The Golden Ratio Proximal ADMM with Norm Independent Step-Sizes for Separable Convex Optimization
Pith reviewed 2026-05-18 09:12 UTC · model grok-4.3
The pith
Two new step-size rules let golden-ratio proximal ADMM solve separable convex problems without operator norm estimates.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
The core discovery is that inexpensive local information suffices to choose step-sizes for the golden ratio proximal ADMM, yielding globally convergent algorithms with sublinear convergence rates for separable convex optimization problems with linear constraints.
What carries the argument
The golden ratio proximal ADMM equipped with the two norm-independent step-size strategies based on local iterate data.
If this is right
- The generated iterates converge globally to an optimal solution.
- Sublinear convergence rates are established for both proposed algorithms.
- Pointwise convergence rates for the iterates are derived.
- The first step-size rule coincides with a fixed-step-size method when the initial step-size is below a certain threshold.
Where Pith is reading between the lines
- These local step-size choices could lower the setup cost for large-scale problems where computing global operator norms is expensive.
- The ability to increase step-sizes over time may lead to faster practical convergence in applications with varying curvature.
- Similar adaptive rules might be developed for other splitting methods that currently depend on norm bounds.
Load-bearing premise
The problem data satisfy the convexity and constraint conditions assumed in the standard convergence theory for proximal ADMM.
What would settle it
Running the algorithms on a simple convex optimization problem with known solution and observing that the iterates do not approach the solution at the claimed rate would falsify the convergence results.
Figures
read the original abstract
In this work, we propose two step-size strategies for the Golden ratio proximal ADMM (GrpADMM) to solve linearly constrained separable convex optimization problems. Both strategies eliminate explicit operator norm estimates by relying on inexpensive local information computed at the current iterate and requiring no backtracking. However, the key difference is that the second step-size strategy allows recovery from poor initial steps and can increase from iteration to iteration. Under standard assumptions, we establish global convergence of the generated iterates and derive sublinear convergence rates for both algorithms. We also obtain pointwise convergence rate results for the iterates of the algorithms. In addition, we show that the first proposed step-size rule for GrpADMM reduces to the fixed-step-size counterpart when the initial step-size is chosen below a certain threshold. Preliminary numerical experiments demonstrate the practical adaptability and effectiveness of the proposed approaches.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript proposes two step-size strategies for the Golden ratio proximal ADMM (GrpADMM) to solve linearly constrained separable convex optimization problems. Both strategies use inexpensive local information at the current iterate to avoid explicit operator norm estimates and do not require backtracking. The second strategy allows the step-size to increase. Under standard assumptions, global convergence and sublinear convergence rates are established for both, along with pointwise convergence rates. The first strategy reduces to fixed-step ADMM below a certain initial step-size threshold. Preliminary numerical experiments are provided to demonstrate effectiveness.
Significance. If the convergence analysis holds, this contribution is significant for practical implementation of ADMM-type methods in large-scale optimization, as it removes the need for computing operator norms which can be costly or difficult. The local information-based rules and the increasing step-size option enhance adaptability. Credit is due for deriving both sublinear and pointwise rates, and for showing the reduction to the fixed-step case. This could be useful for practitioners dealing with separable convex problems.
major comments (1)
- [Convergence analysis section] The global convergence and rate proofs rely on the proposed local step-size rules satisfying the standard descent/Lyapunov inequality from proximal ADMM analyses. Explicitly confirm in the relevant theorem (around the statement of the two algorithms) that the rules meet this condition for all iterates without requiring additional boundedness or hidden assumptions on the local information.
minor comments (3)
- [Algorithm descriptions] Clarify the exact form of the local information used in the step-size rules (e.g., which norm or inner product appears) to make the algorithms fully reproducible from the pseudocode alone.
- [Numerical experiments] In the numerical section, add a brief discussion of how the increasing step-size rule recovers from poor initial choices, with reference to specific iteration counts or objective values in the tables/figures.
- [Main results] The abstract states that one rule 'reduces to the fixed-step-size counterpart'; include a short remark or corollary in the main text showing the threshold condition explicitly.
Simulated Author's Rebuttal
We thank the referee for the careful reading, positive assessment, and constructive comment. We address the major comment below.
read point-by-point responses
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Referee: [Convergence analysis section] The global convergence and rate proofs rely on the proposed local step-size rules satisfying the standard descent/Lyapunov inequality from proximal ADMM analyses. Explicitly confirm in the relevant theorem (around the statement of the two algorithms) that the rules meet this condition for all iterates without requiring additional boundedness or hidden assumptions on the local information.
Authors: We appreciate the suggestion to make this verification more explicit. In the revised manuscript we will add a dedicated remark (or short lemma) immediately after the statements of the two algorithms and prior to the main convergence theorems. The remark will confirm that both proposed local step-size rules satisfy the standard descent/Lyapunov inequality at every iterate. This follows directly from the definition of the rules, which select the step-size using only finite, well-defined local quantities (e.g., residual norms or inner products evaluated at the current iterate) and requires no additional boundedness assumptions on the operators, the local information, or the sequence of iterates beyond the standard convexity, closedness, and constraint-qualification assumptions already stated in the paper. The existing proofs already proceed by direct substitution of these local choices into the Lyapunov function; the added remark will simply isolate and highlight this verification for the reader. revision: yes
Circularity Check
Standard ADMM convergence analysis; no load-bearing reduction to self-fit or self-citation
full rationale
The paper introduces two norm-independent step-size rules for the existing GrpADMM framework and proves global convergence plus sublinear/pointwise rates under standard convex optimization assumptions. These proofs follow the usual Lyapunov descent arguments for proximal ADMM variants; the step-size choices are shown to satisfy the required inequalities directly from local iterate information without backtracking or operator norms. No quoted derivation reduces a claimed rate or convergence result to a fitted parameter or to a prior self-citation by construction. The analysis is self-contained against external benchmarks for this class of methods.
Axiom & Free-Parameter Ledger
free parameters (1)
- initial step-size threshold
axioms (1)
- domain assumption Standard assumptions for convergence of proximal ADMM-type methods (convexity of objective, linear constraints, existence of solutions)
Lean theorems connected to this paper
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IndisputableMonolith/Cost/FunctionalEquation.leanwashburn_uniqueness_aczel echoes?
echoesECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.
ϕ := 1 + √5 / 2 … ψ ∈ (1,ϕ] … uk := (ψ−1)/ψ x^{k−1} + 1/ψ u^{k−1}
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IndisputableMonolith/Foundation/AlphaCoordinateFixation.leanalpha_pin_under_high_calibration refines?
refinesRelation between the paper passage and the cited Recognition theorem.
τ_k = min{τ_{k−1}, μ √λ_min(S) √β / ||A(x^k−x^{k−1})|| … } and the increasing rule with ρ ∈ (1,1/ψ+1/ψ²]
What do these tags mean?
- matches
- The paper's claim is directly supported by a theorem in the formal canon.
- supports
- The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
- extends
- The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
- uses
- The paper appears to rely on the theorem as machinery.
- contradicts
- The paper's claim conflicts with a theorem or certificate in the canon.
- unclear
- Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.
Reference graph
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