, and the tightness examples are independent direct calculations; the self-citations are contextual and not load-bearing.

full rationale

The central derivation chain is self-contained. Lemma 2 derives the time derivative formula (17) by direct differentiation and integration by parts from the Fokker-Planck equations; the formula is not postulated from the desired contraction rate. Theorem 3(ii) then uses the external preservation-of-strong-log-concavity fact (Saumard and Wellner) to get alpha_t = alpha/(1+alpha t), and integrates d/dt FI <= -2*alpha_t*FI, giving exactly (1+alpha t)^(-2); no parameter is fitted to make the conclusion hold. The Proximal Sampler result Theorem 6 composes this SDPI with the reverse-Gaussian-channel DPI; the reverse-step inequality (24b) is obtained by setting b_t = grad log nu_t in (17), where the offending second term vanishes identically, not by assuming the desired contraction. The tightness checks in Examples 2, 4, and 5 evaluate FI in closed form for Gaussian inputs and confirm the stated exponents rather than determining the constants in the theorems. The cited works by the author (Chen et al. 2022, Vempala and Wibisono 2019, etc.) supply standard channel interpretations and comparison results, but the SDPI proof itself does not rest on an unverified self-citation. The manuscript itself flags a regularity hypothesis in Appendix F.1: Lemma 2 assumes smooth densities decaying fast enough for boundary terms to vanish, while Theorems 3 and 6 are stated under FI(rho_0 || nu_0) < infinity and alpha-SLC; this is a gap in rigor or an implicit approximation argument that the paper does not provide, not a circularity, since the missing step would make the proof incomplete rather than making the theorem equivalent to its assumptions. Therefore no circular step satisfying the quoted-reduction standard is present.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central contribution is analytic and uses no fitted parameters. The argument relies on standard results from convex analysis and probability, plus an unstated regularity hypothesis for integration by parts. No new physical entities or fitted constants are introduced.

assumptions (5)

domain assumption The densities rho_t and nu_t are smooth and decay sufficiently fast at infinity so that all integrations by parts and interchanges of derivatives and integrals in Lemma 2 are valid.
Appendix F.1 states 'we assume the density functions rho_t and nu_t are smooth and decay sufficiently fast at infinity.' This regularity premise underlies the time-derivative formula (17) and hence all SDPI results.
standard math Strong log-concavity is preserved under convolution with a Gaussian, with the parameter alpha_t = alpha / (1 + alpha t).
Used in Theorem 3(ii) via Saumard and Wellner, Theorem 3.7, to obtain the contraction factor for the Gaussian channel.
standard math Poincare inequality constants compose under convolution, giving beta_t = beta / (1 + beta t).
Used in Theorem 3(iii) via Chafai, Corollary 3.1, to exploit the second-order Fisher information term in the time-derivative formula.
standard math The heat flow preserves a quantitative strong log-concavity bound for log-Lipschitz perturbations of strongly convex potentials.
Used in Theorem 3(iv) via Brigati and Pedrotti, Theorem 1.3, to establish eventual SDPI along the Gaussian channel.
domain assumption The target potential g is L-log-smooth and we have access to its minimizer x*, and the restricted Gaussian oracle can be implemented by rejection sampling with O(1) expected queries.
Stated in Corollary 7 and Appendix G.1; this is an algorithmic premise for the high-accuracy iteration complexity claim, not for the abstract convergence theorem.

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Cite this review

Pith. "Pith review of Mixing Time of the Proximal Sampler in Relative Fisher Information via Strong Data Processing Inequality." pith.science (2026). https://pith.science/paper/IZIYOSC2

