pith. sign in

arxiv: 2605.15859 · v1 · pith:WDFVIT2Mnew · submitted 2026-05-15 · 💻 cs.DS · cs.LG· math.ST· stat.ML· stat.TH

Complexity of Non-Log-Concave Sampling in Fisher Information

Sinho Chewi , Andre Wibisono This is my paper

Pith reviewed 2026-05-19 19:05 UTC · model grok-4.3

classification 💻 cs.DS cs.LGmath.STstat.MLstat.TH
keywords non-log-concave samplingrelative Fisher informationproximal samplerrestricted Gaussian oracleRényi divergencelog-concave samplingquery complexity
0
0 comments X

The pith

Leveraging log-concave sampling results gives non-log-concave sampling the same dimension dependence in relative Fisher information.

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

The paper establishes that an approximate implementation of the restricted Gaussian oracle, drawn from recent high-accuracy log-concave sampling algorithms in Rényi divergence, can be plugged into the proximal sampler to obtain relative Fisher information guarantees for log-smooth non-log-concave distributions. This approach inherits the dimension dependence of the best known log-concave samplers and improves on earlier bounds for the non-log-concave case. A converse result shows that any advance in the dimension dependence for non-log-concave sampling under this metric would immediately improve high-accuracy log-concave sampling as well. These results frame relative Fisher information as a natural complexity measure that connects the two regimes tightly.

Core claim

By using recent log-concave sampling results with high-accuracy guarantees in Rényi divergence to build an approximate restricted Gaussian oracle, the proximal sampler achieves a complexity bound for relative Fisher information on non-log-concave targets that matches the dimension dependence of log-concave sampling, improving prior non-log-concave bounds, with a matching converse reduction in the other direction.

What carries the argument

The proximal sampler combined with an approximate restricted Gaussian oracle (RGO) constructed from log-concave samplers in Rényi divergence.

If this is right

  • Non-log-concave sampling now has complexity guarantees with the same polynomial dependence on dimension as log-concave sampling for relative Fisher information.
  • Prior algorithms for non-log-concave sampling had worse dimension dependence, which this improves.
  • Improving the dimension dependence for relative Fisher information in the non-log-concave setting would automatically improve high-accuracy sampling for log-concave distributions.
  • The reduction shows equivalence in dimension dependence between the two problems under this guarantee.

Where Pith is reading between the lines

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

  • This tight connection suggests that progress on one sampling regime can directly transfer to the other through this oracle construction.
  • Future work could explore whether similar reductions hold for other sampling guarantees or divergences.
  • If log-concave sampling algorithms continue to improve their Rényi divergence bounds, the non-log-concave results would benefit immediately without new ideas.

Load-bearing premise

The error from the approximate restricted Gaussian oracle does not introduce additional dimension dependence when inserted into the proximal sampler for the non-log-concave target.

What would settle it

A specific non-log-concave distribution where running the described algorithm requires more queries than the log-concave bound predicts, or where the converse reduction fails to hold.

read the original abstract

We study the query complexity of obtaining a relative Fisher information guarantee for sampling from a log-smooth non-log-concave distribution; this is a sampling analog of finding an approximate stationary point in optimization. Our algorithm is based on the proximal sampler, which is an implicit discretization of the Langevin diffusion, and requires an implementation of the backward step known as the restricted Gaussian oracle (RGO). We show that by leveraging the recent results for log-concave sampling with high-accuracy guarantees in R\'enyi divergence, we can obtain an approximate RGO implementation that -- when used with the proximal sampler -- yields a complexity guarantee in relative Fisher information that inherits the same dimension dependence as log-concave sampling, and improves upon prior work for non-log-concave sampling. We also show a converse reduction that any improvement in the dimension dependence in relative Fisher information for non-log-concave sampling will yield an improved dimension dependence for high-accuracy log-concave sampling.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper studies the query complexity of sampling from log-smooth non-log-concave distributions to achieve a relative Fisher information guarantee, which is positioned as a sampling analog of approximate stationarity in optimization. The algorithm relies on the proximal sampler (an implicit discretization of Langevin diffusion) and implements the required restricted Gaussian oracle (RGO) approximately by composing recent high-accuracy log-concave sampling results that provide Rényi divergence guarantees. This composition is claimed to yield a complexity bound whose dimension dependence matches that of log-concave sampling and improves on prior non-log-concave results. A symmetric converse reduction is also presented, showing that any improvement in the non-log-concave Fisher-information setting would imply better dimension dependence for high-accuracy log-concave sampling.

