Frugal forward-backward splitting methods with deviations

Haowen Zheng; Jing Zhao; Qiao-Li Dong; Xiaolong Qin; Yongyu Fu

REVIEW 2 major objections 3 minor 21 references

Frugal forward-backward splitting methods with deviations

T0 review · 2 major / 3 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read The paper unifies deviation-based forward-backward splitting methods under one weak-convergence theorem, with a checkable norm inequality governing the deviations.

desk verdict A genuinely unifying convergence framework for deviation-based frugal forward-backward splitting, with a real gap between the theorem and the numerics. read the letter →

arxiv 2509.03865 v1 pith:5JL5VKRX submitted 2025-09-04 math.OC

Yongyu Fu , Haowen Zheng , Qiao-Li Dong , Xiaolong Qin , Jing Zhao This is my paper · ORCID

classification math.OC MSC 47H0549M2765K1090C25

keywords monotoneinclusionfrugalresolventsplittingforward-backwardalgorithmdeviationsweakconvergenceDouglas-RachfordMarkowitzportfoliooptimizationcocoerciveoperators

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's project is to stop proving convergence of deviation-based splitting algorithms one scheme at a time. It proposes a general frugal forward-backward framework—one that includes Douglas-Rachford splitting with deviations as a special case—and shows that, whenever the deviation vectors satisfy one norm inequality at each step, the whole iteration converges weakly to a zero of a finite sum of maximally monotone and cocoercive operators. The attraction is unification: instead of tailoring a proof of the deviation condition to each algorithm, a user selects matrices once and inherits a theorem. If the claimed result holds, any deviation rule obeying the inequality—learned or hand-designed—can be dropped into any instance of the framework without a fresh convergence analysis.

What carries the argument

The operator T(z) = z - gamma M*x, with x = J_DF(DN x + DM z - DCB(Qx)), encodes the fixed-point form of the algorithm; its fixed points are in bijection with the zeros of the inclusion. The norm inequality (6) is the device that makes the Lyapunov argument work: at each step the next deviations must be dominated by the current residual with a contraction factor xi_k. Causal matrix pairs C, Q^T ensure the cocoercive evaluations can be computed with one evaluation per operator, and the quadratic-dominance condition on S supplies the cancellation needed to remove cross terms. Together they produce the Fejer-type recursion (13), from which weak convergence follows.

What would settle it

Run Algorithm 3.1 on a simple two-operator monotone inclusion using the Douglas-Rachford matrices, and deliberately choose a constant nonzero deviation sequence so that (6) is violated at every step. If {z^k} still converges weakly for many starting points, the inequality is sufficient but not necessary; if divergence appears even once, the condition is load-bearing. A less destructive check: for the CNN-generated deviations used in the Markowitz experiments, evaluate both sides of (6) each iteration—any violation means Theorem 3.1 was not actually applied to those runs.

Watch

Extended reading notes

Core claim

The central claim is Theorem 3.1: under mild assumptions on the parameters and matrices, the sequence {z^k} generated by Algorithm 3.1 converges weakly to a fixed point of an operator T, and the associated {x^k} converges weakly to a vector with equal entries, each entry being a solution of the monotone inclusion 0 in sum F_i + sum B_i. The engine is the deviation inequality (6), which controls both deviation vectors u^k and v^k by the previous step's residual l_k^2. Because Algorithm 3.1 reduces to Douglas-Rachford with deviations when n=2, m=0 and the matrices are chosen specially, the framework subsumes that previously analyzed method, and the norm inequality (6) becomes exactly the inequ

Load-bearing premise

The proof stands on being able to pick deviations at each step so that norm inequality (6) holds with the contraction factor xi_k bounded below 1; without such a selection rule the Lyapunov recursion (13) fails, and the algorithm also needs the diagonal stepsizes d_i = 2/S_ii to be positive for the resolvents to exist.

Editorial extensions

If this is right

Any deviation rule satisfying (6), whether learned or hand-designed, can be inserted into any instance of Algorithm 3.1 and inherits weak convergence; no per-algorithm proof is needed.
Douglas-Rachford splitting with deviations is a special case (n=2, m=0), so the theorem recovers and extends the known convergence result for that method in one stroke.
The residuals l_k^2 are summable and the deviations u_k and v_k converge to zero, so the method asymptotically behaves like the underlying no-deviation splitting.
The step-size condition uses each cocoercivity constant L_i separately, giving a larger allowed step size than frameworks tied to min_i L_i.
In the reported Markowitz portfolio experiments, the deviation-enhanced instances reach the 10^-8 tolerance in fewer iterations than their plain counterparts, with the deviation-augmented adapted graph forward-backward method the fastest.

