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arxiv: 2605.11222 · v1 · submitted 2026-05-11 · 💻 cs.LG

Recognition: 2 theorem links

· Lean Theorem

ADMM-Q: An Improved Hessian-based Weight Quantizer for Post-Training Quantization of Large Language Models

Adam Deng, Mehdi Makni, Rahul Mazumder, Ryan Lucas, Xiang Meng

Authors on Pith no claims yet

Pith reviewed 2026-05-13 02:30 UTC · model grok-4.3

classification 💻 cs.LG
keywords post-training quantizationlarge language modelsADMMweight quantizationreconstruction errorGPTQperplexityHessian-based optimization
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The pith

ADMM-Q uses an ADMM-based splitting procedure to minimize layer-wise reconstruction error while enforcing quantization constraints more effectively than GPTQ.

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

The paper introduces ADMM-Q as a weight quantization algorithm for post-training compression of large language models. It solves the layer-wise problem by alternating continuous weight updates that reduce reconstruction error with gradual enforcement of discrete quantization levels. Enhancements such as penalty scheduling, preconditioning, and local search post-processing make the method practical at LLM scale while preserving convergence guarantees. ADMM-Q is designed as a modular replacement that composes with existing techniques like rotations and activation scaling. On Qwen3-8B it produces lower WikiText-2 perplexity than GPTQ across weight-only, SmoothQuant, and SpinQuant pipelines at 2-4 bit widths.

Core claim

ADMM-Q is a combinatorial variant of the Alternating Direction Method of Multipliers that decouples the continuous minimization of layer reconstruction error from the discrete quantization constraint, updating weights iteratively while progressively tightening the penalty on non-quantized values, with added scheduling and post-processing steps to achieve reliable convergence at the scale of billion-parameter models.

What carries the argument

The ADMM operator-splitting procedure, which alternates between unconstrained weight optimization to minimize Hessian-based reconstruction loss and projection onto the quantization grid with increasing penalty strength.

If this is right

  • Replacing GPTQ with ADMM-Q in weight-only 3-bit quantization reduces WikiText-2 perplexity from 12.85 to 10.06 on Qwen3-8B.
  • ADMM-Q composes with SmoothQuant to lower W4A8 perplexity from 9.29 to 8.68 on the same model.
  • ADMM-Q composes with SpinQuant to lower W2A4KV4 perplexity from 66.11 to 19.42.
  • The method remains compatible with range clipping, random or learned rotations, and activation scaling without architectural changes.

Where Pith is reading between the lines

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

  • The same splitting idea could be tested on activation quantization or joint weight-activation quantization to see if similar error-reduction gains appear.
  • Because the algorithm is Hessian-aware and layer-wise, it may extend naturally to other reconstruction objectives such as attention-map or output-distribution matching.
  • If the convergence behavior proves stable across model families, the approach could reduce the need for per-layer hyperparameter search in production quantization pipelines.

Load-bearing premise

The ADMM procedure with penalty scheduling, preconditioning, and local search converges reliably and efficiently on large language models without hidden instabilities or model-specific retuning.

What would settle it

Applying ADMM-Q to a new large language model and observing either higher final perplexity than GPTQ or failure to converge within reasonable iterations would show the claimed improvements do not hold.

Figures

Figures reproduced from arXiv: 2605.11222 by Adam Deng, Mehdi Makni, Rahul Mazumder, Ryan Lucas, Xiang Meng.

Figure 1
Figure 1. Figure 1: Overview of the proposed ADMM-Q algorithm. (Left) The layerwise quantization problem with a recon￾struction objective; the goal is to approximate the full-precision weight matrix W using quantized weights (Section 3). (Middle) ADMM with diagonal scaling and ρ-update scheme (Algorithm 1) to obtain a high-quality quantized weight matrix (Section 3.3). (Right) Starting from the ADMM solution, a local search p… view at source ↗
Figure 2
Figure 2. Figure 2: Grid refresh updates the projection to match [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Layer-wise reconstruction error of ADMM-Q relative to GPTQ on Qwen3-8B-Base under W4 (top) and W3 (bottom) per-channel weight-only quantization. Each point shows ADMM-Q error/GPTQ error; values below the 100% line indicate ADMM-Q achieves lower error. 8 [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
read the original abstract

Quantization is an effective strategy to reduce the storage and computation footprint of large language models (LLMs). Post-training quantization (PTQ) is a leading approach for compressing LLMs. Popular weight quantization procedures, including GPTQ and RTN, suffer in model utility, especially at aggressive quantization levels (sub-4-bit). We propose ADMM-Q, a novel weight quantization algorithm that considers the layer-wise quantization problem. Our algorithm is based on a combinatorial variant of the Alternating Direction Method of Multipliers (ADMM). Our operator-splitting procedure updates weights continuously to minimize the layer-wise reconstruction error, while gradually enforcing the quantization constraints with convergence guarantees. We propose additional algorithmic enhancements (e.g., penalty scheduling, preconditioning, and a local search post-processing step) to make ADMM-Q efficient at LLM scale. ADMM-Q is modular and can be used as a drop-in replacement for any weight quantizer within existing quantization pipelines: ADMM-Q is fully composable with existing techniques including range clipping, learned or random rotations, and activation scaling. Using ADMM-Q in place of GPTQ on Qwen3-8B, we decrease WikiText-2 perplexity in: (i) the W3A16 weight-only setting (12.85 $\rightarrow$ 10.06); (ii) the W4A8 SmoothQuant procedure (9.29 $\rightarrow$ 8.68); and (iii) the W2A4KV4 SpinQuant procedure (66.11 $\rightarrow$ 19.42).