Significance. If the error-control arguments hold, the result would usefully connect the complexity landscapes of log-concave and non-log-concave sampling under a relative Fisher information metric, inheriting strong dimension dependence from recent Rényi-based log-concave analyses while providing a new hardness implication in the converse direction. The work is technically grounded in the proximal sampler framework and explicitly leverages external high-accuracy guarantees rather than inventing new sampling primitives.

major comments (1)
  1. [Abstract] Abstract and algorithm description: the claim that the approximate RGO obtained from Rényi-divergence log-concave samplers can be plugged into the proximal sampler while preserving the same dimension dependence in relative Fisher information requires a precise error-propagation analysis (including explicit constants and how the approximation error accumulates over proximal steps). The provided sketch does not yet make this composition fully rigorous, leaving open whether additional poly(d) or worse factors appear.
minor comments (2)
  1. Clarify the precise definition of the relative Fisher information guarantee used throughout (e.g., whether it is the standard KL-gradient norm or a smoothed variant) and ensure it is consistent between the forward and converse reductions.
  2. Add a short comparison table or paragraph contrasting the new complexity bound (in terms of dimension d, smoothness L, and target accuracy) against the best prior non-log-concave results cited in the introduction.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thoughtful review and for identifying the need for a more precise error analysis in the composition of the approximate restricted Gaussian oracle (RGO) with the proximal sampler. We address the major comment below and will revise the manuscript to incorporate a fully rigorous error-propagation argument, including explicit constants and accumulation bounds.

read point-by-point responses

  1. Referee: [Abstract] Abstract and algorithm description: the claim that the approximate RGO obtained from Rényi-divergence log-concave samplers can be plugged into the proximal sampler while preserving the same dimension dependence in relative Fisher information requires a precise error-propagation analysis (including explicit constants and how the approximation error accumulates over proximal steps). The provided sketch does not yet make this composition fully rigorous, leaving open whether additional poly(d) or worse factors appear.

    Authors: We agree that the current sketch of the composition requires expansion to be fully rigorous. The proximal sampler framework (as an implicit discretization) allows the total number of steps to be chosen independently of dimension for the target relative Fisher information guarantee, while each RGO call is implemented via a high-accuracy Rényi-divergence sampler for a log-concave target. In the revision we will add a dedicated subsection detailing the error propagation: we convert the Rényi guarantee to a bound on the relative Fisher information error per step using standard inequalities between divergences, then show that the per-step error can be made sufficiently small (with constants depending only on the smoothness parameter and target accuracy) so that the accumulated error over the O(1) proximal steps (in the relevant regime) does not introduce extra polynomial factors in dimension. This preserves the claimed dimension dependence matching recent log-concave Rényi results. We will also include explicit constants in the final complexity bound. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation builds on external log-concave results

full rationale

The paper derives its complexity bound for relative Fisher information in non-log-concave sampling by explicitly composing independent recent high-accuracy Rényi divergence guarantees for log-concave sampling into an approximate restricted Gaussian oracle, then feeding that into the proximal sampler. This composition is presented as an algorithmic reduction that inherits dimension dependence from the external results, without any self-definitional fitting, parameter renaming, or load-bearing self-citation chain internal to the paper. The converse reduction (improvements in non-log-concave Fisher information implying better log-concave Rényi bounds) is a symmetric implication rather than a circular closure. No equations or steps in the abstract or algorithm description reduce the claimed guarantee to the paper's own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard mathematical properties of Langevin diffusions and proximal discretizations plus external results on log-concave sampling; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption The target distribution is log-smooth
    Explicitly stated as the setting in the abstract.
  • domain assumption High-accuracy log-concave sampling algorithms exist with guarantees in Rényi divergence
    Leveraged from recent results as described in the abstract.

pith-pipeline@v0.9.0 · 5702 in / 1495 out tokens · 60926 ms · 2026-05-19T19:05:49.094374+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

32 extracted references · 32 canonical work pages

  1. [1]

    and Salim, Adil and Zhang, Matthew S

    Balasubramanian, Krishna and Chewi, Sinho and Erdogdu, Murat A. and Salim, Adil and Zhang, Matthew S. , booktitle =. Towards a theory of non-log-concave sampling: first-order stationarity guarantees for. 2022 , editor =

    work page 2022

  2. [2]

    Optimal score estimation via empirical

    Wibisono, Andre and Wu, Yihong and Yang, Kaylee Yingxi , booktitle =. Optimal score estimation via empirical. 2024 , series =

    work page 2024

  3. [3]

    2026 , volume=

    Wibisono, Andre , journal=. 2026 , volume=

    work page 2026

  4. [4]

    Sinho Chewi and Alkis Kalavasis and Anay Mehrotra and Omar Montasser , year=

    work page

  5. [5]

    Proceedings of the 34th International Conference on Algorithmic Learning Theory , pages =

    Fisher information lower bounds for sampling , author =. Proceedings of the 34th International Conference on Algorithmic Learning Theory , pages =. 2023 , editor =

    work page 2023

  6. [6]

    and Chewi, Sinho , TITLE =

    Altschuler, Jason M. and Chewi, Sinho , TITLE =. J. ACM , FJOURNAL =. 2024 , NUMBER =

    work page 2024

  7. [7]

    Structured logconcave sampling with a restricted

    Lee, Yin Tat and Shen, Ruoqi and Tian, Kevin , booktitle =. Structured logconcave sampling with a restricted

    work page

  8. [8]

    Improved analysis for a proximal algorithm for sampling , volume =

    Chen, Yongxin and Chewi, Sinho and Salim, Adil and Wibisono, Andre , booktitle =. Improved analysis for a proximal algorithm for sampling , volume =

    work page

  9. [9]

