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arxiv: 2605.06129 · v1 · submitted 2026-05-07 · 🧮 math.OC

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Convergence guarantees for stochastic algorithms solving non-unique problems in metric spaces

Nicholas Pischke , Thomas Powell
Authors on Pith no claims yet

Pith reviewed 2026-05-08 08:19 UTC · model grok-4.3

classification 🧮 math.OC
keywords stochastic algorithmsquasi-Fejér monotonicitymetric spacesconvergence ratesregularity conditionsoptimizationfixed-point problems
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The pith

Stochastic quasi-Fejér monotone sequences in metric spaces have explicit convergence rates under a general regularity condition.

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

The paper proves that an abstract stochastic regularity condition implies quantitative rates of convergence for quasi-Fejér monotone sequences in metric spaces, both in mean and almost surely. This condition unifies many standard assumptions used in optimization such as error bounds and monotonicity properties. A sympathetic reader would care because it provides a uniform way to analyze algorithms for problems with multiple solutions. The rates depend only on limited information about the problem data. The result is then used to derive new rates for three concrete stochastic methods in spaces of nonpositive curvature.

Core claim

The authors establish a general quantitative theorem that constructs explicit rates of convergence for stochastic quasi-Fejér monotone sequences in a broad class of metric spaces, holding both in expectation and with probability one, whenever an abstract stochastic regularity assumption is satisfied. This assumption is designed to extend and unify several common conditions from the literature on optimization and fixed-point problems.

What carries the argument

The abstract stochastic regularity assumption, which ensures controlled progress toward the solution set for quasi-Fejér monotone sequences.

If this is right

  • Stochastic proximal point methods receive new rates of convergence in geodesic metric spaces of nonpositive curvature.
  • Randomized variants of the Krasnoselskii-Mann scheme for stochastic fixed-point equations gain quantitative guarantees.
  • The Busemann subgradient method obtains explicit convergence rates from the general result.

Where Pith is reading between the lines

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

  • This approach could be applied to analyze other stochastic approximation algorithms beyond the three examples.
  • The high uniformity of the rates may allow for practical computation of convergence bounds without extensive parameter tuning.
  • Extending the framework to spaces without curvature assumptions might broaden its applicability to more general metric settings.

Load-bearing premise

The stochastic sequences must satisfy the abstract regularity condition that controls their asymptotic behavior toward the set of solutions.

What would settle it

A counterexample where a stochastic quasi-Fejér monotone sequence satisfies the regularity condition but fails to converge at the constructed rate would disprove the theorem.

read the original abstract

We prove a general quantitative theorem on the asymptotic behavior of stochastic quasi-Fej\'er monotone sequences in a broad metric context. Concretely, our result explicitly constructs a rate of convergence for such process, both in mean and almost surely, under an abstract stochastic regularity assumption, derived from previous work of Kohlenbach, L\'opez-Acedo and Nicolae [Isr. J. Math. 232(1), pp. 261-297, 2019] on such notions in a deterministic context. Our notion of regularity extends and unifies many common conditions from the literature, such as generalized contractivity for self maps, weak sharp minima and error bounds for real-valued functions, uniform monotonicity and global metric subregularity for set-valued operators, related Polyak-{\L}ojasiewicz or Kurdyka-{\L}ojasiewicz conditions, as well as expected sharp growth as e.g. studied by Asi and Duchi [SIAM J. Optim. 29(3), pp. 2257-2290, 2019]. The rate is moreover highly uniform, depending only on very few data of the surrounding objects. We also discuss special cases which allow for the construction of fast rates in the form of linear non-asymptotic guarantees. We conclude by presenting three concrete methods from stochastic approximation where our results yield new rates of convergence, including the classical example of the stochastic proximal point method, a randomized variant of the Krasnoselskii-Mann scheme for solving stochastic fixed-point equations, and a Busemann subgradient method recently introduced by Goodwin, Lewis, L\'opez-Acedo and Nicolae [Math. Program., to appear], all of which make use of our metric generality by being formulated over complete geodesic metric spaces of nonpositive curvature.

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

0 major / 3 minor

Summary. The paper proves a general quantitative theorem on the asymptotic behavior of stochastic quasi-Fejér monotone sequences in complete geodesic metric spaces of non-positive curvature. It explicitly constructs rates of convergence both in mean and almost surely under an abstract stochastic regularity condition that extends the deterministic regularity notion of Kohlenbach, López-Acedo and Nicolae. The regularity condition unifies several standard assumptions (contractivity, error bounds, Polyak-Łojasiewicz, etc.), and the result is applied to obtain new rates for the stochastic proximal point method, a randomized Krasnoselskii-Mann scheme, and a Busemann subgradient method.

