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Upper and Lower Bounds on Expected Soft Maxima of Gaussian Processes

T0 review · 2 major / 1 minor · reviewed 2026-06-26 · grok-4.3

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Pith's one-line read Upper and lower bounds are obtained for smoothed expected suprema of centered Gaussian processes, recovering the majorizing measure theorem at zero temperature.

desk verdict Finite-beta soft-max bounds via Liu tensorization recover majorizing measures and give finite-N Parisi for SK, but the tensorization step needs verification. read the letter →

arxiv 2606.22611 v1 pith:YTTH6WMB submitted 2026-06-21 math.PR

Yifeng Chu , Maxim Raginsky This is my paper · ORCID
classification math.PR
keywords GaussianprocessessoftmaximaexpectedsupremummajorizingmeasuresgenericchainingSherrington-KirkpatrickmodelParisiformulatensorization
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 derives upper and lower bounds on expected soft maxima of centered Gaussian processes indexed by finite or countable sets. These soft maxima are defined via expected values of random Gibbs averages at positive inverse temperature beta and serve as smoothed approximations to the usual expected supremum. The bounds keep the multiscale structure familiar from generic chaining but incorporate an additional truncation term that depends on beta. As beta tends to infinity the bounds reduce exactly to the majorizing measure theorem. The same machinery yields a Parisi formula for the quenched free energy of the Sherrington-Kirkpatrick model that holds at finite system size.

What carries the argument

The soft maximum, defined as the expectation of a random Gibbs average of the Gaussian process at finite inverse temperature beta, which approximates the supremum while enabling tensorization.

What would settle it

Explicit computation of the true expected soft maximum for a two-point Gaussian process, together with the proposed upper and lower bounds, showing that the true value lies outside the interval.

Watch

Extended reading notes

Core claim

We obtain upper and lower bounds for smoothed versions of the expected supremum of centered Gaussian processes with finite or countable index sets. These so-called soft maxima are computed in terms of expected values of random Gibbs averages at inverse temperature β > 0 and reduce to expected suprema in the zero-temperature limit β → ∞. The bounds retain the same multiscale structure as in the expressions for the expected supremum derived using the method of generic chaining, with a truncation term governed by the inverse temperature β. In the zero-temperature limit, we recover the majorizing measure theorem. As an illustrative example, we apply our results to the analysis of the quenched fr

Load-bearing premise

The tensorization technique applies without modification to the Gibbs-averaged soft-maxima functionals considered here.

Editorial extensions

If this is right

  • The majorizing measure theorem emerges exactly as the zero-temperature limit of the soft-maxima bounds.
  • A Parisi formula for the quenched free energy holds at finite system size for the Sherrington-Kirkpatrick model.
  • The bounds preserve the multiscale chaining structure of generic chaining, modified only by a beta-dependent truncation term.

Reading between the lines

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

  • Finite-beta smoothing may permit numerical approximation of expected suprema for processes too complex for direct supremum analysis.
  • The same tensorization approach could be tested on other smoothed functionals arising in high-dimensional probability.
  • The finite-size Parisi formula offers a concrete starting point for studying convergence rates to the thermodynamic limit in spin-glass models.
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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

2 major / 1 minor

Summary. The paper derives upper and lower bounds on smoothed versions of the expected supremum (soft maxima) of centered Gaussian processes on finite or countable index sets. These soft maxima are defined via expected random Gibbs averages at finite inverse temperature β > 0 and reduce to ordinary expected suprema as β → ∞. The bounds preserve the multiscale chaining structure of generic chaining, with an additional β-dependent truncation term. The derivation relies on the tensorization inequality from Liu (2025). In the zero-temperature limit the majorizing-measure theorem is recovered. The results are illustrated by deriving a finite-N Parisi formula for the quenched free energy of the Sherrington-Kirkpatrick model.

