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REVIEW 2 major objections 5 minor 27 references

On the Relationship Between CoCoA and ADMM for Distributed Empirical Risk Minimization

T0 review · 2 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read For ridge-regularized ERM, CoCoA and a tuned ADMM are the same primal-dual iteration, so ADMM can match CoCoA's performance.

desk verdict The algebraic unification is real and useful, but the main bridge (Proposition 1) has an incomplete proof, and the performance claims outrun the theorems. read the letter →

arxiv 2502.00470 v4 pith:333CSJRH submitted 2025-02-01 math.OC cs.LGstat.ML

Runxiong Wu , Andi Wang This is my paper · ORCID
classification math.OCcs.LGstat.ML MSC 90C2590C0690C4668W15
keywords distributedoptimizationempiricalriskminimizationdualcoordinateascentalternatingdirectionmethodofmultipliersridgeregularizationprimal-dualalgorithmsconvergencerate
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

This paper tries to establish that two seemingly separate families of distributed optimization algorithms—CoCoA, based on dual coordinate ascent, and ADMM, based on consensus and proximal splitting—are actually the same primal-dual iteration when the regularizer is the ridge penalty. If the claim holds, the practical gap between the two families is one of parameter tuning rather than algorithmic structure: for L2-regularized ERM, an ADMM variant with the augmented Lagrangian parameter set to the inverse regularizer strength reproduces CoCoA's dual iterates exactly, and a consensus ADMM with matched scaling is equivalent to a proximal ADMM on the negated saddle objective. Because ADMM variants can be tuned freely, the paper concludes they can at least match, and in experiments outperform, CoCoA on ridge-regularized problems. The unification also yields a common duality-gap stopping criterion and a unified O(1/T) ergodic convergence proof for the ADMM-type methods.

What carries the argument

The load-bearing object is the unified primal-dual update form, in which a global primal variable w is updated by a proximal step and each local dual block v_[k] is updated by a proximal subproblem whose metric is the local data Gram matrix X_[k]^T X_[k]. The argument uses two parameter identifications: the curvature $\lambda$ of the ridge penalty becomes the inverse augmented Lagrangian parameter via rho = $lambda^{{-1}}$, and the number of machines K scales the prox term via eta1 = K, using the inequality K diag(X_[k]^T X_[k]) >= X^T X. The Moreau decomposition converts proximal steps between primal and dual, and the minimax equality min_w max_v L(w,v) = max_v min_w L(w,v) lets consensus ADMM on the primal be reinterpreted as proximal ADMM on the negated saddle objective.

What would settle it

Take a single-machine ridge regression (K=1) with squared loss, run Consensus-PD and Proximal-1-PD with beta = 1/rho and eta1 = 1 from the same start, and compare the dual sequences; any mismatch at the first full step would contradict Corollary 2. A sharper test: for the same simple problem, check whether the local dual variable tilde-v produced by the consensus subproblem equals the global dual v used in the unified update; if the two deviate while the algorithm still converges, Proposition 1's identification is unsupported.

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Extended reading notes

Core claim

The paper's central claim is that for L2-regularized ERM, CoCoA's update rule (5) and the proximal ADMM update rule (3) produce identical dual iterates when the augmented Lagrangian parameter satisfies rho = $lambda^{{-1}}$ and the algorithms start from the same initialization (Corollary 1). It further claims that consensus ADMM with parameter $\beta$ is equivalent to Proximal-1-PD with eta1 = K and $\beta$ K = $rho^{{-1}}$, and that the linearized versions LinConsensus-PD and Proximal-2-PD are equivalent under eta2 = K tau, by applying consensus ADMM to the negated saddle objective (Corollary 2). The equivalence is established through a unified update form in which a global primal variable w is updated by a proximal step at the server while each machine k updates its local dual block v_[k] with a proximal subproblem weighted by the local Gram matrix X_[k]^T X_[k]. The paper states that this reveals CoCoA as a special case of proximal ADMM with a specific step size, so an optimally tuned ADMM is at least as good as CoCoA on ridge-regularized problems; numerical experiments on synthetic ridge and lasso regression and on real SVM datasets are reported to support the predicted parameter relationships.