Significance. If the central construction holds, the work is significant for providing explicit, highly uniform rates of convergence for stochastic algorithms solving non-unique problems in metric spaces. The unification of regularity conditions and the explicit rate extraction (both mean and a.s.) under minimal data dependence represent a clear advance over qualitative results. The applications to three concrete methods in CAT(0) spaces demonstrate practical utility and strengthen the contribution.

minor comments (3)
  1. The abstract claims the rate depends on 'very few data of the surrounding objects'; the precise list of parameters and the uniformity statement should be stated explicitly in the main theorem (likely Theorem 3.1 or 4.1) rather than left to the reader to extract from the proof.
  2. In the applications section, verify that the stochastic regularity condition is checked with explicit constants for each of the three algorithms; if the verification is only sketched, add a short table or paragraph summarizing the data dependence for each example.
  3. Notation for the stochastic regularity assumption (Definition 2.x) should be cross-referenced clearly when it is instantiated for the proximal point and subgradient methods to avoid ambiguity between the abstract and concrete settings.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our manuscript, including the recognition of its significance in providing explicit uniform rates for stochastic quasi-Fejér monotone sequences and the unification of regularity conditions. The recommendation for minor revision is noted, but the report contains no specific major comments requiring point-by-point rebuttal.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper constructs explicit rates of convergence for stochastic quasi-Fejér monotone sequences from an abstract stochastic regularity condition that extends the external deterministic result of Kohlenbach, López-Acedo and Nicolae (2019). This extension unifies standard conditions and is applied to three concrete algorithms, but the rates are derived directly from the stated assumptions without reduction to fitted parameters, self-definitions, or load-bearing self-citations. The cited prior work is independent and external; no equation or step equates a claimed output to an input by construction. The derivation remains self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The theorem rests on the definition of stochastic quasi-Fejér monotonicity and the abstract stochastic regularity assumption; both are domain assumptions extending prior deterministic concepts. No free parameters or new entities are introduced in the abstract statement.

axioms (3)
  • domain assumption Stochastic quasi-Fejér monotonicity of the sequence
    Stated as the starting point for the asymptotic analysis.
  • domain assumption Abstract stochastic regularity condition extending Kohlenbach et al.
    The load-bearing hypothesis that enables the rate construction.
  • domain assumption Complete geodesic metric space of nonpositive curvature for the concrete applications
    Required for the proximal-point, Krasnoselskii-Mann, and Busemann subgradient instantiations.

pith-pipeline@v0.9.0 · 5625 in / 1559 out tokens · 48171 ms · 2026-05-08T08:19:10.250723+00:00 · methodology

discussion (0)

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Works this paper leans on

92 extracted references · 7 canonical work pages

  1. [1]

    Aleksandrov

    A.D. Aleksandrov. A theorem on triangles in a metric space and some of its applications.Trudy Matem- aticheskogo Instituta imeni V.A. Steklova, 38:5–23, 1951

    1951

  2. [2]

    Alexander, V

    S. Alexander, V. Kapovitch, and A. Petrunin.Alexandrov Geometry: Foundations, volume 236 ofGraduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2024

    2024

  3. [3]

    Aliprantis and K.C

    C.D. Aliprantis and K.C. Border.Infinite Dimensional Analysis: A Hitchhiker’s Guide. Springer Berlin, Heidelberg, 2006

    2006

  4. [4]

    H. Asi, K. Chadha, G. Cheng, and J.C. Duchi. Minibatch stochastic approximate proximal point methods. InProceedings of the 34th International Conference on Neural Information Processing Systems (NeurIPS 2020), pages 21958–21968. Curran Associates Inc., USA, 2020

    2020

  5. [5]

    Asi and J.C

    H. Asi and J.C. Duchi. Stochastic (approximate) proximal point methods: Convergence, optimality, and adaptivity.SIAM Journal on Optimization, 29(3):2257–2290, 2019

    2019

  6. [6]

    Aubin and H

    J.-P. Aubin and H. Frankowska.Set-Valued Analysis. Springer, New York, 2009

    2009

  7. [7]

    R.J. Aumann. Measurable utility and the measurable choice problem. In G.T. Guilbaud, editor,La D´ ecision, pages 15–26. Editions du Centre National de la Recherche Scientifique, Paris, 1969

    1969

  8. [8]

    Bauschke and P.L

    H.H. Bauschke and P.L. Combettes.Convex Analysis and Monotone Operator Theory in Hilbert Spaces. CMS Books in Mathematics. Springer Cham, 2nd edition, 2017

    2017

  9. [9]

    Bauschke and A.S

    H.H. Bauschke and A.S. Lewis. Dykstras algorithm with Bregman projections: A convergence proof.Op- timization, 48(4):409–427, 2000

    2000

  10. [10]

    Baˇ c´ ak

    M. Baˇ c´ ak. The proximal point algorithm in metric spaces.Israel Journal of Mathematics, 194(2):689–701, 2013

    2013

  11. [11]

    Baˇ c´ ak

    M. Baˇ c´ ak. Computing medians and means in Hadamard spaces.SIAM Journal of Optimization, 24(3):1542– 1566, 2014

    2014

  12. [12]

    Baˇ c´ ak.Convex analysis and optimization in Hadamard spaces, volume 22 ofDe Gruyter Series in Nonlinear Analysis and Applications

    M. Baˇ c´ ak.Convex analysis and optimization in Hadamard spaces, volume 22 ofDe Gruyter Series in Nonlinear Analysis and Applications. Walter de Gruyter GmbH, Berlin/Boston, 2014. CONVERGENCE GUARANTEES FOR STOCHASTIC ALGORITHMS IN METRIC SPACES 33

    2014

  13. [13]

    Baˇ c´ ak

    M. Baˇ c´ ak. A variational approach to stochastic minimization of convex functionals.Pure and Applied Functional Analysis, 3(2):287–295, 2018

    2018

  14. [14]

    Baˇ c´ ak

    M. Baˇ c´ ak. Old and new challenges in Hadamard spaces.Japanese Journal of Mathematics, 18(2):115–168, 2023

    2023

  15. [15]

    Bertsekas

    D.P. Bertsekas. Incremental proximal methods for large scale convex optimization.Mathematical Program- ming. Series B, 129:163–195, 2011

    2011

  16. [16]

    Bertsekas

    D.P. Bertsekas. Incremental gradient, subgradient, and proximal methods for convex optimization: A sur- vey. In S. Sra, S. Nowozin, and S.J. Wright, editors,Optimization for Machine Learning, Neural Information Processing Series, pages 85–120. The MIT Press, Cambridge, Massachusetts, 2012

    2012

  17. [17]

    P. Bianchi. Ergodic convergence of a stochastic proximal point algorithm.SIAM Journal on Optimization, 26(4):2235–2260, 2016

    2016

  18. [18]

    Billera, S.P

    L.J. Billera, S.P. Holmes, and K. Vogtmann. Geometry of the space of phylogenetic trees.Advances in Applied Mathematics, 27(4):733–767, 2001

    2001

  19. [19]

    Bot ¸ and C

    R.I. Bot ¸ and C. Schindler. On a stochastic differential equation with correction term governed by a monotone and Lipschitz continuous operator.Evolution Equations and Control Theory, 14(3):463–493, 2025

    2025

  20. [20]

    Bot ¸ and C

    R.I. Bot ¸ and C. Schindler. Long-Time Analysis of Stochastic Heavy Ball Dynamics for Convex Optimization and Monotone Equations.Discrete and Continuous Dynamical Systems, 51:439–474, 2026

    2026

  21. [21]

    Bolte, A

    J. Bolte, A. Daniilidis, O. Ley, and L. Mazet. Characterizations of Lojasiewicz inequalities: subgradient flows, talweg, convexity.Transactions of the American Mathematical Society, 362(6):3319–3363, 2010

    2010

  22. [22]

    Bolte, T.P

    J. Bolte, T.P. Nguyen, J. Peypouquet, and B.W. Suter. From error bounds to the complexity of first-order descent methods for convex functions.Mathematical Programming, 165:471–507, 2017

    2017

  23. [23]

    Borwein, G

    J.M. Borwein, G. Li, and M.K. Tam. Convergence rate analysis for averaged fixed point iterations in common fixed point problems.SIAM Journal on Optimization, 27:1–33, 2017

    2017

  24. [24]

    Br´ ezis and P.L

    H. Br´ ezis and P.L. Lions. Produits infinis de resolvantes.Israel Journal of Mathematics, 29(4):329–345, 1978

    1978

  25. [25]

    Bridson and A

    M.R. Bridson and A. Haefliger.Metric Spaces of Non-Positive Curvature, volume 319 ofGrundlehren der mathematischen Wissenschaften. Springer Berlin, Heidelberg, 1999

    1999

  26. [26]

    Bruhat and J

    F. Bruhat and J. Tits. Groupes r´ eductifs sur un corps local. I. Donn´ ees radicielles valu´ ees.Publications Math´ ematiques de l’Institut des Hautes ´Etudes Scientifiques, 41:5–251, 1972

    1972

  27. [27]

    Burke and S

    J.V. Burke and S. Deng. Weak sharp minima revisited. I. Basic theory.Control and Cybernetics, 31:439–469, 2002

    2002

  28. [28]

    Burke and M.C

    J.V. Burke and M.C. Ferris. Weak sharp minima in mathematical programming.SIAM Journal on Control and Optimization, 31:1340–1359, 1993

    1993

  29. [29]

    Butnariu and A.N

    D. Butnariu and A.N. Iusem.Totally Convex Functions for Fixed Points Computation and Infinite Dimen- sional Optimization, volume 40 ofApplied Optimization. Springer Dordrecht, 2000

    2000

  30. [30]

    Butnariu and E

    D. Butnariu and E. Resmerita. Bregman distances, totally convex functions and a method for solving operator equations in Banach spaces.Abstract and Applied Analysis, 2006. 84919, 39pp

    2006

  31. [31]

    Castaing and M

    C. Castaing and M. Valadier.Convex Analysis and Measurable Multifunctions, volume 580 ofLecture Notes in Mathematics. Springer Berlin, Heidelberg, 1977

    1977

  32. [32]

    Chaipunya, F

    P. Chaipunya, F. Kohsaka, and P. Kumam. Monotone vector fields and generation of nonexpansive semi- groups in complete CAT(0) spaces.Numerical Functional Analysis and Optimization, 42(9):989–1018, 2021

    2021

  33. [33]

    Combettes

    P.L. Combettes. Quasi-Fej´ erian Analysis of Some Optimization Algorithms. In D. Butnariu, Y. Censor, and S. Reich, editors,Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, volume 8 ofStudies in Computational Mathematics, pages 115–152. North-Holland, Amsterdam, 2001

    2001

  34. [34]

    Combettes

    P.L. Combettes. Fej´ er monotonicity in convex optimization. In C.A. Floudas and P.M. Pardalos, editors, Encyclopedia of Optimization, pages 1016–1024. Springer, Boston, MA, 2009

    2009

  35. [35]

    Combettes and J.I

    P.L. Combettes and J.I. Madariaga. A geometric framework for stochastic iterations.Mathematics of Com- putation, 2026. To appear

    2026

  36. [36]

    Combettes and J.C

    P.L. Combettes and J.C. Pesquet. Stochastic quasi-Fej´ er block-coordinate fixed point iterations with ran- dom sweeping.SIAM Journal on Optimization, 25(2):1221–1248, 2015

    2015

  37. [37]

    Combettes and J.C

    P.L. Combettes and J.C. Pesquet. Stochastic quasi-Fej´ er block-coordinate fixed point iterations with ran- dom sweeping II: mean-square and linear convergence.Mathematical Programming, 174(1):433–451, 2019

    2019

  38. [38]

    Dontchev and R.T

    A.L. Dontchev and R.T. Rockafellar.Implicit functions and solution mappings. A view from variational analysis. Springer Monographs in Mathematics. Springer, Dordrecht, 2009

    2009

  39. [39]

    Drusvyatskiy and A.S

    D. Drusvyatskiy and A.S. Lewis. Error bounds, quadratic growth, and linear convergence of proximal methods.Mathematics of Operations Research, 43(3):919–948, 2018. 34 N. PISCHKE AND T. POWELL

    2018

  40. [40]

    Ermol’ev

    Y.M. Ermol’ev. On the method of generalized stochastic gradients and quasi-Fej´ er sequences.Cybernetics, 5:208–220, 1969

    1969

  41. [41]

    Ermol’ev

    Y.M. Ermol’ev. On convergence of random quasi-Fej´ er sequences.Cybernetics, 7:655–656, 1971

    1971

  42. [42]

    Ermol’ev and A.D

    Y.M. Ermol’ev and A.D. Tuniev. Random fej´ er and quasi-fej´ er sequences.Theory of Optimal Solutions – Akademiya Nauk Ukrainsko ˘i, SSR Kiev, 2:76–83, 1968. in Russian; English translation in Amer. Math. Soc. Select. Translat. Math. Statist. Probab., 13 (1973), pp. 143–148

    1968

  43. [43]

    M.C. Ferris. Finite termination of the proximal point algorithm.Mathematical Programming, Series A, 50:359–366, 1991

    1991

  44. [44]

    Freund and N

    A. Freund and N. Pischke. Effective rates for continuous-time quasi-Fej´ er monotone dynamical systems,

  45. [45]

    Preprint, available athttps://arxiv.org/abs/2603.23708

    work page arXiv

  46. [46]

    Goodwin, A.S

    A. Goodwin, A.S. Lewis, G. L´ opez-Acedo, and A. Nicolae. Stochastic and incremental subgradient methods for convex optimization on Hadamard spaces.Mathematical Programming, 2026. To appear

    2026

  47. [47]

    Grasmair

    M. Grasmair. Generalized Bregman distances and convergence rates for non-convex regularization methods. Inverse Problems, 26(11), 2010. 115014, 16pp

    2010

  48. [48]

    M. Gromov. Hyperbolic groups. In S.M. Gersten, editor,Essays in group theory, volume 8 ofMathematical Sciences Research Institute Publications, pages 75–263. Springer, New York, 1987

    1987

  49. [49]

    O. G¨ uler. On the convergence of the proximal point algorithm for convex minimization.SIAM Journal on Control and Optimization, 29:403–419, 1991

    1991

  50. [50]

    Hermer, D.R

    N. Hermer, D.R. Luke, and A. Sturm. Random function iterations for consistent stochastic feasibility. Numerical Functional Analysis and Optimization, 40(4):386–420, 2019

    2019

  51. [51]

    J. Jost. Convex functionals and generalized harmonic maps into spaces of nonpositive curvature.Commen- tarii Mathematici Helvetici, 70:659–673, 1995

    1995

  52. [52]

    Kohlenbach.Applied Proof Theory: Proof Interpretations and their Use in Mathematics

    U. Kohlenbach.Applied Proof Theory: Proof Interpretations and their Use in Mathematics. Springer Mono- graphs in Mathematics. Springer-Verlag Berlin Heidelberg, 2008

    2008

  53. [53]

    Kohlenbach

    U. Kohlenbach. Proof-theoretic Methods in Nonlinear Analysis. In B. Sirakov, P. Ney de Souza, and M. Viana, editors,Proceedings of ICM 2018, volume 2, pages 61–82. World Scientific, Singapure, 2019

    2018

  54. [54]

    Kohlenbach, G

    U. Kohlenbach, G. L´ opez-Acedo, and A. Nicolae. Moduli of regularity and rates of convergence for Fej´ er monotone sequences.Israel Journal of Mathematics, 232:261–297, 2019

    2019

  55. [55]

    Krasnoselskii

    M.A. Krasnoselskii. Two remarks on the method of successive approximations.Uspekhi Matematicheskikh Nauk, 10(1(63)):123–127, 1955

    1955

  56. [56]

    Leu¸ stean and A

    L. Leu¸ stean and A. Sipo¸ s. Effective strong convergence of the proximal point algorithm in CAT(0) spaces. Journal of Nonlinear and Variational Analysis, 2(2):219–228, 2018

    2018

  57. [57]

    Leventhal

    D. Leventhal. Metric subregularity and the proximal point method.Journal of Mathematical Analysis and Applications, 360:681–688, 2009

    2009

  58. [58]

    C. Li, G. L´ opez, and V. Mart´ ın-M´ arquez. Monotone vector fields and the proximal point algorithm on Hadamard manifolds.Journal of the London Mathematical Society, 79(3):663–683, 2009

    2009

  59. [59]

    C. Li, B.S. Mordukhovich, J.H. Wang, and J.C. Yao. Weak sharp minima on Riemannian manifolds.SIAM Journal on Optimization, 21:1523–1560, 2011

    2011

  60. [60]

    G. Li, B. Mordukhovich, and J. Zhu. Generalized metric subregularity with applications to high-order regularized Newton methods.Mathematics of Operations Research, 2026. To appear

    2026

  61. [61]

    Liu, X.J

    J. Liu, X.J. Long, X.S. Li, and N.J. Huang. Stochastic dual dynamical systems for linear equality con- strained convex optimization problems.Communications in Nonlinear Science and Numerical Simulation, 154, 2026. 109538, 15pp

