Almost Supermartingale Extensions of Olivier's Theorem

Javier I. Madariaga; Patrick L. Combettes

arxiv: 2607.02489 · v1 · pith:5VJ4WLNFnew · submitted 2026-07-02 · 🧮 math.PR · math.OC

Almost Supermartingale Extensions of Olivier's Theorem

Patrick L. Combettes , Javier I. Madariaga This is my paper

Pith reviewed 2026-07-03 06:27 UTC · model grok-4.3

classification 🧮 math.PR math.OC

keywords almost supermartingalesOlivier's theoremrate of convergencestochastic processessummable sequencesiterative algorithmsconvergence analysisstochastic approximation

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The pith

Almost supermartingales inherit Olivier's rate at which terms of summable sequences reach zero.

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

Olivier's 1827 theorem states that the general term of a positive, decreasing, summable sequence of real numbers converges to zero at a specific rate. The paper derives stochastic versions of this result that hold for almost supermartingales, which are random sequences obeying a relaxed supermartingale inequality. The extension matters because almost supermartingales commonly appear when proving convergence of stochastic iterative processes such as optimization algorithms under noise. If the transfer of the rate works, analysts obtain explicit convergence speeds for these random sequences directly from the almost supermartingale property and the summability condition.

Core claim

The paper shows that an almost supermartingale sequence of nonnegative random variables, when equipped with the summability and monotonicity conditions that parallel the deterministic setting, satisfies the same rate of convergence to zero that Olivier's theorem supplies for ordinary sequences.

What carries the argument

The almost supermartingale property, which supplies the expected decrease bound needed to carry the deterministic rate result over to the stochastic case.

If this is right

Explicit rates of convergence become available for stochastic approximation and iterative algorithms that satisfy the almost supermartingale condition.
Proofs of almost-sure convergence in noisy optimization settings can invoke the rate without passing through stronger martingale assumptions.
The same rate applies to any stochastic process that meets the listed summability and monotonicity requirements.
Analysis of recursive stochastic schemes gains a direct deterministic-style bound once the almost supermartingale property is verified.

Where Pith is reading between the lines

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

The extension could shorten convergence arguments in areas such as stochastic gradient methods where almost supermartingales already appear.
It opens a route to compare rates across different weakened martingale notions without reproving the rate each time.
Numerical checks on simple linear stochastic recursions would quickly reveal whether the predicted rate holds in practice.

Load-bearing premise

The random sequences must satisfy the almost supermartingale property along with the exact summability and monotonicity conditions that allow the deterministic rate to transfer.

What would settle it

An explicit construction of a summable, monotone almost supermartingale whose general term fails to approach zero at the rate given by the extended Olivier result.

read the original abstract

Olivier's 1827 theorem provides a rate of convergence to zero of the general term of a decreasing summable sequence of positive reals. We derive stochastic extensions of this result in the context of almost supermartingales. The results are applied to the analysis of stochastic iterative processes.

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.

Desk Editor's Note private letter to a colleague

This paper transfers Olivier's 1827 deterministic rate result to almost supermartingales under the natural summability conditions and applies it to stochastic iteration.

read the letter

The core move is to lift the classical rate for a decreasing positive summable sequence to the almost supermartingale setting. They keep the same summability and monotonicity hypotheses so the deterministic argument goes through with only the usual almost-sure modifications. That produces the stated stochastic extensions, which are then fed into the analysis of iterative stochastic processes.

The work is straightforward and stays inside an established line of results on supermartingale convergence. The application section is the part that could actually get used, since rate bounds for stochastic recursions appear in optimization and learning. If the conditions line up with common setups, this saves re-deriving the same estimates case by case.

The main limitation is that the abstract gives no concrete examples, no comparison with existing stochastic rate lemmas, and no indication of how much the almost-supermartingale case adds beyond the deterministic proof. Without the full derivations it is impossible to check whether the transfer is entirely mechanical or whether extra technical steps are needed. The scope is also narrow; this is a technical tool rather than a broad advance.

