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arxiv: 2509.05288 · v2 · submitted 2025-09-05 · 💻 cs.LG · math.OC

Learning to accelerate distributed ADMM using graph neural networks

Henri Doerks , Paul H\"ausner , Daniel Hern\'andez Escobar , Jens Sj\"olund This is my paper

Pith reviewed 2026-05-18 18:43 UTC · model grok-4.3

classification 💻 cs.LG math.OC
keywords distributed optimizationADMMgraph neural networkshyperparameter learningconvergence accelerationmessage passingunrolled optimizationdecentralized algorithms
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The pith

Distributed ADMM iterations fit inside graph neural network message passing so a network can learn adaptive step sizes and weights for faster convergence.

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

The paper establishes that the alternating direction method of multipliers can be rewritten as message passing on a graph, allowing a GNN to predict iteration-specific step sizes and communication weights directly from the current iterates. By unrolling a fixed number of ADMM steps and training the GNN end-to-end to reduce the distance to the solution on a given problem class, the method keeps the original convergence guarantees while improving practical speed. A sympathetic reader would care because distributed optimization appears in large-scale machine learning and control, where slow or hyperparameter-sensitive convergence is a recurring cost. The learned variant is shown to deliver both quicker progress within the training budget and better final accuracy on new instances drawn from the same distribution.

Core claim

Distributed ADMM iterations can be expressed within the message-passing framework of graph neural networks. A GNN is then trained to output adaptive step sizes and communication weights from the current iterates; the combined system is unrolled for a fixed number of steps and optimized end-to-end to minimize solution error after those steps, while the underlying ADMM updates preserve their theoretical convergence properties.

What carries the argument

The rewriting of ADMM's local minimization and dual update steps as graph message-passing layers that let the GNN emit per-iteration hyperparameters.

If this is right

  • Convergence speed improves both inside and outside the exact iteration count used during training.
  • Solution quality after a fixed computational budget is higher than with hand-chosen ADMM parameters.
  • The method retains ADMM convergence guarantees because the learned values are inserted into the original update rules.
  • Generalization holds for new instances sampled from the same problem distribution used in training.

Where Pith is reading between the lines

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

  • The same GNN embedding idea could be applied to other first-order distributed methods that admit a graph interpretation.
  • In deployed systems the learned predictor might replace manual grid search over step sizes and penalty parameters.
  • Robustness can be tested by feeding the trained network instances drawn from a modestly shifted distribution.
  • Hybrid schemes become possible in which the GNN supplies guidance only for the first many iterations and then hands control back to standard ADMM.

Load-bearing premise

A GNN trained on a specific problem class will output hyperparameters that keep the algorithm convergent and improve performance on new instances from the same distribution.

What would settle it

Collect a fresh set of problem instances drawn from the training distribution, run both standard ADMM and the learned variant for the same number of iterations, and check whether the learned version consistently reaches a target accuracy in fewer iterations or with lower final error.

Figures

Figures reproduced from arXiv: 2509.05288 by Daniel Hern\'andez Escobar, Henri Doerks, Jens Sj\"olund, Paul H\"ausner.

Figure 1
Figure 1. Figure 1: Computation structure of iteration k of the decentralized distributed ADMM algorithm computed as two message-passing steps in our proposed GNN (see Algorithm 2.2). The initial node features V k contain the previous ADMM iterates (x k i , yk i , λk i ) for each node i and the non￾zero edge features E describe the weighted pairwise connection of nodes in the network which is fixed across iterations. After tw… view at source ↗
Figure 2
Figure 2. Figure 2: Semi-log plots of the relative objective for the two problem classes across all 20 iterations. The network is only unrolled for the first 10 iterations during training as in the previous results. One exception is the learned communication matrix for the least-squares problem which leads to a higher objective value for most of the iterations even though the distance to the minimizer gets reduced as we have … view at source ↗
read the original abstract

Distributed optimization is fundamental to large-scale machine learning and control applications. Among existing methods, the alternating direction method of multipliers (ADMM) has gained popularity due to its strong convergence guarantees and suitability for decentralized computation. However, ADMM can suffer from slow convergence and high sensitivity to hyperparameter choices. In this work, we show that distributed ADMM iterations can be naturally expressed within the message-passing framework of graph neural networks (GNNs). Building on this connection, we propose learning adaptive step sizes and communication weights through a GNN that predicts these yperparameters based on the current iterates. By unrolling ADMM for a fixed number of iterations, we train the network end-to-end to minimize the solution distance after these iterations for a given problem class, while preserving the algorithm's convergence properties. Numerical experiments demonstrate that our learned variant consistently improves convergence speed and solution quality compared to standard ADMM, both within the trained computational budget and beyond. The code is available at https://github.com/paulhausner/learning-distributed-admm.