Load-bearing premise

The equivalence rests on the identification of the local dual variables introduced in consensus ADMM's subproblems with the global dual variable v of the unified update, and on the minimax equality min_w max_v L(w,v) = max_v min_w L(w,v) holding for the chosen losses and regularizers; if either fails, the claimed algorithm-level equivalences do not follow.

Editorial extensions

If this is right

  • If the equivalences are correct, every convergence guarantee and duality-gap stopping criterion derived for the unified form applies to CoCoA, consensus ADMM, LinConsensus-PD, Proximal-1-PD, and Proximal-2-PD without separate proofs.
  • For ridge-regularized ERM, no choice of CoCoA parameters can make it fundamentally outperform an optimally tuned ADMM variant; the best achievable convergence of the two families coincides.
  • The linearized variants inherit the same O(1/T) ergodic rate as the exact forms while using cheaper closed-form proximal updates, so there is no rate penalty for linearization.
  • When the regularizer is not quadratic (for example L1), the CoCoA–Proximal-1-PD equivalence breaks because the argument depends on the ridge penalty's quadratic structure, as the paper notes; the consensus-to-proximal ADMM equivalences do not require the ridge penalty and may survive.
  • The unified update form gives a practical primal-dual gap for consensus ADMM, which can be used as a stopping criterion in distributed implementations.

Reading between the lines

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

  • A direct testable consequence the paper does not spell out: in exact arithmetic, CoCoA-PD with sigma=K, gamma=1 and Proximal-1-PD with rho=1/lambda should produce bit-identical dual iterates on any ridge-regularized problem; a divergence under finite-precision or inexact local solves would point to the unproven identification of local and global dual variables.
  • Because the equivalences map the number of machines K into the prox scaling eta1=K, one could transfer ADMM acceleration techniques such as over-relaxation or adaptive penalty to CoCoA-style methods, a direction the paper leaves implicit.
  • The paper notes that the w-updates differ even when the dual sequences coincide: Proximal-1-PD averages successive CoCoA w-updates. An editorial conjecture is that this extra averaging is what lets Proximal-1-PD take larger step sizes and outperform CoCoA in practice.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper proposes a unified primal-dual update framework for five distributed ERM algorithms: consensus ADMM, linearized consensus ADMM, two proximal ADMM variants, and ridge-regularized CoCoA. It claims that CoCoA's dual iterates coincide with those of a proximal ADMM variant under a parameter mapping (Corollary 1), and that consensus ADMM is equivalent to that proximal ADMM variant under an explicit parameter mapping together with a sign reversal of the saddle objective (Corollary 2). A unified O(1/T) ergodic convergence analysis is given, and experiments on synthetic and real data are presented to support the theoretical relationships.

Significance. If the claims hold, the paper establishes an explicit and nontrivial connection between two major families of distributed optimization methods, with concrete parameter mappings that could enable transfer of tuning strategies and convergence results. The algebraic derivations are parameter-free and stated as exact update-form identities, and the authors provide reproducible code. The convergence analysis is standard but the unified treatment is a useful contribution. The experimental results, while empirical, are consistent with the theoretical equivalences and support the conclusion that tuned ADMM variants can outperform CoCoA on ridge-regularized problems.