Reading between the lines

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

Because (6) is checkable online, one could build a 'safe' learned optimizer by projecting each proposed deviation pair onto the set defined by (6); the paper does not propose this projection step.
The proof only uses monotonicity, cocoercivity, and a Young/parallelogram identity, so analogous unified results likely hold for other splitting families that satisfy similar matrix identities; the paper does not show this.
The numerical comparison could be sharpened by zeroing u and v in the deviation-augmented variants: if the iteration-count improvement vanishes, the gains are attributable to the deviations themselves, and if it persists, the learned rule is effectively re-tuned through matrix behavior alone. The paper does not report this ablation.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Read this if you work on operator splitting or learning-to-optimize. The paper's contribution is a single weak convergence theorem for a whole family of frugal forward-backward methods with deviations, generalizing Åkerman et al.'s matrix framework. The central proof is sound: I checked the Lyapunov argument, the Young estimate, the absorption step, and the equivalence with Douglas–Rachford with deviations. This genuinely unifies what was previously done algorithm-by-algorithm. The deviation condition (6) is a contraction condition, not a fit to the solution, so the convergence proof is not circular—that's a real plus.

The soft spots are real but not fatal. Condition (6) is load-bearing: each new deviation pair must satisfy it, and the paper gives no constructive rule for choosing nonzero deviations that work. The CNN in Section 4 is never checked against (6) for the actual runs. So the reported speedups are evidence about an unanalyzed heuristic, not about the convergent framework. The authors should either verify (6) or clearly label the experiments as heuristic.

Some smaller issues: Lemma 3.1 is delegated to a preprint [12] and a published paper [20], so the fixed-point/zero equivalence is not self-contained; the proof of Theorem 3.1(iv)-(v), especially the Phi operator and inclusion (20), is compressed and took real effort to verify; positivity of the stepsizes d_i is not stated explicitly; and the experiments report iteration counts rather than wall-clock time, which hides CNN overhead. These are reporting gaps, not mathematical errors.

Who is this for? Specialists in monotone operator theory and people building learned optimizers with convergence certificates. It gives a clean template for proving convergence of deviation-based methods, and the framework is likely to be reused.

Verdict: this deserves a serious referee. The main theorem is important and appears correct; the numerical claims need to be brought in line with the theory. I'd recommend acceptance after the authors verify (6) in the experiments, expand the proof of (iv)-(v), and cite [12] as a preprint rather than a settled source.

Referee Report

2 major / 3 minor

Summary. The paper proposes a unified framework (Algorithm 3.1) for frugal forward-backward splitting with deviation vectors, aimed at finding zeros of a sum of finitely many maximally monotone operators and cocoercive operators. Deviations u^k, v^k are admissible provided they satisfy the norm inequality (6). Under Assumptions 3.1 and 3.2, Theorem 3.1 establishes weak convergence of the sequence {z^k} to a fixed point of the operator T and of {x^k} to a common solution of the monotone inclusion. The framework is shown in Section 3.1 to specialize to the Douglas-Rachford splitting method with deviations, recovering the earlier deviation inequality (3) as (9). Section 4 reports numerical experiments on a Markowitz portfolio problem, comparing SFB+ and aGFB with CNN-selected deviations, and reports iteration-count improvements.

Significance. If the claims hold, the paper offers a genuine unifying convergence analysis for deviation-based splitting methods, eliminating the algorithm-by-algorithm treatment found in the recent literature. The central derivation in Lemma 3.2 is explicit and checkable: the monotonicity of the resolvent product, Young's inequality (17), and Assumption 3.2(c) close inequality (12), leading to the Fejer-type estimate (13). The equivalence with DR-with-deviations is demonstrated exactly, and the deviation condition (6) is a contraction-type energy constraint rather than an a posteriori fit to the solution, so the argument is not circular. The main limitation is that the numerical section does not verify that the CNN-generated deviations satisfy (6), so the experimental claims are not yet tied to the convergence theorem.

major comments (2)

[Section 4.2 / Algorithm 3.1, condition (6)] The paper uses a three-layer CNN to select deviation vectors but never reports whether the generated u^{k+1}, v^{k+1} satisfy the key inequality (6). Theorem 3.1 applies only to sequences satisfying (6) at every iteration via Lemma 3.2, eq. (13); a single violation invalidates the convergence guarantee for the algorithm actually run. Table 1 and Figure 1 therefore currently demonstrate an unverified heuristic, not the proposed convergent framework. Please add a direct check (e.g., the maximum over k of the ratio of the left-hand side of (6) to ξ_k l_k^2, with the chosen θ and ξ_k) or clearly state that the numerical results are heuristic.
[Section 3.1, Algorithm 3.1, step 6] The only explicitly identified admissible deviations are u^k = v^k = 0. Since the stated contribution is the additional degrees of freedom provided by nonzero deviations, the paper should provide at least one constructive family of nonzero choices satisfying (6), or a selection rule that enforces (6). The DR-with-deviations reduction in Section 3.1 gives one example in the n=2,m=0 case, but no analogous constructive rule is offered for the general framework. As written, the theorem is conditional on an abstract condition, and the practical applicability of the method is not demonstrated.

minor comments (3)

[Section 3.1, Lemma 3.1] The fixed-point characterization of Fix T is load-bearing for Theorem 3.1(iv)-(v), but its proof is omitted and delegated to [12, Lemma 3.5] and [20, Lemma 3.1]. Including a proof or a precise statement of the imported result would make the framework more self-contained.
[Section 4.2] The termination criterion uses 'the exact solution x*', but the paper does not state how x* is computed. For reproducibility, please specify the oracle used to obtain x*.
[Figure 1] The legend labels 'de_aGFB' and 'de_SFB+' are inconsistent with the text's 'dev_aGFB' and 'dev_SFB+'. Please unify the notation.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: convergence proof is a direct Lyapunov argument; deviation condition (6) is a stated constraint, not a fitted output.

full rationale

The derivation chain is self-contained. Theorem 3.1 rests on Lemma 3.2, whose key inequality (12)/(13) is obtained directly from monotonicity of the resolvents, Young's inequality, the cocoercivity bound (5), and Assumption 3.2(c); it never invokes the target convergence result. Condition (6) is an explicit contractive constraint on the deviation sequences, not a quantity fitted to the solution, and the paper notes the feasible choice u=v=0. The claimed inclusion of Douglas–Rachford with deviations is verified by explicit algebra in equations (8)–(9), reducing to the scheme (2)–(3); this is a special-case calculation, not an imported convergence theorem. The self-citations [16]–[18] (sharing author Dong) are references to existing deviation methods and are not load-bearing for any proof step. Two flagged items are support gaps rather than circularity: Lemma 3.1's proof is omitted and delegated to [12,20], and Section 4 does not report checking whether the CNN-selected deviations satisfy condition (6). Neither gap makes the proof reduce to its own assumptions; the theoretical claims stand independently of the numerical verification.

Assumptions & free parameters 4 free parameters · 7 assumptions · 0 invented entities

The theory is nearly self-contained: given standard monotone operator results ([19]) and satisfiable matrix conditions, the convergence proof closes without fitting. The main external dependencies are Lemma 3.1 (proof delegated to [12,20]) and the numerical pipeline (CNN from [15], matrices from [11]). theta, xi_k, gamma_k and the CNN weights are the selection/tuning parameters; none are fitted in the derivation, but the experimental speed-ups depend on them.

free parameters (4)

theta = not reported ('tuned manually to a nearly optimal value')
Free constant in Assumption 3.1(i), in the matrix inequality Assumption 3.2(c), and in the deviation inequality (6). In the theory it can be any positive number; in the numerics it is tuned per problem, and the reported speed-ups depend on this choice.
xi_k = not reported
User-selected shrinkage factors in [0, 1-epsilon] controlling how much deviation is allowed at each step through inequality (6). Values used in the experiments are not stated.
gamma_k step sizes = gamma_k = 0.9 (numerics)
Step size sequence in [epsilon, 1-epsilon]; fixed at 0.9 in the experiments. Whether 0.9 satisfies Assumption 3.2(c) for the chosen matrices and theta is not verified in the paper.
CNN weights for deviation selection = learned, not given
Numerical deviations are produced by a three-layer CNN trained following [15]; the weights and training details are external to the paper, so the reported iteration counts are not reproducible from the manuscript.

assumptions (7)

domain assumption Solution set of (1) is nonempty: zer(sum F_i + sum B_i) != empty
Stated at the start of Section 3. Standard nonemptiness/consistency assumption for convergence claims.
standard math Resolvents J_{d_i F_i} of maximally monotone operators are single-valued and cocoercive [19, Corollary 23.11]
Used throughout Algorithm 3.1 and in the proof of Theorem 3.1(iv)-(v).
domain assumption Assumption 3.2(c): S - MM^T - (1/2)(1+1/theta)(C^T-Q)^T L^{-1}(C^T-Q) >= 0
The matrix inequality that makes Lemma 3.2 close. Always satisfiable by construction (Remark 3.1(1)) but it restricts the admissible algorithms and couples the stepsizes to the cocoercivity constants L_i and to theta.
standard math The operator Phi in (21) is maximally monotone and its graph is closed in the weak-strong topology [19, Proposition 20.32, Corollary 24.4(i)]
Background results used in the cluster-point argument of Theorem 3.1(v), where the linear part has full domain.
ad hoc to paper Lemma 3.1: Fix T corresponds to zer(sum F_i + sum B_i); proof omitted, delegated to [12, Lemma 3.5] and [20, Lemma 3.1]
Load-bearing for the interpretation of the weak limit as a solution of the original problem; not proven in the paper.
domain assumption Resolvent parameters d_i = 2/(S_ii) are positive, so J_{d_i F_i} is well-defined for arbitrary maximally monotone F_i
Implicit: Algorithm 3.1 writes J_{d_i F_i} without stating d_i > 0. For general monotone F_i, resolvents with nonpositive parameters are not standard.
ad hoc to paper The deviation control inequality (6) holds at every iteration with the stated gamma, theta, xi and L^{-1} weights
The convergence theorem is conditional on the deviations satisfying (6) at each step; existence is shown by u=v=0, but the numerical learning rule's compliance is asserted, not demonstrated.

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Pith. "Pith review of Frugal forward-backward splitting methods with deviations." pith.science (2026). https://pith.science/paper/5JL5VKRX

@misc{pith2026250903865,
  author       = {Pith},
  title        = {Pith review of: Frugal forward-backward splitting methods with deviations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5JL5VKRX}},
  note         = {Machine review of arXiv:2509.03865}
}

read the original abstract

The deviation vectors provide additional degrees of freedom and effectively enhance the flexibility of algorithms. In the literature, the iterative schemes with deviations are constructed and their convergence analyses are performed on an inefficient, algorithm-by-algorithm basis. In this paper, we address these by providing a general framework of frugal forward-backward splitting methods with deviations for finding zeros in the sum of a finite number of maximally monotone operators and cocoercive operators. Our framework encompasses the Douglas--Rachford splitting method with deviations. A unified weak convergence analysis is made under mild conditions. Numerical experiments on Markowitz portfolio optimization problem are given to demonstrate the effectiveness of deviations.

Figures

Figures reproduced from arXiv: 2509.03865 by the authors.

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Reference graph

Works this paper leans on

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [12]

A general approach to distribu ted oper- ator splitting

Dao, M.N., Tam, M.K., Truong, T.D. A general approach to distribu ted oper- ator splitting. https://arxiv.org/abs/2504.14987

arXiv

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Frugal and decentraised resolvent splitting deﬁned b y nonexpansive operators

Tam, M.K. Frugal and decentraised resolvent splitting deﬁned b y nonexpansive operators. Optimization Letters. 18(7), 1-19 (2023)

work page 2023

[3] [1]

Arag´ on-Artacho, F.J., Malitsky, Y., Tam, M.K. et al. Distributed f orward- backward methods for ring networks. Computational Optimization and Appli- cations. 86, 845-870 (2023)

work page 2023

[4] [2]

The forward-backward algorithm and the normal pr oblem

Moursi, W.M. The forward-backward algorithm and the normal pr oblem. Jour- nal of Optimization Theory and Applications. 176, 605-624 (2018)

work page 2018

[5] [3]

Convergence rates in forward-bac kward splitting

Chen, G.H., Rockafellar, R.T. Convergence rates in forward-bac kward splitting. SIAM Journal on Optimization. 7(2), 421-444 (1997)

work page 1997

[6] [4]

Uniqueness of DRS as the 2 operator resolvent-splittin g and im- possibility of 3 operator resolvent-splitting

Ryu, E.K. Uniqueness of DRS as the 2 operator resolvent-splittin g and im- possibility of 3 operator resolvent-splitting. Mathematical Progra mming. 182, 233-273 (2020). 15

work page 2020

[7] [5]

Forward-backward algorithms devised by graphs

Arag´ on-Artacho, F.J., Campoy, R., L´ opez-Pastor, C. Forward-backward algo- rithms devised by graphs. https://arxiv.org/abs/2406.03309

work page Pith review arXiv

[8] [6]

Frugal splitting operators: r epresentation, minimal lifting, and convergence

Morin, M., Banert, S., Giselsson, P. Frugal splitting operators: r epresentation, minimal lifting, and convergence. SIAM Journal on Optimization. 34( 2), 1187- 2168 (2024)

work page 2024