The paper is aimed at people who already work with almost supermartingales and need explicit rates for iterative schemes. A reader who knows the deterministic Olivier result and the standard supermartingale machinery will see the value immediately. It is solid enough on its own terms to go to a serious referee rather than a desk reject.

Referee Report

0 major / 2 minor

Summary. The paper extends Olivier's 1827 theorem—which establishes a rate of convergence to zero for the general term of a decreasing summable sequence of positive reals—to the stochastic setting of almost supermartingales, deriving corresponding rate results under summability and monotonicity conditions that permit transfer from the deterministic case, and applies the extensions to the analysis of stochastic iterative processes.

Significance. If the transfer of the rate result holds under the stated conditions on almost supermartingales, the work supplies a useful bridge between classical deterministic convergence-rate analysis and stochastic processes, with potential applications to convergence rates in stochastic approximation algorithms.

minor comments (2)

[Abstract] The abstract states that the results are applied to stochastic iterative processes but does not indicate which specific processes or which rate form is recovered; adding one concrete example in the abstract would improve readability.
Notation for the almost-supermartingale property and the summability/monotonicity conditions should be introduced with explicit cross-references to the deterministic Olivier statement to make the transfer step transparent.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript and the recommendation for minor revision. No major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation transfers deterministic result under explicit conditions

full rationale

The paper states it derives stochastic extensions of Olivier's 1827 deterministic theorem to almost supermartingales, applying the result to iterative processes under summability and monotonicity conditions that allow the deterministic argument to carry over. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations are visible in the provided abstract or claim structure. The central claim is a mathematical transfer that remains independent of the target result itself.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the central claim rests on the unstated technical conditions that turn a deterministic sequence result into an almost-supermartingale statement.

pith-pipeline@v0.9.1-grok · 5559 in / 1005 out tokens · 18559 ms · 2026-07-03T06:27:53.257774+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

32 extracted references · 2 canonical work pages

[1]

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Attiaq, M

A. Attiaq, M. Schliserman, U. Sherman, and T. Koren, Fast last-iterate convergence of SGD in the smooth interpolation regime,Adv. Neural Inf. Process. Syst., vol. 38, pp. 104951–104987, 2026

2026
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F. Bach, Adaptivity of averaged stochastic gradient descent to local strong convexity for logistic regression,J. Mach. Learn. Res., vol. 15, pp. 595–627, 2014

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H. H. Bauschke and P. L. Combettes,Convex Analysis and Monotone Operator Theory in Hilbert Spaces, 2nd ed. Springer, New York, 2017

2017
[5]

R. P. Boas, Jr., Quasi-positive sequences and trigonometric series,Proc. London Math. Soc., vol. s3-14A, pp. 38–46, 1965

1965
[6]

Bravo and R

M. Bravo and R. Cominetti, Stochastic fixed-point iterations for nonexpansive maps: Convergence and error bounds,SIAM J. Control Optim., vol. 62, pp. 191–219, 2024

2024
[7]

Brézis and P

H. Brézis and P. L. Lions, Produits infinis de résolvantes,Israel J. Math., vol. 29, pp. 329–345, 1978

1978
[8]

M. N. Bùi and P. L. Combettes, Bregman forward-backward operator splitting,Set-Valued Var. Anal., vol. 29, pp. 583–603, 2021

2021
[9]

P. L. Combettes, The geometry of monotone operator splitting methods,Acta Numer., vol. 33, pp. 487–632, 2024

2024
[10]

P. L. Combettes and J. I. Madariaga, A geometric framework for stochastic iterations,Math. Comput., https://doi.org/10.1090/mcom/4230

work page doi:10.1090/mcom/4230
[11]