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

3 major / 2 minor

Summary. The paper proposes ADMM-Q, a combinatorial ADMM-based algorithm for post-training weight quantization of LLMs. It minimizes layer-wise reconstruction error via operator splitting with gradual enforcement of quantization constraints, augmented by penalty scheduling, preconditioning, and local-search post-processing. The method is presented as modular and composable with existing pipelines (e.g., SmoothQuant, SpinQuant). On Qwen3-8B it reports large WikiText-2 perplexity reductions when substituted for GPTQ: 12.85→10.06 (W3A16), 9.29→8.68 (W4A8), and 66.11→19.42 (W2A4KV4).

Significance. If the reported gains prove robust, ADMM-Q would supply a stronger drop-in solver for the non-convex layer-wise quantization problem than current Hessian-based methods such as GPTQ, improving utility at aggressive bit-widths while preserving composability with rotation and scaling techniques. The explicit algorithmic enhancements and claimed convergence properties are potentially valuable contributions to PTQ methodology.

major comments (3)
  1. [§3.2] §3.2 (ADMM formulation and convergence claim): the manuscript asserts convergence guarantees for the combinatorial ADMM, yet the quantization subproblem is non-convex and the standard ADMM convergence theory does not apply directly. No explicit proof or Lyapunov argument is supplied that shows the penalty schedule plus preconditioner reliably reaches a stationary point of the layer-wise objective rather than cycling or stalling on some layers.
  2. [§4.1] §4.1 and Table 2 (experimental results on Qwen3-8B): the headline perplexity improvements (especially the 66.11→19.42 drop in the W2A4KV4 SpinQuant setting) are presented without error bars, multiple random seeds, or layer-wise reconstruction-error histograms. This leaves open whether the gains are statistically reliable or sensitive to the specific ADMM penalty/preconditioner hyper-parameters listed in the free-parameter ledger.
  3. [§3.3–3.4] §3.3–3.4 (penalty scheduling and preconditioning): these are introduced as algorithmic enhancements required for LLM-scale stability, yet no ablation is reported that isolates their contribution versus a plain ADMM baseline or versus GPTQ on the same layers. Without such controls it is difficult to attribute the observed perplexity reductions to the core ADMM splitting rather than to the added heuristics.
minor comments (2)
  1. [Abstract and §3] The abstract and §4 mention “convergence guarantees” but the precise statement (e.g., to a stationary point of the non-convex problem) should be clarified in the main text.
  2. [§3.3] Notation for the preconditioning matrix and the ADMM penalty schedule parameters should be introduced with explicit symbols and ranges in §3.3 so that readers can reproduce the procedure.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We address each major point below and indicate planned revisions to strengthen the paper.

read point-by-point responses

  1. Referee: [§3.2] §3.2 (ADMM formulation and convergence claim): the manuscript asserts convergence guarantees for the combinatorial ADMM, yet the quantization subproblem is non-convex and the standard ADMM convergence theory does not apply directly. No explicit proof or Lyapunov argument is supplied that shows the penalty schedule plus preconditioner reliably reaches a stationary point of the layer-wise objective rather than cycling or stalling on some layers.

    Authors: We agree that standard ADMM theory requires convexity and does not directly apply to the non-convex quantization subproblem. The phrase 'convergence guarantees' in the manuscript was meant to describe the empirical stability achieved by our penalty scheduling and preconditioning, which prevent cycling in practice. We will revise §3.2 to qualify this claim explicitly, noting the lack of a rigorous proof and adding a brief discussion of how the increasing penalty and Hessian preconditioner promote descent toward stationary points of the layer-wise objective. revision: partial

  2. Referee: [§4.1] §4.1 and Table 2 (experimental results on Qwen3-8B): the headline perplexity improvements (especially the 66.11→19.42 drop in the W2A4KV4 SpinQuant setting) are presented without error bars, multiple random seeds, or layer-wise reconstruction-error histograms. This leaves open whether the gains are statistically reliable or sensitive to the specific ADMM penalty/preconditioner hyper-parameters listed in the free-parameter ledger.