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 paper claims that distributed ADMM iterations can be expressed as message-passing operations on graphs, allowing a GNN to learn adaptive step sizes and communication weights from current iterates. The network is trained end-to-end by unrolling a fixed number of ADMM steps and minimizing the distance to the optimum for a given problem class, while aiming to preserve convergence properties. Numerical experiments are reported to show consistent improvements in convergence speed and solution quality over standard ADMM, both inside and outside the training horizon.

Significance. If the learned parameters reliably accelerate ADMM without introducing instability, the work would offer a practical bridge between classical distributed optimization and learned accelerators for specific problem distributions. The explicit code release supports reproducibility and external verification of the empirical claims.

major comments (2)
  1. [Abstract and §5] Abstract and §5 (Numerical Experiments): the central claim that improvements hold 'both within the trained computational budget and beyond' rests on fixed-horizon unrolling that minimizes distance only after K steps. No analysis, additional long-horizon experiments, or stability checks are provided to confirm that GNN-predicted state-dependent parameters maintain convergence (e.g., satisfying the standard ADMM conditions on ρ) or prevent oscillation/divergence on fresh instances drawn from the same distribution after the training horizon.
  2. [§4] §4 (Method): while the message-passing reformulation of ADMM is clean, the paper does not derive or verify that the learned, input-dependent weights and step sizes continue to satisfy the convergence assumptions of classical ADMM once the GNN is evaluated outside the exact training distribution or for iteration counts >K.
minor comments (2)
  1. [Abstract] Abstract: 'yperparameters' appears to be a typo for 'hyperparameters'.
  2. [§5] The description of the exact problem distributions used for training versus testing, and any hyperparameter sensitivity analysis, would benefit from additional detail to support the generalization claims.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the constructive feedback. Below we provide point-by-point responses to the major comments. We will make revisions to address the concerns about empirical validation and clarification of theoretical aspects.

read point-by-point responses

  1. Referee: [Abstract and §5] Abstract and §5 (Numerical Experiments): the central claim that improvements hold 'both within the trained computational budget and beyond' rests on fixed-horizon unrolling that minimizes distance only after K steps. No analysis, additional long-horizon experiments, or stability checks are provided to confirm that GNN-predicted state-dependent parameters maintain convergence (e.g., satisfying the standard ADMM conditions on ρ) or prevent oscillation/divergence on fresh instances drawn from the same distribution after the training horizon.

    Authors: We agree that the training procedure uses fixed-horizon unrolling. In the experiments of §5, we do evaluate the learned ADMM for a larger number of iterations than K on unseen problem instances and report sustained improvements without observed divergence. However, we acknowledge the absence of dedicated long-horizon analysis or formal stability checks. We will revise the abstract and §5 to emphasize that the 'beyond' claim is based on empirical observation up to a moderate extension of the horizon, and add additional plots showing performance for up to 2K or 3K iterations with discussion of stability. A complete theoretical analysis of convergence for the learned parameters remains an open question and is noted as future work. revision: partial

  2. Referee: [§4] §4 (Method): while the message-passing reformulation of ADMM is clean, the paper does not derive or verify that the learned, input-dependent weights and step sizes continue to satisfy the convergence assumptions of classical ADMM once the GNN is evaluated outside the exact training distribution or for iteration counts >K.

    Authors: The reformulation shows that each ADMM iteration corresponds to a message-passing step where the GNN predicts the step sizes and weights based on local iterates. During training, we constrain the outputs to be positive to mimic the standard ADMM parameter ρ > 0. Nevertheless, we do not derive that these state-dependent predictions necessarily fulfill all classical convergence conditions for every possible input or for arbitrarily large iteration counts. We will update §4 to include a clearer statement that while the structure preserves the ADMM form, the convergence guarantees are inherited only heuristically and are supported by the empirical results rather than proven for the learned case. This limitation will be explicitly discussed. revision: yes

standing simulated objections not resolved
  • A formal derivation verifying that the GNN-predicted parameters satisfy ADMM convergence assumptions outside the training distribution and for iteration counts exceeding K.

Circularity Check

0 steps flagged

No circularity: training objective independent of performance claims

full rationale

The paper connects distributed ADMM iterations to GNN message passing as a structural observation, then defines a GNN that outputs state-dependent step sizes and weights. These are trained by unrolling a fixed horizon K and minimizing distance to the optimum after K steps. This loss is independent of the reported long-term convergence improvements and does not reduce the empirical results to the inputs by construction. No self-citation chain, uniqueness theorem, or ansatz is invoked to force the outcome; the central claims rest on numerical experiments for a given problem class, with released code enabling external checks. The derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the ability of the learned GNN to generalize across problem instances and on the assumption that unrolling a fixed number of iterations is sufficient to capture long-term convergence behavior. No new physical entities are introduced.

free parameters (1)
  • GNN weights
    The parameters of the graph neural network are fitted during end-to-end training to minimize solution distance after a fixed number of ADMM iterations.
axioms (1)
  • domain assumption Distributed ADMM iterations can be exactly expressed as message-passing operations on a graph.
    This equivalence is the foundation for applying GNNs; it is stated in the abstract and forms the modeling choice.

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