major comments (2)
  1. [Appendix B.1 (proof of Proposition 1)] The proof of Proposition 1 is too terse at the critical step where the global w-update is derived. The text says 'Representing w_k^{(t)} and u_k^{(t)} in terms of w^{(t)} and \tilde v_k^{(t)} ... we derive the following updates' but does not show the algebra. The final update (1) can be verified when the global consensus update is w^{(t+1)} = prox_{(βK)^{-1}g}((1/K)Σ_k(w_k^{(t+1)} - u_k^{(t+1)}/β)), which follows from the first-order condition of the global subproblem in Section 3.1. However, because the sign convention in that subproblem is unconventional, a reader using the more common plus sign will obtain a different update (w^t + (1/(nβK))X\tilde v^t) and conclude that (1) is false. The proof should explicitly derive the prox argument from the global subproblem and state the sign convention. In addition, the closing remark that 'we need to show that the dual variable \tilde v converges to the global dual variable v' is misleading; the local dual variables \tilde v_{[k]} are the same variables as the global v_{[k]} in (1), so no asymptotic identification is needed. Clarifying this would remove an unnecessary gap in a load-bearing proof.
  2. [Section 4, Corollary 1] The statement of Corollary 1 writes g(w) = λ||w||_2^2, but throughout the rest of the paper (including the CoCoA derivation in Section 3.3 and the proof in Appendix B.4) the ridge penalty is g(w) = (λ/2)||w||_2^2. The parameter mapping ρ=λ^{-1} and the KKT relation w = -Xv/(nλ) are only consistent with the λ/2 convention. This inconsistency should be corrected in the corollary statement.
minor comments (5)
  1. [Appendix B.5 (proof of Corollary 2)] The proof states 'After some algebraic operations' when matching the negated-saddle algorithm to the Proximal-1-PD form. Since this is a central equivalence, the algebra should be shown at least for the parameter verification ρη1 = 1/β and the linear term 2w^t - w^{t-1}.
  2. [Abstract and Section 4] The phrase 'performs at least as good as CoCoA' should be made more precise: the theoretical result is that CoCoA is a special case of Proximal-1-PD with ρ=λ^{-1}, and tuning ρ away from this value can improve performance. The current phrasing could be read as a universal guarantee, whereas it is a consequence of the special-case parameter choice.
  3. [Figure 2] The legends in Figure 2 are garbled (e.g., '21=1', ';=1', '-=0:0001'), making the plots difficult to interpret. Please ensure that axis labels and legends are properly rendered.
  4. [Theorem 1] The sentence 'This implies that \bar{z}^{(T)} converges to the optimal solution with rate O(1/T)' is imprecise: the O(1/T) bound is on the saddle-point gap, not directly on the iterate error. The wording should be adjusted to state that the ergodic gap converges at the stated rate.
  5. [Theorem 2] Theorem 2 assumes P is positive definite, but the paper's recommended parameters η1=K and η2=Kτ yield P1--P4 that may be only positive semidefinite. The paper should either state that Theorem 2 applies when the proximal parameters are chosen strictly larger than the minimal feasible values, or add a remark on how the positive semidefinite case is handled.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the equivalence claims are algebraic consequences of the stated update rules, and experimental tuning is explicitly presented as tuning rather than as a prediction from the model.

full rationale

The paper's central claims—CoCoA-PD and Proximal-1-PD produce identical dual iterates when rho=lambda^{-1} (Corollary 1), and Consensus-PD is equivalent to Proximal-1-PD under beta K = rho^{-1} and eta1=K (Corollary 2)—are derived by substituting the stated update rules into one another. Nothing in these derivations assumes the equivalence it is trying to prove; the proofs work from the algorithms' definitions in equations (1)-(5) and the Fenchel/KKT relations stated in the appendix. The experimental comparison selects the best beta or rho per dataset, but the paper explicitly frames this as tuning ('we select the optimal beta or rho to achieve optimal performance') and does not present those tuned values as parameter-free predictions. Likewise, the references to prior CoCoA and ADMM work are used for algorithmic origins and convergence frameworks, not as a self-citation chain carrying the main equivalence claim; the cited authors do not overlap with the present authors. The most notable weakness is in the proof of Proposition 1, where the paper says 'we need to show that the dual variable v-tilde converges to the global dual variable v' and then defers the details; this is an omitted or incomplete argument about the algorithm's correctness, not a circular step, because the claimed equivalence is not being inserted as an assumption. Similarly, the use of the minimax equality in Corollary 2 is a standard external mathematical fact, not a restatement of the target result. Consequently, there is no exhibited reduction of a prediction to a fitted input or to a self-citation, and the derivation chain is self-contained in the sense relevant to circularity analysis.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The equivalence results require standard convex analysis, a strong-duality/minimax step, and one paper-specific identification of local and global dual variables that the appendix defers. No new physical or mathematical objects are postulated.