    Authors: The algorithm is deterministic for fixed hyperparameters, which is why single-run results were reported. We acknowledge that error bars and additional diagnostics would improve confidence in the gains, particularly the large improvement in the W2A4KV4 setting. In the revision we will add results over multiple random seeds (via small perturbations to the initial weight scaling) and include layer-wise reconstruction-error histograms comparing ADMM-Q to GPTQ on the same layers. revision: yes

  3. Referee: [§3.3–3.4] §3.3–3.4 (penalty scheduling and preconditioning): these are introduced as algorithmic enhancements required for LLM-scale stability, yet no ablation is reported that isolates their contribution versus a plain ADMM baseline or versus GPTQ on the same layers. Without such controls it is difficult to attribute the observed perplexity reductions to the core ADMM splitting rather than to the added heuristics.

    Authors: We agree that isolating the contribution of penalty scheduling and preconditioning would strengthen attribution. The revised manuscript will include a new ablation subsection comparing (i) plain ADMM without scheduling or preconditioning, (ii) ADMM-Q with the enhancements, and (iii) GPTQ on representative layers of Qwen3-8B, reporting both reconstruction error and downstream perplexity. revision: yes

Circularity Check

0 steps flagged

No significant circularity; algorithmic procedure validated on external benchmarks

full rationale

The paper introduces ADMM-Q as a combinatorial ADMM-based algorithm for layer-wise weight quantization, with enhancements for LLM scale, and demonstrates its use as a modular replacement for GPTQ within existing pipelines. Reported gains are measured via perplexity on WikiText-2 for Qwen3-8B under multiple quantization settings, which are independent external benchmarks. No equations reduce outputs to inputs by construction, no fitted parameters are relabeled as predictions, and no load-bearing self-citations or uniqueness theorems from the authors are invoked to justify the core claims. The derivation chain remains self-contained against external data and baselines.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The claim rests on standard ADMM convergence theory plus three algorithmic choices (penalty scheduling, preconditioning, local search) whose effectiveness is demonstrated empirically rather than derived from first principles.

free parameters (2)
  • ADMM penalty schedule parameters
    The rate and form of penalty increase over iterations is chosen to balance convergence speed and quantization enforcement.
  • Preconditioning matrix parameters
    Preconditioner design involves choices that affect the conditioning of the sub-problems.
axioms (2)
  • domain assumption The layer-wise reconstruction error objective admits a useful operator splitting under quantization constraints
    Invoked when the paper states the ADMM procedure updates weights continuously while enforcing quantization.
  • domain assumption Standard ADMM convergence guarantees extend to the combinatorial quantization setting with the proposed enhancements
    The abstract claims convergence guarantees without detailing the proof conditions.

pith-pipeline@v0.9.0 · 5598 in / 1450 out tokens · 55696 ms · 2026-05-13T02:30:49.840419+00:00 · methodology

discussion (0)

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Works this paper leans on

17 extracted references · 17 canonical work pages · 8 internal anchors

  1. [1]

    GPT-4 Technical Report

    Josh Achiam, Steven Adler, Sandhini Agarwal, Lama Ahmad, Ilge Akkaya, Florencia Leoni Aleman, Diogo Almeida, Janko Altenschmidt, Sam Altman, Shyamal Anadkat, et al. Gpt-4 technical report.arXiv preprint arXiv:2303.08774,

    work page internal anchor Pith review Pith/arXiv arXiv

  2. [2]

    Careful selection of knowledge to solve open book question answering.arXiv preprint arXiv:1907.10738,

    Pratyay Banerjee, Kuntal Kumar Pal, Arindam Mitra, and Chitta Baral. Careful selection of knowledge to solve open book question answering.arXiv preprint arXiv:1907.10738,

    work page arXiv 1907

  3. [3]

    Fast and optimal weight update for pruned large language models.arXiv preprint arXiv:2401.02938,

    Vladimír Boža. Fast and optimal weight update for pruned large language models.arXiv preprint arXiv:2401.02938,

    work page arXiv

  4. [4]

    BoolQ: Exploring the surprising difficulty of natural yes/no questions

    Christopher Clark, Kenton Lee, Ming-Wei Chang, Tom Kwiatkowski, Michael Collins, and Kristina Toutanova. BoolQ: Exploring the surprising difficulty of natural yes/no questions. In Jill Burstein, Christy Doran, and Thamar Solorio, editors,Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Huma...

    work page 2019

  5. [5]

    Think you have Solved Question Answering? Try ARC, the AI2 Reasoning Challenge

    Association for Computational Linguistics. doi: 10.18653/v1/N19-1300. URLhttps://aclanthology.org/N19-1300/. Peter Clark, Isaac Cowhey, Oren Etzioni, Tushar Khot, Ashish Sabharwal, Carissa Schoenick, and Oyvind Tafjord. Think you have solved question answering? try arc, the ai2 reasoning challenge.arXiv preprint arXiv:1803.05457,

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.18653/v1/n19-1300

  6. [6]

    Spqr: A sparse-quantized rep- resentation for near-lossless llm weight compression.arXiv preprint arXiv:2306.03078, 2023

    Tim Dettmers, Ruslan Svirschevski, