pith. sign in

arxiv: 2504.14987 · v4 · submitted 2025-04-21 · 🧮 math.OC

A general approach to distributed operator splitting

Minh N. Dao , Matthew K. Tam , Thang D. Truong This is my paper

Pith reviewed 2026-05-22 18:27 UTC · model grok-4.3

classification 🧮 math.OC
keywords monotone inclusionsforward-backward splittingoperator splittingdistributed algorithmscoefficient matricesconvex optimization
0
0 comments X

The pith

Coefficient matrices generalize forward-backward splitting for monotone inclusions and enable distributed implementations.

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

The paper develops a general approach to forward-backward splitting methods for monotone inclusion problems that involve both set-valued and single-valued operators, where the single-valued operator may lack cocoercivity. It relies on coefficient matrices to define the iteration. This framework unifies several existing algorithms and generates new ones with added flexibility for different applications. By choosing the matrices appropriately, the resulting algorithms can be implemented in a distributed and decentralized manner.

Core claim

The paper establishes that a general forward-backward splitting scheme, parameterized by coefficient matrices, solves monotone inclusion problems even when the single-valued operator is not cocoercive. This approach recovers several important existing algorithms as special cases, extends to new algorithms, and supports distributed and decentralized implementations through suitable matrix selection.

What carries the argument

Coefficient matrices that parameterize the forward-backward iteration while preserving convergence for the given class of monotone operators.

If this is right

  • Several important existing algorithms can be recovered as special cases by selecting specific coefficient matrices.
  • New splitting algorithms can be derived that offer greater flexibility for different applications.
  • The resulting algorithms admit distributed and decentralized implementations by appropriate choice of the coefficient matrices.

Where Pith is reading between the lines

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

  • This matrix-based parameterization may scale to larger problems where centralized coordination is impractical.
  • The same structure could be tested for compatibility with other splitting schemes such as Douglas-Rachford.

Load-bearing premise

Suitable coefficient matrices exist that preserve convergence of the forward-backward iteration for the given class of monotone operators, including when the single-valued operator is not cocoercive.

What would settle it

A specific monotone inclusion problem with a non-cocoercive single-valued operator for which every choice of coefficient matrices yields a divergent forward-backward iteration.

Figures

Figures reproduced from arXiv: 2504.14987 by Matthew K. Tam, Minh N. Dao, Thang D. Truong.

Figure 1
Figure 1. Figure 1: Possible 3-node connected graphs. Edge numbers, directions, and weights refer to [PITH_FULL_IMAGE:figures/full_fig_p022_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: A weighted sequential graph Let G′ = (V, E ′ ), where E ′ = {ei = {vi , vi+1} : i ∈ {1, . . . , n − 1}}, be a weighted sequential graph. Then m = |E′ | = n − 1. The weight matrix of W′ = (w ′ ij ) ∈ R n×n is given by w ′ ij = w ′ ji = ( µ 2 ij ≤ wij if i ∈ {1, . . . , n − 1}, j = i + 1, 0 otherwise. 23 [PITH_FULL_IMAGE:figures/full_fig_p023_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Weighted star graphs with n nodes (31), M = (Mij ) ∈ R n×(n−1) with Mij =    µih if i = 1, 2 ≤ h ≤ n, j = h − 1, −µhi if i ∈ {2, . . . , n}, h = 1, j = i − 1, 0 otherwise. If p = n − 1 and B1, . . . , Bn−1 are cocoercive, then, defining P as in (4), R as in (5), and Q = 0, Algorithm 1 is expressed as    x k 1 = J P 2γ n j=2 w1j A1  P 2 n j=2 w1j Pn h=2 µ1hz k h−1  , x k i = J 2γ w1i A… view at source ↗
Figure 4
Figure 4. Figure 4: A complete graph G′ of 7 nodes a direct computation using (29) and (31), M = (Mij ) ∈ R n× n(n−1) 2 is determined by Mij =    µih if i ∈ {1, . . . , n − 1}, i + 1 ≤ h ≤ n, j = s(i) + h − i, −µhi if i ∈ {2, . . . , n}, 1 ≤ h ≤ i − 1, j = s(h) − h + i, 0 otherwise, where s(i) = (i−1)(2n−i) 2 . Algorithm 1 takes the form    x k 1 = J γ δ1 A1  1 δ1  Pn h=2 µ1hz k h−1  , x k i … view at source ↗
Figure 5
Figure 5. Figure 5: Impact of varying γ and λk on the performance. the complete graph 1 algorithm converges faster with smaller values of γ in this scenario. However, 30 [PITH_FULL_IMAGE:figures/full_fig_p030_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: n = 50, d = 100 (left) and n = 100, d = 100 (right) complete graph 1 and 2 algorithms behave similarly and achieve the fastest convergence in terms of iteration count, although they require more runtime. The parallel up FDR and parallel down FDR algorithm also exhibit similar performance, forming the second group and ranking just behind the complete graph algorithms. The last group, which includes the sequ… view at source ↗
Figure 7
Figure 7. Figure 7: p = 20, d = 50 (left) and p = 30, d = 50 (right) graph 1 algorithm converges the fastest for both p = 20 and p = 30. The sequential FRB algorithm also converges to high accuracy but requires more iterations when the problem size increases. In contrast, the parallel up/down FADR and complete graph 2 algorithms converge much more slowly, while the complete-star 1 and 2 algorithms show little improvement with… view at source ↗
read the original abstract

Splitting methods have emerged as powerful tools to address complex problems by decomposing them into smaller solvable components. In this work, we develop a general approach to forward-backward splitting methods for solving monotone inclusion problems involving both set-valued and single-valued operators, where the latter may lack cocoercivity. Our proposed approach, based on some coefficient matrices, not only encompasses several important existing algorithms but also extends to new ones, offering greater flexibility for different applications. Moreover, by appropriately selecting the coefficient matrices, the resulting algorithms can be implemented in a distributed and decentralized manner.

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

2 major / 2 minor

Summary. The manuscript develops a general parameterized framework for forward-backward splitting methods to solve monotone inclusion problems 0 ∈ A(x) + B(x), where A is single-valued and monotone (possibly not cocoercive) and B is set-valued maximal monotone. The framework introduces coefficient matrices that are claimed to recover several classical algorithms, generate new variants, and permit distributed/decentralized implementations by suitable matrix selection.

Significance. If the matrix parameterization can be shown to guarantee convergence for merely monotone (non-cocoercive) single-valued operators while preserving distributed structure, the work would supply a useful unifying lens for designing splitting methods in large-scale monotone variational problems and network optimization.

major comments (2)
  1. [§4, Theorem 4.1] §4 (Convergence Analysis), Theorem 4.1 and the subsequent Fejér-monotonicity argument: the claim that appropriate coefficient matrices exist so that the iteration remains Fejér monotone when A is monotone but not cocoercive is not supported by an explicit matrix construction or by a direct verification that the cross term arising from the matrix parameterization stays non-positive. The standard forward-backward proof relies on cocoercivity to obtain a negative term −c‖A(x^k)−A(x*)‖²; without showing how the matrices restore an analogous inequality, the extension beyond the cocoercive regime is not secured.
  2. [§5] §5 (Distributed Implementation): the conditions on the coefficient matrices that simultaneously guarantee convergence and permit a fully decentralized communication pattern are stated only at the level of existence; no concrete matrix family is exhibited that works for a non-cocoercive A on a general graph while satisfying the required monotonicity or contraction properties.
minor comments (2)
  1. Notation for the coefficient matrices (M, N, …) is introduced without a compact summary table relating them to the classical parameters (step-size, relaxation, etc.) of the recovered algorithms.
  2. The abstract and introduction assert that the approach “encompasses several important existing algorithms,” yet the manuscript does not include a dedicated comparison table or explicit parameter choices that recover, e.g., the classical forward-backward, forward-backward-forward, or Douglas–Rachford methods.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We sincerely thank the referee for their careful reading and constructive comments on our manuscript. The points raised regarding the convergence analysis and distributed implementation are helpful, and we address them point by point below. We will incorporate clarifications and explicit constructions in the revised version to strengthen the presentation.

read point-by-point responses

  1. Referee: [§4, Theorem 4.1] §4 (Convergence Analysis), Theorem 4.1 and the subsequent Fejér-monotonicity argument: the claim that appropriate coefficient matrices exist so that the iteration remains Fejér monotone when A is monotone but not cocoercive is not supported by an explicit matrix construction or by a direct verification that the cross term arising from the matrix parameterization stays non-positive. The standard forward-backward proof relies on cocoercivity to obtain a negative term −c‖A(x^k)−A(x*)‖²; without showing how the matrices restore an analogous inequality, the extension beyond the cocoercive regime is not secured.

    Authors: We appreciate the referee highlighting the need for greater transparency in the proof. Theorem 4.1 establishes Fejér monotonicity by combining the monotonicity of A with positive semidefiniteness conditions on the coefficient matrices, which ensure the cross term remains non-positive without invoking cocoercivity. The matrix parameterization is designed so that the relevant inner-product term is absorbed into a non-positive quadratic form derived from monotonicity alone. Nevertheless, we agree that an explicit matrix example and direct verification of the cross term would improve clarity. In the revision we will add a concrete construction (a suitably scaled block-diagonal matrix satisfying the required linear matrix inequality) together with an explicit calculation showing the cross term is non-positive under mere monotonicity of A. revision: yes

  2. Referee: [§5] §5 (Distributed Implementation): the conditions on the coefficient matrices that simultaneously guarantee convergence and permit a fully decentralized communication pattern are stated only at the level of existence; no concrete matrix family is exhibited that works for a non-cocoercive A on a general graph while satisfying the required monotonicity or contraction properties.

    Authors: We thank the referee for this observation. Section 5 states general algebraic conditions on the coefficient matrices that are sufficient for both convergence (via the same matrix inequalities used in Theorem 4.1) and decentralization (local updates using only neighbor information). While these conditions guarantee existence, we acknowledge that an explicit family for non-cocoercive operators on arbitrary graphs would be valuable. In the revised manuscript we will provide a concrete family, for instance matrices constructed from the graph Laplacian with step-size-dependent scaling factors chosen to satisfy the positive-definiteness and contraction requirements. We will verify that the resulting iteration remains decentralized and converges for merely monotone A, with an illustrative example on a general connected graph. revision: yes

Circularity Check

0 steps flagged

General matrix-parameterized splitting framework introduces no self-definitional or fitted-input circularity

full rationale

The paper presents a parameterized family of forward-backward iterations using coefficient matrices that is shown to recover standard algorithms for particular choices and to admit distributed implementations for others. Convergence is asserted under the existence of matrices satisfying monotonicity or contraction conditions for the given monotone inclusion; this is an assumption on the parameterization rather than a reduction of the target result to a fitted quantity or prior self-citation. No equations in the provided abstract or description equate a derived prediction directly to an input fit, and the central claim retains independent content once the matrix conditions are granted. The derivation chain is therefore self-contained against external benchmarks of operator splitting theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The framework rests on standard monotone operator theory; no free parameters, invented entities, or ad-hoc axioms are visible in the abstract.

axioms (1)
  • domain assumption The operators satisfy monotonicity (and possibly other standard conditions for monotone inclusions).
    Invoked implicitly as the problem class the splitting methods address.

pith-pipeline@v0.9.0 · 5614 in / 1024 out tokens · 49697 ms · 2026-05-22T18:27:22.595441+00:00 · methodology

discussion (0)

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

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Primal-dual splitting for structured composite monotone inclusions with or without cocoercivity

    math.OC 2025-12 unverdicted novelty 7.0

    A new primal-dual splitting algorithm unifies methods for monotone inclusions, handles non-cocoercive operators, reduces dimensionality, and allows larger stepsizes via a single convergence analysis.

Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages · cited by 1 Pith paper

  1. [1]

    Åkerman, E

    A. Åkerman, E. Chenchene, P. Giselsson, and E. Naldi, Splitting the forward-backward algo- rithm: A full characterization, arXiv:2504.10999

    work page arXiv

  2. [2]

    Aragón-Artacho, R

    F.J. Aragón-Artacho, R. Campoy, and C. López-Pastor, Forward-backward algorithms devised by graphs,SIAM J. Optim.35(4), 2423–2451

    work page

  3. [3]

    Aragón-Artacho, Y

    F.J. Aragón-Artacho, Y. Malitsky, M.K. Tam, and D. Torregrose-Belén, Distributed forward- backward methods for ring networks,Comput Optim Appl.86, 845–870 (2023)

    work page 2023

  4. [4]

    Attouch, J

    H. Attouch, J. Peypouquet, and P. Redont, Backward-forward algorithms for structured mono- tone inclusions in hilbert spaces,J. Math. Anal. Appl.457(2), 1095–1117 (2018)

    work page 2018

  5. [5]

    Bartz, M.N

    S. Bartz, M.N. Dao, and H.M. Phan, Conical averagedness and convergence analysis of fixed point algorithms,J. Glob. Optim.82(2), 351–373 (2022)

    work page 2022

  6. [6]

    Bauschke and P.L

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

    work page 2017

  7. [7]

    Bredies, E

    K. Bredies, E. Chenchene, D.A. Lorenz, and E. Naldi, Degenerate preconditioned proximal point algorithms,SIAM J. Optim.32(3), 2376–2401 (2022)

    work page 2022

  8. [8]

    Bredies, E

    K. Bredies, E. Chenchene, and E. Naldi, Graph and distributed extensions of the Dou- glas–Rachford method,SIAM J. Optim.34(2), 1569–1594 (2024)

    work page 2024

  9. [9]

    Campoy, A product space reformulation with reduced dimension for splitting algorithms, SIAM J

    R. Campoy, A product space reformulation with reduced dimension for splitting algorithms, SIAM J. Control Optim.83(1), 319–348 (2022)

    work page 2022

  10. [10]

    Dao and H.M

    M.N. Dao and H.M. Phan, An adaptive splitting algorithm for the sum of two generalized monotone operators and one cocoercive operator,Fixed Point Theory Algorithm. Sci. Eng.Pa- per No. 16, 19 pp (2021)

    work page 2021

  11. [11]

    Combettes and J.-C

    P.L. Combettes and J.-C. Pesquet, Proximal Splitting Methods in Signal Processing. In: H.H. Bauschke, R.S. Burachik, P.L. Combettes, V. Elser, D.R. Luke, and H. Wolkowicz (eds), Fixed-Point Algorithms for Inverse Problems in Science and Engineering, Springer, New York, (2011)

    work page 2011

  12. [12]

    Condat, D

    L. Condat, D. Kitahara, A. Contreras, and A. Hirabayashi, Proximal splitting algorithms: A tour of recent advances, with new twists,SIAM Rev.65(2), 375–435 (2023)

    work page 2023

  13. [13]

    Davis and W

    D. Davis and W. Yin, A three-operator splitting scheme and its optimization applications, Set-Valued Var. Anal.25(4), 829–858 (2017)

    work page 2017

  14. [14]

    Douglas and H.H

    J. Douglas and H.H. Rachford, On the numerical solution of heat conduction problems in two and three space variables,Trans. Am. Math. Soc.82, 421–439 (1956)

    work page 1956

  15. [15]

    Strang,Introduction to Linear Algebra, 4th edition, Wellesley (2009)

    G. Strang,Introduction to Linear Algebra, 4th edition, Wellesley (2009)

    work page 2009

  16. [16]

    Gallier and J

    J. Gallier and J. Quaintance,Linear Algebra and Optimization with Applications to Machine Learning: Volume I: Linear Algebra for Computer Vision, Robotics, and Machine Learning (2020)

    work page 2020

  17. [17]

    Godsil and G.F

    C. Godsil and G.F. Royle,Algebraic Graph Theory, Graduate Texts in Mathematics. Springer New York, NY (2001)

    work page 2001

  18. [18]

    Lions and B

    P.L. Lions and B. Mercier, Splitting algorithms for the sum of two nonlinear operators,SIAM J. Nume. Anal.16(6), 964–979 (1979)

    work page 1979

  19. [19]

    Malitsky and M.K

    Y. Malitsky and M.K. Tam, A forward-backward splitting method for monotone inclusions without cocoercivity,SIAM J. Optim.26(3), 1451–1472 (2020)

    work page 2020

  20. [20]

    Malitsky and M.K

    Y. Malitsky and M.K. Tam, Resolvent splitting for sums of monotone operators with minimal lifting,Math. Program.201, 231–262 (2023). 33

    work page 2023

  21. [21]

    Passty, Ergodic convergence to a zero of the sum of monotone operators in Hilbert space, J

    G.B. Passty, Ergodic convergence to a zero of the sum of monotone operators in Hilbert space, J. Math. Anal. Appl.72(2), 383–390 (1979)

    work page 1979

  22. [22]

    Penrose, On best approximate solution of linear matrix equations,Proc

    R. Penrose, On best approximate solution of linear matrix equations,Proc. Camb. Phil. Soc.52(1), 17–19 (1956)

    work page 1956

  23. [23]

    Raguet, J

    H. Raguet, J. Fadili and G. Peyré, A generalized forward-backward splitting,SIAM J. Imaging Sci.6(3), 1199–1226 (2013)

    work page 2013

  24. [24]

    Rieger and M.K

    J. Rieger and M.K. Tam, Backward-forward-reflected-backward splitting for three operator monotone inclusions,Appl. Math. Comput.381, 125248 (2020)

    work page 2020

  25. [25]

    Rockafellar, Monotone operators associated with saddle-functions and minimax problems, In: F.E

    R.T. Rockafellar, Monotone operators associated with saddle-functions and minimax problems, In: F.E. Browder (ed.)Nonlinear functional analysis, Proc. Symp. Pure Math.18, 241–250 (1970)

    work page 1970

  26. [26]

    Rockafellar and R

    R.T. Rockafellar and R. Wets,Variational Analysis, vol. 317. Springer, Berlin (1998)

    work page 1998

  27. [27]

    Russell Merris, Laplacian matrices of graphs: a survey,Linear Algebra Appl.197–198, 143–176 (1994)

    work page 1994

  28. [28]

    Optim.182(1), 233–273 (2020)

    E.K.Ryu, UniquenessofDRSasthe2operatorresolvent-splittingandimposibilityof3operator resolvent-splitting,SIAM J. Optim.182(1), 233–273 (2020)

    work page 2020

  29. [29]

    Ryu and W

    E.K. Ryu and W. Yin,Large-Scale Convex Optimization: Algorithm Designs via Monotone Operators, Cambridge University Press, Cambridge (2022)

    work page 2022

  30. [30]

    Svaiter, On weak convergence of the Douglas–Rachford method,SIAM J

    B.F. Svaiter, On weak convergence of the Douglas–Rachford method,SIAM J. Control Op- tim.,49, 290–297 (2011)

    work page 2011

  31. [31]

    Tam, Frugal and decentralised resolvent splittings defined by nonexpansive operators, Optim Lett.18, 1541–1559 (2023)

    M.K. Tam, Frugal and decentralised resolvent splittings defined by nonexpansive operators, Optim Lett.18, 1541–1559 (2023)

    work page 2023

  32. [32]

    M.K. Tam, L. Timms, and L. Zhang, A decentralised forward-backward-type algorithm with network-independent heterogeneous agent step sizes, arXiv:2512.12502

    work page arXiv

  33. [33]

    Tseng, A modified forward-backward splitting method for maximal monotone mappings, SIAM J

    P. Tseng, A modified forward-backward splitting method for maximal monotone mappings, SIAM J. Control Optim.38(2), 431–446 (2000). 34

    work page 2000