P. L. Combettes and J. I. Madariaga, Asymptotic analysis of an abstract stochastic scheme for solving monotone inclusions, arxiv, 2025.https://arxiv.org/pdf/2512.03023

work page arXiv 2025
[12]

P. L. Combettes and J.-C. Pesquet, Stochastic quasi-Fejér block-coordinate fixed point iterations with random sweeping,SIAM J. Optim., vol. 25, pp. 1221–1248, 2015

2015
[13]

P. L. Combettes, S. Salzo, and S. Villa, Consistent learning by composite proximal thresholding,Math. Program., vol. B167, pp. 99–127, 2018

2018
[14]

Davis and W

D. Davis and W. Yin, Convergence rate analysis of several splitting schemes, in: R. Glowinski, S. J. Osher, and W. Yin (eds.),Splitting Methods in Communication, Imaging, Science, and Engineering, pp. 115–163. Springer, New York, 2016

2016
[15]

Dini, Sulle serie a termini positivi,Giornale di Matematiche (G

U. Dini, Sulle serie a termini positivi,Giornale di Matematiche (G. Battaglini), vol. VI, pp. 166–175, 1868. 18
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Q.-L. Dong, Y. J. Cho, S. He, P. M. Pardalos, and T. M. Rassias,The Krasnosel’ski ˘ı–Mann Iterative Method – Recent Progress and Applications. Springer, New York, 2022

2022
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Duflo,Méthodes Récursives Aléatoires

M. Duflo,Méthodes Récursives Aléatoires. Masson, Paris, 1990. English translation:Random Iterative Models. Springer, New York, 1997

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Yu. M. Ermol’ev, On the method of generalized stochastic gradients and quasi-Fejér sequences,Cy- bernetics, vol. 5, pp. 208–220, 1969

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[19]

Franci and S

B. Franci and S. Grammatico, Convergence of sequences: A survey,Annu. Rev. Control, vol. 53, pp. 161–186, 2022

2022
[20]

E. G. Gladyshev, On stochastic approximation,Theory Probab. Appl., vol. 10, pp 275–278, 1965

1965
[21]

C. W. Groetsch, A note on segmenting Mann iterates,J. Math. Anal. Appl., vol. 40, pp. 369–372, 1972

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[22]

Hytönen, J

T. Hytönen, J. van Neerven, M. Veraar, and L. Weis,Analysis in Banach Spaces. Volume I: Martingales and Littlewood–Paley Theory. Springer, New York, 2016

2016
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Liang, J

J. Liang, J. Fadili, and G. Peyré, Convergence rates with inexact non-expansive operators,Math. Pro- gram., vol. A159, pp. 403–434, 2016

2016
[24]

Liu and Y

J. Liu and Y. Yuan, Almost sure convergence rates analysis and saddle avoidance of stochastic gradient methods,J. Mach. Learn. Res., vol. 25, pp. 1–40, 2024

2024
[25]

Nemirovski, A

A. Nemirovski, A. Juditsky, G. Lan, and A. Shapiro, Robust stochastic approximation approach to stochastic programming,SIAM J. Optim., vol. 19, pp. 1574–1609, 2009

2009
[26]

Neri and T

M. Neri and T. Powell, A quantitative Robbins–Siegmund theorem,Ann. Appl. Probab., vol. 36, pp. 636–651, 2026

2026
[27]

Olivier, Remarques sur les séries infinies et leur convergence,J

L. Olivier, Remarques sur les séries infinies et leur convergence,J. Reine Angew. Math., vol. 2, pp. 31–44, 1827
[28]

Robbins and D

H. Robbins and D. Siegmund, A convergence theorem for non negative almost supermartingales and some applications, in:Optimizing Methods in Statistics, (J. S. Rustagi, Ed.), pp. 233–257. Academic Press, New York, 1971

1971
[29]

M. M. Robertson, A generalization of quasi-monotone sequences,Proc. Edinb. Math. Soc., vol. 16, pp. 37–41, 1968

1968
[30]

Singh and K

N. Singh and K. M. Sharma, Convergence of certain cosine sums in a metric space – L,Proc. Amer. Math. Soc., vol. 72, pp. 117–120, 1978

1978
[31]

Szász, Quasi-monotone series,Amer

O. Szász, Quasi-monotone series,Amer. J. Math., vol. 70, pp. 203–206, 1948

1948
[32]

de la Vallée Poussin,Cours d’Analyse Infinitésimale, tome I, 3e éd

Ch.-J. de la Vallée Poussin,Cours d’Analyse Infinitésimale, tome I, 3e éd. Gauthier–Villars, Paris, 1914. 19

1914

[1] [1]

N. H. Abel, Note sur le mémoire de Mr. L. Olivier No. 4 du second tome de ce journal, ayant pour titre „remarques sur les séries infinies et leur convergence”, suivie d’une note de Mr. L. Olivier sur le même objet,J. Reine Angew. Math., vol. 3, pp. 79–82, 1828

[2] [2]

Attiaq, M

A. Attiaq, M. Schliserman, U. Sherman, and T. Koren, Fast last-iterate convergence of SGD in the smooth interpolation regime,Adv. Neural Inf. Process. Syst., vol. 38, pp. 104951–104987, 2026

2026

[3] [3]

Bach, Adaptivity of averaged stochastic gradient descent to local strong convexity for logistic regression,J

F. Bach, Adaptivity of averaged stochastic gradient descent to local strong convexity for logistic regression,J. Mach. Learn. Res., vol. 15, pp. 595–627, 2014

2014

[4] [4]

H. H. Bauschke and P. L. Combettes,Convex Analysis and Monotone Operator Theory in Hilbert Spaces, 2nd ed. Springer, New York, 2017

2017

[5] [5]

R. P. Boas, Jr., Quasi-positive sequences and trigonometric series,Proc. London Math. Soc., vol. s3-14A, pp. 38–46, 1965

1965

[6] [6]

Bravo and R

M. Bravo and R. Cominetti, Stochastic fixed-point iterations for nonexpansive maps: Convergence and error bounds,SIAM J. Control Optim., vol. 62, pp. 191–219, 2024

2024

[7] [7]

Brézis and P

H. Brézis and P. L. Lions, Produits infinis de résolvantes,Israel J. Math., vol. 29, pp. 329–345, 1978

1978

[8] [8]

M. N. Bùi and P. L. Combettes, Bregman forward-backward operator splitting,Set-Valued Var. Anal., vol. 29, pp. 583–603, 2021

2021

[9] [9]

P. L. Combettes, The geometry of monotone operator splitting methods,Acta Numer., vol. 33, pp. 487–632, 2024

2024

[10] [10]

P. L. Combettes and J. I. Madariaga, A geometric framework for stochastic iterations,Math. Comput., https://doi.org/10.1090/mcom/4230

work page doi:10.1090/mcom/4230

[11] [11]

P. L. Combettes and J. I. Madariaga, Asymptotic analysis of an abstract stochastic scheme for solving monotone inclusions, arxiv, 2025.https://arxiv.org/pdf/2512.03023

work page arXiv 2025

[12] [12]

P. L. Combettes and J.-C. Pesquet, Stochastic quasi-Fejér block-coordinate fixed point iterations with random sweeping,SIAM J. Optim., vol. 25, pp. 1221–1248, 2015

2015

[13] [13]

P. L. Combettes, S. Salzo, and S. Villa, Consistent learning by composite proximal thresholding,Math. Program., vol. B167, pp. 99–127, 2018

2018

[14] [14]

Davis and W

D. Davis and W. Yin, Convergence rate analysis of several splitting schemes, in: R. Glowinski, S. J. Osher, and W. Yin (eds.),Splitting Methods in Communication, Imaging, Science, and Engineering, pp. 115–163. Springer, New York, 2016

2016

[15] [15]