@misc{pith2026250205623,
  author       = {Pith},
  title        = {Pith review of: Mixing Time of the Proximal Sampler in Relative Fisher Information via Strong Data Processing Inequality},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IZIYOSC2}},
  note         = {Machine review of arXiv:2502.05623}
}

read the original abstract

We study the mixing time guarantee for sampling in relative Fisher information via the Proximal Sampler algorithm, which is an approximate proximal discretization of the Langevin dynamics. We show that when the target probability distribution is strongly log-concave, the relative Fisher information converges exponentially fast along the Proximal Sampler; this matches the exponential convergence rate of the relative Fisher information along the continuous-time Langevin dynamics for strongly log-concave target. When combined with a standard implementation of the Proximal Sampler via rejection sampling, this exponential convergence rate provides a high-accuracy iteration complexity guarantee for the Proximal Sampler in relative Fisher information when the target distribution is strongly log-concave and log-smooth. Our proof proceeds by establishing a strong data processing inequality for relative Fisher information along the Gaussian channel under strong log-concavity, and a data processing inequality along the reverse Gaussian channel for a special distribution. The forward and reverse Gaussian channels compose to form the Proximal Sampler, and these data processing inequalities imply the exponential convergence rate of the relative Fisher information along the Proximal Sampler.

Figures

Figures reproduced from arXiv: 2502.05623 by the authors.

**Figure 2.** An example where DPI in relative Fisher information does not hold along the OU channel. [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

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Reference graph

Works this paper leans on

15 extracted references · 10 canonical work pages · cited by 1 Pith paper

[7]

Coupled Wasserstein gradient flows for min-max and cooperative games

Lauren Conger, Franca Hoffmann, Eric Mazumdar, and Lillian J Ratliff. Coupled Wasserstein gradient flows for min-max and cooperative games. arXiv preprint arXiv:2411.07403,

arXiv
[10]

Private convex optimization via exponential mecha- nism

Sivakanth Gopi, Yin Tat Lee, and Daogao Liu. Private convex optimization via exponential mecha- nism. In Conference on Learning Theory, pages 1948–1989. PMLR,

work page 1948
[11]

A proximal algorithm for sampling from non-smooth potentials

Jiaming Liang and Yongxin Chen. A proximal algorithm for sampling from non-smooth potentials. In 2022 Winter Simulation Conference (WSC), pages 3229–3240. IEEE,

work page 2022
[12]

Jiaming Liang and Yongxin Chen

ISSN 2835-8856. Jiaming Liang and Yongxin Chen. Proximal oracles for optimization and sampling. arXiv preprint arXiv:2404.02239,

work page arXiv
[338]

Andre Wibisono

Springer, 2009b. Andre Wibisono. Sampling as optimization in the space of measures: The Langevin dynamics as a composite optimization problem. In Conference on Learning Theory, pages 2093–3027. PMLR,

work page
[1965]

Heat flow, log-concavity, and Lipschitz transport maps

Giovanni Brigati and Francesco Pedrotti. Heat flow, log-concavity, and Lipschitz transport maps. arXiv preprint arXiv:2404.15205v2,

arXiv
[1998]

Convexity of mutual information along the Fokker-Planck flow

42 MIXING TIME OF THE PROXIMAL SAMPLER IN RELATIVE FISHER INFORMATION VIA SDPI Jiayang Zou, Luyao Fan, Jiayang Gao, and Jia Wang. Convexity of mutual information along the Fokker-Planck flow. arXiv preprint arXiv:2501.05094,

work page arXiv
[2006]

Further and stronger analogy between sampling and optimization: Langevin Monte Carlo and gradient descent

Arnak Dalalyan. Further and stronger analogy between sampling and optimization: Langevin Monte Carlo and gradient descent. In Satyen Kale and Ohad Shamir, editors, Proceedings of the 2017 Conference on Learning Theory, volume 65 ofProceedings of Machine Learning Research, pages 678–689. PMLR, 07–10 Jul 2017a. Arnak S Dalalyan. Theoretical guarantees for a...

work page 2017

Show all 15 references

[2016]

Fisher information dissipation for time inhomogeneous stochastic differential equations

40 MIXING TIME OF THE PROXIMAL SAMPLER IN RELATIVE FISHER INFORMATION VIA SDPI Qi Feng, Xinzhe Zuo, and Wuchen Li. Fisher information dissipation for time inhomogeneous stochastic differential equations. arXiv preprint arXiv:2402.01036,

arXiv
[2018]

Convexity of mutual information along the heat flow

Andre Wibisono and Varun Jog. Convexity of mutual information along the heat flow. In 2018 IEEE International Symposium on Information Theory (ISIT), pages 1615–1619. IEEE, 2018a. Andre Wibisono and Varun Jog. Convexity of mutual information along the Ornstein–Uhlenbeck flow. ...

