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arxiv: 2606.08993 · v1 · pith:GZHDMUW7new · submitted 2026-06-08 · 💻 cs.LG · cs.SY· eess.SY· math.OC

LEAF: A Learning-Enabled ADMM Framework for Accelerated Convex Optimization

Binh Nguyen , Trinh Tran , Truong X. Nghiem This is my paper

Pith reviewed 2026-06-27 17:35 UTC · model grok-4.3

classification 💻 cs.LG cs.SYeess.SYmath.OC
keywords ADMMconvex optimizationinput convex neural networksMoreau envelopelearning-enabled optimizationaccelerationconvergence analysis
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The pith

Approximating the Moreau envelope with an input convex neural network accelerates ADMM while preserving convergence rates.

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

The paper presents LEAF, a framework that learns an approximation to the Moreau envelope of a convex objective using an input convex neural network. This scalar-valued model is inserted directly into the ADMM updates to produce the MEL-ADMM and sMEL-ADMM variants. Because the network is trained to respect convexity and smoothness, the resulting iterations retain the structural properties needed for standard convergence analysis. The approach targets a wide range of convex problems that mix smooth and nonsmooth terms and reports lower per-iteration cost than classical ADMM or existing operator-learning methods.

Core claim

LEAF approximates the Moreau envelope of the objective with an input convex neural network, yielding a learned scalar model that preserves convexity and smoothness. Substituting this model into the ADMM iteration produces MEL-ADMM and its splitting form sMEL-ADMM, both equipped with convergence and feasibility guarantees under the learned operator. The framework applies to convex problems with smooth or nonsmooth objectives and achieves rates comparable to classical ADMM at reduced per-iteration cost.

What carries the argument

The ICNN approximation of the Moreau envelope, a scalar-valued convex function learned from data and substituted into the ADMM proximal step to lower computational cost per iteration.

If this is right

  • MEL-ADMM and sMEL-ADMM achieve convergence rates comparable to classical ADMM.
  • Per-iteration computational cost is lower than that of standard ADMM or direct operator-learning baselines.
  • The methods remain feasible when the learned model replaces the exact Moreau envelope.
  • Numerical tests show up to an order-of-magnitude reduction in runtime while keeping optimality gaps low across a range of convex problems.

Where Pith is reading between the lines

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

  • The scalar form of the learned envelope may allow a single trained network to be reused across problem instances that share the same objective structure but differ in dimension or data.
  • If the same ICNN construction can be applied to other first-order methods that rely on proximal mappings, the framework could reduce tuning effort for a broader family of splitting algorithms.
  • The explicit convexity constraint inside the network architecture may make it easier to certify safety properties when the learned optimizer is deployed inside a control loop.

Load-bearing premise

The learned ICNN must approximate the true Moreau envelope accurately enough that the substituted model still satisfies the convexity and smoothness conditions required by the convergence proofs.

What would settle it

A convex test problem on which the learned model produces iterates whose optimality gap grows or whose feasibility violation exceeds a fixed tolerance, while the same problem solved with the exact Moreau envelope converges normally.

Figures

Figures reproduced from arXiv: 2606.08993 by Binh Nguyen, Trinh Tran, Truong X. Nghiem.

Figure 1
Figure 1. Figure 1: Schematic overview of the Learning-Enabled ADMM [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Example 1 of energy management in a microgrid. [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Optimality gap over solving time for Example 1. [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Example 1: Prediction over 24-hour (96-step) horizon [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: provides a statistical breakdown of the solving time with m = 55 under optimality gaps of 1% and 0.1%. While Table IV summarizes average performance, these box plots reveal the consistency of sMEL-ADMM. The interquartile range of sMEL-ADMM is significantly narrower and lower than those of Mosek and Clarabel. Even in the more stringent optimality gap scenario, sMEL-ADMM maintains a much tighter distribution… view at source ↗
read the original abstract

We propose LEAF, a learning-enabled ADMM framework for accelerated convex optimization. The key idea is to approximate the Moreau envelope of the objective function using an Input Convex Neural Network (ICNN), resulting in a learned model that preserves convexity and smoothness. This leads to the proposed Moreau Envelope Learning ADMM (MEL-ADMM) and its splitting variant sMEL-ADMM. Unlike existing approaches that learn high-dimensional operators directly, LEAF learns a scalar-valued Moreau envelope, significantly reducing model complexity and improving data efficiency. The framework accommodates a broad class of convex problems with smooth and non-smooth objectives. By embedding convexity explicitly through the ICNN architecture, the proposed approach maintains high approximation accuracy while preserving key structural properties of the optimization problem. Both MEL-ADMM and sMEL-ADMM are developed with theoretical guarantees of convergence and feasibility under the learned model. Rigorous analysis shows that the proposed methods achieve convergence rates comparable to classical ADMM while reducing per-iteration computational cost. Numerical experiments demonstrate up to an order-of-magnitude speedup over state-of-the-art solvers while maintaining low optimality gaps

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 proposes LEAF, a learning-enabled ADMM framework that approximates the Moreau envelope of convex objectives via Input Convex Neural Networks (ICNNs) to obtain MEL-ADMM and its splitting variant sMEL-ADMM. The central claims are that the ICNN approximation preserves convexity and smoothness, that both variants admit convergence guarantees comparable to classical ADMM (including O(1/k) rates), and that numerical experiments show up to an order-of-magnitude speedup over state-of-the-art solvers while keeping low optimality gaps. The approach is positioned as more data-efficient than direct operator learning because it targets a scalar-valued envelope.

Significance. If the error-propagation analysis can be completed, the work would offer a principled route to data-driven acceleration of proximal algorithms for a broad class of convex problems, trading per-iteration cost for a learned model whose structural properties are architecturally enforced. The reduction from learning high-dimensional operators to learning a scalar Moreau envelope is a concrete complexity improvement that could be reusable beyond ADMM.

major comments (2)
  1. [Theoretical Analysis] Theoretical Analysis section: the convergence theorem for MEL-ADMM under the learned model asserts rates comparable to classical ADMM, yet provides no explicit tolerance on the ICNN approximation error ||f_learned - f_true|| (or on the induced proximal-operator error) that is sufficient to preserve the contraction or feasibility properties used in the proof. The argument therefore reduces to the classical case plus an unquantified perturbation claim.
  2. [Theoretical Analysis] § on sMEL-ADMM convergence: the splitting variant inherits the same unquantified error assumption; because the learned proximal step appears inside the ADMM iteration, an explicit propagation bound through the augmented-Lagrangian update is required to justify that the O(1/k) rate (or a controlled degradation) still holds.
minor comments (2)
  1. [Problem Formulation] The abstract states that the framework 'accommodates a broad class of convex problems with smooth and non-smooth objectives,' but the precise regularity assumptions (e.g., strong convexity modulus, Lipschitz constants) under which the ICNN training and convergence guarantees apply should be stated explicitly in the problem formulation section.
  2. [Experiments] Numerical experiments: the reported speedups are given relative to 'state-of-the-art solvers,' but the precise solver versions, stopping tolerances, and hardware configuration are not listed in the experimental protocol; adding a table of these parameters would improve reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on the theoretical analysis. The observations correctly identify that the convergence statements would be strengthened by explicit error tolerances. We address each point below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Theoretical Analysis] Theoretical Analysis section: the convergence theorem for MEL-ADMM under the learned model asserts rates comparable to classical ADMM, yet provides no explicit tolerance on the ICNN approximation error ||f_learned - f_true|| (or on the induced proximal-operator error) that is sufficient to preserve the contraction or feasibility properties used in the proof. The argument therefore reduces to the classical case plus an unquantified perturbation claim.

    Authors: We agree that an explicit tolerance on the approximation error is needed to make the perturbation argument rigorous. The current proof invokes the preserved convexity and smoothness of the ICNN but does not quantify how large the error ||f_learned - f_true|| may be while still guaranteeing the O(1/k) rate. In the revised manuscript we will insert a new lemma that derives the admissible error bound in terms of the ADMM penalty parameter and the strong-convexity modulus (when present), together with the resulting degradation in the convergence constant. revision: yes

  2. Referee: [Theoretical Analysis] § on sMEL-ADMM convergence: the splitting variant inherits the same unquantified error assumption; because the learned proximal step appears inside the ADMM iteration, an explicit propagation bound through the augmented-Lagrangian update is required to justify that the O(1/k) rate (or a controlled degradation) still holds.

    Authors: We concur that the error introduced by the learned proximal operator must be tracked through the augmented-Lagrangian updates for sMEL-ADMM. The revised version will contain an additional proposition that propagates the proximal error through one full ADMM iteration and shows that the same O(1/k) rate is retained provided the per-iteration error satisfies a summable bound derived from the dual step-size and the Lipschitz constant of the linear operator. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation relies on external data-driven approximation and standard ADMM analysis

full rationale

The paper's core steps are (1) training an ICNN on data to approximate the Moreau envelope while enforcing convexity/smoothness via architecture, (2) substituting the learned model into ADMM iterations to obtain MEL-ADMM/sMEL-ADMM, and (3) providing convergence analysis under the learned model. None of these reduce by construction to their own inputs: the ICNN parameters are fitted from external samples rather than defined in terms of the target rates, the convergence claims are stated as holding for the approximate operator (with the approximation error treated as an external perturbation), and no self-citation chain or uniqueness theorem is invoked to force the result. The framework therefore remains self-contained against external benchmarks and data.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that ICNNs can produce sufficiently accurate convex approximations of the Moreau envelope. The number and nature of free parameters inside the ICNN are not stated in the abstract.

free parameters (1)
  • ICNN weights
    Parameters of the input convex neural network are fitted to data to approximate the Moreau envelope; exact count and regularization are unknown from the abstract.
axioms (1)
  • domain assumption Input convex neural networks preserve convexity when used to model the Moreau envelope
    The framework requires this architectural property to transfer convergence guarantees from classical ADMM to the learned variants.

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