Significance. If the tensorization step applies as stated, the work supplies a finite-temperature extension of chaining bounds that recovers a classical theorem in the appropriate limit and yields an explicit finite-size formula for the SK model. The multiscale structure and explicit β-dependence are potentially useful for statistical-physics applications where zero-temperature limits are not the only regime of interest.

major comments (2)
  1. [Abstract (analysis paragraph)] Abstract (paragraph on analysis): the assertion that Liu (2025) tensorization applies directly to the Gibbs-averaged soft-maxima functionals is load-bearing for all stated bounds, yet the abstract supplies no verification that the β-dependent averaging preserves the subadditivity or product-measure Lipschitz hypotheses required by that inequality. If an extra dependence on the underlying measure appears, the constants in the resulting multiscale bounds would change and the zero-temperature recovery would require a separate justification.
  2. [Application to SK model] Application to SK model (final paragraph): the finite-system-size Parisi formula is obtained by feeding the quenched free-energy functional into the same tensorization step. Without an explicit check that this functional satisfies the hypotheses of Liu (2025), the claimed finite-N formula rests on an unverified extension and cannot be regarded as established.
minor comments (1)
  1. The abstract states that the index set may be countable, but does not indicate whether the truncation term or the tensorization constants remain uniform when the index set is infinite.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the constructive major comments. We address each point below.

read point-by-point responses

  1. Referee: [Abstract (analysis paragraph)] Abstract (paragraph on analysis): the assertion that Liu (2025) tensorization applies directly to the Gibbs-averaged soft-maxima functionals is load-bearing for all stated bounds, yet the abstract supplies no verification that the β-dependent averaging preserves the subadditivity or product-measure Lipschitz hypotheses required by that inequality. If an extra dependence on the underlying measure appears, the constants in the resulting multiscale bounds would change and the zero-temperature recovery would require a separate justification.

    Authors: The abstract is intentionally concise. The verification that the soft-maxima functional satisfies the subadditivity and product-measure Lipschitz conditions of Liu (2025) without introducing extraneous measure dependence is carried out in the proof of the main chaining bound (Section 3). The β-dependent truncation arises only from the explicit form of the functional and does not alter the constants or the zero-temperature argument, which is handled separately by monotone convergence in Section 4. We will add a one-sentence pointer in the abstract to this verification. revision: partial

  2. Referee: [Application to SK model] Application to SK model (final paragraph): the finite-system-size Parisi formula is obtained by feeding the quenched free-energy functional into the same tensorization step. Without an explicit check that this functional satisfies the hypotheses of Liu (2025), the claimed finite-N formula rests on an unverified extension and cannot be regarded as established.

    Authors: The quenched free energy of the SK model is exactly the soft-maxima functional applied to the centered Gaussian process given by the SK Hamiltonian. Because the general theorem already establishes that every soft-maxima functional of this form meets the hypotheses of Liu (2025), the finite-N Parisi formula follows directly. To make the application fully self-contained we will insert a short explicit verification paragraph immediately before the statement of the finite-N formula. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: bounds derived from external tensorization applied to new functionals

full rationale

The derivation applies the tensorization inequality from the independent reference Liu (2025) to the expected Gibbs-averaged soft-maxima. No step reduces a claimed bound to a fitted parameter or prior result by the paper's own equations. The zero-temperature recovery of the majorizing measure theorem is stated as a limit case rather than an input. The paper is therefore self-contained against the external benchmark provided by Liu (2025).

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

No free parameters, invented entities, or non-standard axioms are mentioned in the abstract; the derivation relies on the external Liu (2025) tensorization result.

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Cite this review

Pith. "Pith review of Upper and Lower Bounds on Expected Soft Maxima of Gaussian Processes." pith.science (2026). https://pith.science/paper/YTTH6WMB

@misc{pith2026260622611,
  author       = {Pith},
  title        = {Pith review of: Upper and Lower Bounds on Expected Soft Maxima of Gaussian Processes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YTTH6WMB}},
  note         = {Machine review of arXiv:2606.22611}
}
abstract

We obtain upper and lower bounds for "smoothed" versions of the expected supremum of centered Gaussian processes with finite or countable index sets. These so-called soft maxima are computed in terms of expected values of random Gibbs averages at inverse temperature $\beta > 0$ and reduce to expected suprema in the zero-temperature limit $\beta \to \infty$. Our analysis builds on ideas from statistical physics and information theory, and relies crucially on the tensorization technique introduced recently by Liu (2025). The bounds retain the same multiscale structure as in the expressions for the expected supremum derived using the method of generic chaining, with a truncation term governed by the inverse temperature $\beta$. In the zero-temperature limit, we recover the majorizing measure theorem. As an illustrative example, we apply our results to the analysis of the quenched free energy in the Sherrington-Kirkpatrick model and obtain a Parisi formula in the finite system size setting.