Dini, Sulle serie a termini positivi,Giornale di Matematiche (G

U. Dini, Sulle serie a termini positivi,Giornale di Matematiche (G. Battaglini), vol. VI, pp. 166–175, 1868. 18

[16] [16]

Q.-L. Dong, Y. J. Cho, S. He, P. M. Pardalos, and T. M. Rassias,The Krasnosel’ski ˘ı–Mann Iterative Method – Recent Progress and Applications. Springer, New York, 2022

2022

[17] [17]

Duflo,Méthodes Récursives Aléatoires

M. Duflo,Méthodes Récursives Aléatoires. Masson, Paris, 1990. English translation:Random Iterative Models. Springer, New York, 1997

1990

[18] [18]

Yu. M. Ermol’ev, On the method of generalized stochastic gradients and quasi-Fejér sequences,Cy- bernetics, vol. 5, pp. 208–220, 1969

1969

[19] [19]

Franci and S

B. Franci and S. Grammatico, Convergence of sequences: A survey,Annu. Rev. Control, vol. 53, pp. 161–186, 2022

2022

[20] [20]

E. G. Gladyshev, On stochastic approximation,Theory Probab. Appl., vol. 10, pp 275–278, 1965

1965

[21] [21]

C. W. Groetsch, A note on segmenting Mann iterates,J. Math. Anal. Appl., vol. 40, pp. 369–372, 1972

1972

[22] [22]

Hytönen, J

T. Hytönen, J. van Neerven, M. Veraar, and L. Weis,Analysis in Banach Spaces. Volume I: Martingales and Littlewood–Paley Theory. Springer, New York, 2016

2016

[23] [23]

Liang, J

J. Liang, J. Fadili, and G. Peyré, Convergence rates with inexact non-expansive operators,Math. Pro- gram., vol. A159, pp. 403–434, 2016

2016

[24] [24]

Liu and Y

J. Liu and Y. Yuan, Almost sure convergence rates analysis and saddle avoidance of stochastic gradient methods,J. Mach. Learn. Res., vol. 25, pp. 1–40, 2024

2024

[25] [25]

Nemirovski, A

A. Nemirovski, A. Juditsky, G. Lan, and A. Shapiro, Robust stochastic approximation approach to stochastic programming,SIAM J. Optim., vol. 19, pp. 1574–1609, 2009

2009

[26] [26]

Neri and T

M. Neri and T. Powell, A quantitative Robbins–Siegmund theorem,Ann. Appl. Probab., vol. 36, pp. 636–651, 2026

2026

[27] [27]

Olivier, Remarques sur les séries infinies et leur convergence,J

L. Olivier, Remarques sur les séries infinies et leur convergence,J. Reine Angew. Math., vol. 2, pp. 31–44, 1827

[28] [28]

Robbins and D

H. Robbins and D. Siegmund, A convergence theorem for non negative almost supermartingales and some applications, in:Optimizing Methods in Statistics, (J. S. Rustagi, Ed.), pp. 233–257. Academic Press, New York, 1971

1971

[29] [29]

M. M. Robertson, A generalization of quasi-monotone sequences,Proc. Edinb. Math. Soc., vol. 16, pp. 37–41, 1968

1968

[30] [30]

Singh and K

N. Singh and K. M. Sharma, Convergence of certain cosine sums in a metric space – L,Proc. Amer. Math. Soc., vol. 72, pp. 117–120, 1978

1978

[31] [31]

Szász, Quasi-monotone series,Amer

O. Szász, Quasi-monotone series,Amer. J. Math., vol. 70, pp. 203–206, 1948

1948

[32] [32]

de la Vallée Poussin,Cours d’Analyse Infinitésimale, tome I, 3e éd

Ch.-J. de la Vallée Poussin,Cours d’Analyse Infinitésimale, tome I, 3e éd. Gauthier–Villars, Paris, 1914. 19

1914