    2026

  62. [62]

    Luke, J.C

    D.R. Luke, J.C. Schnebel, M. Staudigl, J. Peypouquet, and S. Qu. Asymptotic behaviour of coupled random dynamical systems with multiscale aspects, 2026. Preprint, available athttps://arxiv.org/abs/2601. 15411

    2026

  63. [63]

    W.R. Mann. Mean value methods in iteration.Proceedings of the American Mathematical Society, 4:506– 510, 1953

    1953

  64. [64]

    Martinet

    B. Martinet. R´ egularisation din´ equations variationnelles par approximations successives.Revue fran¸ caise d’informatique et de recherche op´ erationnelle, 4:154–159, 1970

    1970

  65. [65]

    Maulen-Soto, J

    R. Maulen-Soto, J. Fadili, and H. Attouch. An stochastic differential equation perspective on stochastic convex optimization.Mathematics of Operations Research, 50(4):3190–3221, 2025

    2025

  66. [66]

    Maulen-Soto, J

    R. Maulen-Soto, J. Fadili, H. Attouch, and P. Ochs. Stochastic inertial dynamics via time scaling and averaging.Stochastic Systems, 16(1):61–89, 2026

    2026

  67. [67]

    U. Mayer. Gradient flows on nonpositively curved metric spaces and harmonic maps.Communications in Analysis and Geometry, 6:199–253, 1998. CONVERGENCE GUARANTEES FOR STOCHASTIC ALGORITHMS IN METRIC SPACES 35

    1998

  68. [68]

    Moulton and A

    V. Moulton and A. Spillner. Spaces of ranked tree-child networks.Journal of Mathematical Biology, 91(3),

  69. [69]

    Nemirovski, A

    A. Nemirovski, A. Juditsky, G. Lan, and A. Shapiro. Robust stochastic approximation approach to sto- chastic programming.SIAM Journal of Optimization, 19(4):1574–1609, 2009

    2009

  70. [70]

    M. Neri, N. Pischke, and T. Powell. An abstract effective convergence theorem for stochastic processes, with applications to stochastic approximation, 2026. Preprint, available athttps://arxiv.org/abs/2504. 12922

    2026

  71. [71]

    M. Neri, N. Pischke, and T. Powell. Generalized fluctuation bounds for stochastic algorithms in the presence of compactness, 2026. Preprint, available athttps://arxiv.org/abs/2602.22741

    work page arXiv 2026

  72. [72]

    Neri and T

    M. Neri and T. Powell. A quantitative Robbins-Siegmund theorem.The Annals of Applied Probability, 36(1):636–651, 2026

    2026

  73. [73]

    E. Neumann. Computational problems in metric fixed point theory and their Weihrauch degrees.Logical Methods in Computer Science, 11(4:20), 2015. 44pp

    2015

  74. [74]

    Ohta and M

    S.I. Ohta and M. P´ alfia. Discrete-time gradient flows and law of large numbers in Alexandrov spaces. Calculus of Variations and Partial Differential Equations, 54(2):1591–1610, 2015

    2015

  75. [75]

    Pinto and N

    P. Pinto and N. Pischke. On Dykstra’s algorithm with Bregman projections, 2026. Preprint, available at https://nicholaspischke.github.io

    2026

  76. [76]

    N. Pischke. Generalized Fej´ er monotone sequences and their finitary content.Optimization, 74(14):3771– 3838, 2025

    2025

  77. [77]

    N. Pischke. Mean-square and linear convergence of a stochastic proximal point algorithm in metric spaces of nonpositive curvature, 2025. Preprint, available athttps://arxiv.org/abs/2510.10697

    work page arXiv 2025

  78. [78]

    N. Pischke. On Busemann subgradient methods for stochastic minimization in Hadamard spaces, 2026. Preprint, available athttps://arxiv.org/abs/2602.08127

    work page arXiv 2026

  79. [79]

    Pischke and U

    N. Pischke and U. Kohlenbach. Effective rates for iterations involving Bregman strongly nonexpansive operators.Set-Valued and Variational Analysis, 32(4), 2024. 33, 58pp

    2024

  80. [80]

    Pischke and T

    N. Pischke and T. Powell. Asymptotic regularity of a generalised stochastic Halpern scheme, 2024. Preprint, available athttps://arxiv.org/abs/2411.04845

    work page arXiv 2024

Showing first 80 references.