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Works this paper leans on

32 extracted references · 3 canonical work pages

  1. [1]

    Adler and Jonathan E

    Robert J. Adler and Jonathan E. Taylor. Random Fields and Geometry. Springer, 2007

    2007

  2. [2]

    The Parisi formula has a unique minimizer

    Antonio Auffinger and Wei-Kuo Chen. The Parisi formula has a unique minimizer. Communications in Mathematical Physics, 335 0 (3): 0 1429--1444, 2015

    2015

  3. [3]

    Bogachev, and Stanislav A

    G \'e rard Ben Arous, Leonid V. Bogachev, and Stanislav A. Molchanov. Limit theorems for sums of random exponentials. Probability Theory and Related Fields, 132 0 (4): 0 579--612, 2005

    2005

  4. [4]

    Statistical Mechanics of Disordered Systems : A Mathematical Perspective

    Anton Bovier. Statistical Mechanics of Disordered Systems : A Mathematical Perspective . Cambridge University Press, 2006

    2006

  5. [5]

    Convex Optimization

    Stephen Boyd and Lieven Vandenberghe. Convex Optimization. Cambridge University Press, 2004

    2004

  6. [6]

    Superconcentration and Related Topics

    Sourav Chatterjee. Superconcentration and Related Topics. Springer, 2014

    2014

  7. [7]

    A unified framework for information-theoretic generalization bounds

    Yifeng Chu and Maxim Raginsky. A unified framework for information-theoretic generalization bounds. In A. Oh, T. Naumann, A. Globerson, K. Saenko, M. Hardt, and S. Levine, editors, Advances in Neural Information Processing Systems, volume 36, pages 79260--79278. Curran Associates, Inc., 2023

    2023

  8. [8]

    Talagrand meets Talagrand : Upper and lower bounds on expected soft maxima of Gaussian processes with finite index sets

    Yifeng Chu and Maxim Raginsky. Talagrand meets Talagrand : Upper and lower bounds on expected soft maxima of Gaussian processes with finite index sets. In Conference on Algorithmic Learning Theory, 2026. URL https://arxiv.org/abs/2502.06709

    work page arXiv 2026

Show all 32 references
  1. [9]

    Cover and Joy A

    Thomas M. Cover and Joy A. Thomas. Elements of Information Theory. Wiley, 2nd edition, 2006

    2006

  2. [10]

    Information Theory : Coding Theorems for Discrete Memoryless Systems

    Imre Csisz \'a r and J \'a nos K \"o rner. Information Theory : Coding Theorems for Discrete Memoryless Systems . Cambridge University Press, 2011

    2011

  3. [11]

    Dobrushin

    Roland L. Dobrushin. Passage to the limit under the information and entropy signs. Theory of Probability and Its Applications, 5 0 (1): 0 25--32, 1960

    1960

  4. [12]

    Regularit\'e des trajectoires des fonctions al\'eatoires gaussiennes

    Xavier Fernique. Regularit\'e des trajectoires des fonctions al\'eatoires gaussiennes. In Ecole d'Et \' e de Probabilit \' e s de Saint-Flour IV 1974 , pages 1--96. Springer, 1975

    1974

  5. [13]

    Caract\'erisation de processus \`a trajectoires major\'ees ou continues

    Xavier Fernique. Caract\'erisation de processus \`a trajectoires major\'ees ou continues. S\'eminaire de probabilit\'es de Strasbourg, 12: 0 691--706, 1978

    1978

  6. [14]

    Supremum of a Process in Terms of Trees

    Olivier Gu\'edon and Artem Zvavitch. Supremum of a Process in Terms of Trees . In Geometric Aspects of Functional Analysis , pages 136--147. Springer , 2003

    2003

  7. [15]