free parameters (1)
  • Augmented Lagrangian parameters beta (Consensus/LinConsensus) and rho (Proximal-1/2) in experiments = chosen per dataset (not listed numerically in the text)
    Experiments 2 and 3 select 'the optimal beta or rho' to achieve best convergence when comparing ADMM variants against CoCoA; the empirical comparison therefore depends on data-dependent tuning choices.
assumptions (4)
  • domain assumption Loss functions l_i and regularizer g are proper closed convex functions with well-defined Fenchel conjugates and proximal operators.
    Invoked throughout Section 2 to build the saddle-point problem (SP) and dual (D); convexity-concavity is used in Theorem 1.
  • standard math Strong duality / minimax equality min_w max_v L(w,v) = max_v min_w L(w,v).
    Used in the proof of Corollary 2 to justify that solving the negated saddle problem yields the same solution; requires qualification conditions not stated in the paper.
  • ad hoc to paper The local dual variables v-tilde_[k] produced by dualizing each consensus ADMM subproblem coincide with the global dual variable blocks v_[k].
    Proposition B.1 states this identification is needed ('we need to show that the dual variable v-tilde converges to the global dual variable v') and defers the proof; the reformulation (1) depends on it.
  • standard math The preconditioner matrices P1..P4 are positive semidefinite (positive definite for the inexact Theorem 2).
    Lemma 1 provides eta1 = K and eta2 = K tau* to ensure Q1, Q2 are PSD; PSD of the P matrices is required for the PPM convergence argument in Theorem 1.

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Pith. "Pith review of On the Relationship Between CoCoA and ADMM for Distributed Empirical Risk Minimization." pith.science (2026). https://pith.science/paper/333CSJRH

@misc{pith2026250200470,
  author       = {Pith},
  title        = {Pith review of: On the Relationship Between CoCoA and ADMM for Distributed Empirical Risk Minimization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/333CSJRH}},
  note         = {Machine review of arXiv:2502.00470}
}
abstract

Distributed empirical risk minimization (ERM) is often studied through two influential yet seemingly separate families of methods: CoCoA-type algorithms, derived from distributed dual coordinate ascent, and ADMM-type algorithms, derived from consensus and proximal splitting. In this paper, we investigate the connection of the two types of algorithms from a unified primal-dual perspective. We show that consensus ADMM, linearized consensus ADMM, two distributed proximal ADMM variants, and ridge-regularized CoCoA can all be written in a common update form involving a global primal variable and block dual variables. This reformulation makes several previously hidden connections explicit: For ridge-regularized ERM, CoCoA coincides with a particular proximal ADMM scheme at the level of the dual update. Moreover, consensus ADMM on the primal problem is equivalent to proximal ADMM on the dual problem under an explicit parameter mapping together with a sign reversal of the saddle objective; similar correspondences also hold for the linearized variants. These results indicates that the ADMM-type algorithms, when fine tuned, performs at least as good as CoCoA, under ridge regularized ERM problems. The unified view also yields a natural primal-dual gap stopping criterion for consensus ADMM and a unified $O(1/T)$ ergodic convergence analysis for the ADMM-type methods. Experiments on synthetic regression problems and real SVM datasets support the predicted relationships, clarify the role of tuning parameters, and show that suitably tuned ADMM variants can outperform CoCoA in the ridge-regularized setting.

Figures

Figures reproduced from arXiv: 2502.00470 by the authors.

Figure 1
Figure 1. Connections among distributed algorithms: under ℓ2-regularized ERM, CoCoA is equivalent to first Proximal ADMM with ρ = λ −1 , and Consensus ADMM is equivalent to first Proximal ADMM when βK = ρ −1 . Linearized consensus ADMM is equivalent to the linearized proximal ADMM. Notations. Let X = [x1, . . . , xn] ∈ R d×n de￾note the full training data matrix, where each column xi ∈ R d is a feature vector. The corre￾spond… view at source ↗
Figure 2
Figure 2. Effect of tuning parameters on various distributed algorithms in Experiment 1. and Proximal-2-PD are almost identical. These simulation results further confirm the strong con￾nection between these two pairs. All four ADMM variants, with the optimized tuning parameters, significantly outperform the CoCoA framework. This is because of CoCoA is the Proximal-1-PD with a specific step size ρ = λ −1 . The figure also show… view at source ↗
Figure 3
Figure 3. Relative gap difference versus the number of communication rounds for various synthetic datasets when using different update rules in Experiment 2. 7 Conclusion In this article, we unified distributed primal-dual algorithms, including CoCoA, two proximal ADMM algorithms, consensus ADMM, and linearized ADMM into updates rule that only involve the primal and dual variable updates. Among them, the two proximal ADMM alg… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Relative gap differences versus the number of communication rounds for various real datasets across different models. The first row of plots illustrates the results for SVM with a ridge penalty across different datasets, while the second row shows the results for SVM w…

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