    2026 , NOTE =

    Chewi, Sinho , TITLE =. 2026 , NOTE =

    work page 2026

  10. [10]

    Beyond log-concavity: provable guarantees for sampling multi-modal distributions using simulated tempering

    Lee, Holden and Risteski, Andrej and Ge, Rong , booktitle =. Beyond log-concavity: provable guarantees for sampling multi-modal distributions using simulated tempering

    work page

  11. [11]

    How to make the gradients small , author=. Optima. Mathematical Optimization Society Newsletter , number=

    work page

  12. [12]

    and Hinder, Oliver and Sidford, Aaron , TITLE =

    Carmon, Yair and Duchi, John C. and Hinder, Oliver and Sidford, Aaron , TITLE =. Math. Program. , FJOURNAL =. 2020 , NUMBER =

    work page 2020

  13. [13]

    Jordan, Richard and Kinderlehrer, David and Otto, Felix , TITLE =. SIAM J. Math. Anal. , FJOURNAL =. 1998 , NUMBER =

    work page 1998

  14. [14]

    Sampling as optimization in the space of measures: the

    Wibisono, Andre , booktitle =. Sampling as optimization in the space of measures: the. 2018 , editor =

    work page 2018

  15. [15]

    Proceedings of Thirty Third Conference on Learning Theory , pages =

    How to trap a gradient flow , author =. Proceedings of Thirty Third Conference on Learning Theory , pages =. 2020 , editor =

    work page 2020

  16. [16]

    Proceedings of the 34th International Conference on Algorithmic Learning Theory , pages =

    On the complexity of finding stationary points of smooth functions in one dimension , author =. Proceedings of the 34th International Conference on Algorithmic Learning Theory , pages =. 2023 , editor =

    work page 2023

  17. [17]

    Proceedings of Thirty Sixth Conference on Learning Theory , pages =

    The computational complexity of finding stationary points in non-convex optimization , author =. Proceedings of Thirty Sixth Conference on Learning Theory , pages =. 2023 , editor =

    work page 2023

  18. [18]

    Proceedings of Thirty Sixth Conference on Learning Theory , pages =

    Improved dimension dependence of a proximal algorithm for sampling , author =. Proceedings of Thirty Sixth Conference on Learning Theory , pages =. 2023 , editor =

    work page 2023

  19. [19]

    2026 , journal=

    High-accuracy sampling for diffusion models and log-concave distributions , author=. 2026 , journal=

    work page 2026

  20. [20]

    2026 , journal=

    Reweighted information inequalities , author=. 2026 , journal=

    work page 2026

  21. [21]

    Fast conditional mixing of

    Cheng, Xiang and Wang, Bohan and Zhang, Jingzhao and Zhu, Yusong , booktitle =. Fast conditional mixing of

    work page

  22. [22]

    A convergence theory for

    Salim, Adil and Sun, Lukang and Richtarik, Peter , booktitle =. A convergence theory for. 2022 , editor =

    work page 2022

  23. [23]

    A finite-particle convergence rate for

    Shi, Jiaxin and Mackey, Lester , booktitle =. A finite-particle convergence rate for

    work page

  24. [24]

    Convergence of

    Sun, Lukang and Karagulyan, Avetik and Richtarik, Peter , booktitle =. Convergence of. 2023 , editor =

    work page 2023

  25. [25]

    Finite-particle rates for regularized

    Ye He and Krishnakumar Balasubramanian and Sayan Banerjee and Promit Ghosal , year=. Finite-particle rates for regularized

    work page

  26. [26]

    Improved finite-particle convergence rates for

    Krishna Balasubramanian and Sayan Banerjee and Promit Ghosal , booktitle=. Improved finite-particle convergence rates for

    work page

  27. [27]

    and Lu, Jianfeng , TITLE =

    He, Ye and Balasubramanian, Krishnakumar and Sriperumbudur, Bharath K. and Lu, Jianfeng , TITLE =. Found. Comput. Math. , FJOURNAL =. 2025 , NUMBER =

    work page 2025

  28. [28]

    Stein variational gradient descent as gradient flow , volume =

    Liu, Qiang , booktitle =. Stein variational gradient descent as gradient flow , volume =

    work page

  29. [29]

    Chewi, Sinho and de Dios Pont, Jaume and Li, Jerry and Lu, Chen and Narayanan, Shyam , TITLE =. J. ACM , FJOURNAL =. 2024 , NUMBER =

    work page 2024

  30. [30]

    The adaptive complexity of minimizing relative

    Zhou, Huanjian and Sugiyama, Masashi , booktitle =. The adaptive complexity of minimizing relative

    work page

  31. [31]

    Proceedings of Thirty Eighth Conference on Learning Theory , pages =

    On the query complexity of sampling from non-log-concave distributions (extended abstract) , author =. Proceedings of Thirty Eighth Conference on Learning Theory , pages =. 2025 , editor =

    work page 2025

  32. [32]

    Provable benefit of annealed

    Guo, Wei and Tao, Molei and Chen, Yongxin , booktitle =. Provable benefit of annealed

    work page