2018
[2019]

Entropy contractions in Markov chains: Half-step, full-step and continuous-time

Pietro Caputo, Zongchen Chen, Yuzhou Gu, and Yury Polyanskiy. Entropy contractions in Markov chains: Half-step, full-step and continuous-time. arXiv preprint arXiv:2409.07689,

arXiv
[2021]

The query complexity of sampling from strongly log-concave distributions in one dimension

Sinho Chewi, Patrik R Gerber, Chen Lu, Thibaut Le Gouic, and Philippe Rigollet. The query complexity of sampling from strongly log-concave distributions in one dimension. InConference on Learning Theory, pages 2041–2059. PMLR,
[2022]

A simple proof of the mixing of Metropolis-Adjusted Langevin Algorithm under smoothness and isoperimetry.arXiv preprint arXiv:2304.04095,

Yuansi Chen and Khashayar Gatmiry. A simple proof of the mixing of Metropolis-Adjusted Langevin Algorithm under smoothness and isoperimetry.arXiv preprint arXiv:2304.04095,

arXiv
[2023]

Diffusions hypercontractives

Dominique Bakry and Michel ´Emery. Diffusions hypercontractives. In S´eminaire de Probabilit´es XIX 1983/84, pages 177–206. Springer,

1983
[2024]

On the convergence of min-max Langevin dynamics and algorithm

Yang Cai, Siddharth Mitra, Xiuyuan Wang, and Andre Wibisono. On the convergence of min-max Langevin dynamics and algorithm. arXiv preprint arXiv:2412.20471v2,

arXiv

Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.

[1] [7]

Coupled Wasserstein gradient flows for min-max and cooperative games

Lauren Conger, Franca Hoffmann, Eric Mazumdar, and Lillian J Ratliff. Coupled Wasserstein gradient flows for min-max and cooperative games. arXiv preprint arXiv:2411.07403,

arXiv

[2] [10]

Private convex optimization via exponential mecha- nism

Sivakanth Gopi, Yin Tat Lee, and Daogao Liu. Private convex optimization via exponential mecha- nism. In Conference on Learning Theory, pages 1948–1989. PMLR,

work page 1948

[3] [11]

A proximal algorithm for sampling from non-smooth potentials

Jiaming Liang and Yongxin Chen. A proximal algorithm for sampling from non-smooth potentials. In 2022 Winter Simulation Conference (WSC), pages 3229–3240. IEEE,

work page 2022

[4] [12]

Jiaming Liang and Yongxin Chen

ISSN 2835-8856. Jiaming Liang and Yongxin Chen. Proximal oracles for optimization and sampling. arXiv preprint arXiv:2404.02239,

work page arXiv

[5] [338]

Andre Wibisono

Springer, 2009b. Andre Wibisono. Sampling as optimization in the space of measures: The Langevin dynamics as a composite optimization problem. In Conference on Learning Theory, pages 2093–3027. PMLR,

work page

[6] [1965]

Heat flow, log-concavity, and Lipschitz transport maps

Giovanni Brigati and Francesco Pedrotti. Heat flow, log-concavity, and Lipschitz transport maps. arXiv preprint arXiv:2404.15205v2,

arXiv

[7] [1998]

Convexity of mutual information along the Fokker-Planck flow

42 MIXING TIME OF THE PROXIMAL SAMPLER IN RELATIVE FISHER INFORMATION VIA SDPI Jiayang Zou, Luyao Fan, Jiayang Gao, and Jia Wang. Convexity of mutual information along the Fokker-Planck flow. arXiv preprint arXiv:2501.05094,

work page arXiv

[8] [2006]

Further and stronger analogy between sampling and optimization: Langevin Monte Carlo and gradient descent

Arnak Dalalyan. Further and stronger analogy between sampling and optimization: Langevin Monte Carlo and gradient descent. In Satyen Kale and Ohad Shamir, editors, Proceedings of the 2017 Conference on Learning Theory, volume 65 ofProceedings of Machine Learning Research, pages 678–689. PMLR, 07–10 Jul 2017a. Arnak S Dalalyan. Theoretical guarantees for a...

work page 2017