    Broken Replica Symmetry Bounds in the Mean Field Spin Glass Model

    Francesco Guerra. Broken Replica Symmetry Bounds in the Mean Field Spin Glass Model . Communications in Mathematical Physics, 233 0 (1): 0 1--12, 2003

    2003

  8. [16]

    John C. Kieffer. Sample converses in source coding theory. IEEE Transactions on Information Theory, 37 0 (2): 0 263--268, 1991

    1991

  9. [17]

    Arbitrary source models and Bayesian codebooks in rate-distortion theory

    Ioannis Kontoyiannis and Junshan Zhang. Arbitrary source models and Bayesian codebooks in rate-distortion theory. IEEE Transactions on Information Theory, 48 0 (8): 0 2276--2290, 2002

    2002

  10. [18]

    Simple and Sharp Generalization Bounds via Lifting , September 2025

    Jingbo Liu. Simple and Sharp Generalization Bounds via Lifting , September 2025. URL https://arxiv.org/abs/2508.18682

    2025 arXiv

  11. [19]

    MacWilliams and Neil J.A

    Florence J. MacWilliams and Neil J.A. Sloane. The Theory of Error-Correcting Codes. North-Holland, Amsterdam, 1977

    1977

  12. [20]

    On the uniform continuity of the rate-distortion function

    Hari Palaiyanur and Anant Sahai. On the uniform continuity of the rate-distortion function. In 2008 IEEE International Symposium on Information Theory, pages 857--861, 2008

    2008

  13. [21]

    The Sherrington-Kirkpatrick Model

    Dmitry Panchenko. The Sherrington-Kirkpatrick Model . Springer Monographs in Mathematics . Springer , 2013

    2013

  14. [22]

    The Parisi formula for mixed p -spin models

    Dmitry Panchenko. The Parisi formula for mixed p -spin models. The Annals of Probability, 42 0 (3), 2014

    2014

  15. [23]

    Information Theory : From Coding to Learning

    Yury Polyanskiy and Yihong Wu. Information Theory : From Coding to Learning . Cambridge University Press, 2024

    2024

  16. [24]

    Controlling bias in adaptive data analysis using information theory

    Daniel Russo and James Zou. Controlling bias in adaptive data analysis using information theory. In Arthur Gretton and Christian C. Robert, editors, Proceedings of the 19th International Conference on Artificial Intelligence and Statistics, volume 51 of Proceedings of Machine ...

    2016

  17. [25]

    Entropy Bounds for Discrete Random Variables via Maximal Coupling

    Igal Sason. Entropy Bounds for Discrete Random Variables via Maximal Coupling . IEEE Transactions on Information Theory, 59 0 (11): 0 7118--7131, 2013

    2013

  18. [26]

    Regularity of gaussian processes

    Michel Talagrand. Regularity of gaussian processes. Acta Mathematica, 159: 0 99--149, 1987

    1987

  19. [27]

    A simple proof of the majorizing measure theorem

    Michel Talagrand. A simple proof of the majorizing measure theorem. Geometric and Functional Analysis, 2 0 (1): 0 118--125, March 1992

    1992

  20. [28]

    The Parisi formula

    Michel Talagrand. The Parisi formula. Annals of Mathematics, 163 0 (1): 0 221--263, 2006

    2006

  21. [29]

    Mean Field Models for Spin Glasses, Volume I: Basic Examples

    Michel Talagrand. Mean Field Models for Spin Glasses, Volume I: Basic Examples. Springer, 2011

    2011

  22. [30]

    Upper and Lower Bounds for Stochastic Processes: Modern Methods and Classical Problems

    Michel Talagrand. Upper and Lower Bounds for Stochastic Processes: Modern Methods and Classical Problems. Springer, 2014

    2014

  23. [31]

    On the subgaussian comparison theorem, 2025

    Ramon van Handel. On the subgaussian comparison theorem, 2025. URL https://arxiv.org/abs/2512.18588

    2025

  24. [32]

    Information-theoretic analysis of generalization capability of learning algorithms

    Aolin Xu and Maxim Raginsky. Information-theoretic analysis of generalization capability of learning algorithms. In I. Guyon, U. Von Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett, editors, Advances in Neural Information Processing Systems, volume 3...

    2017

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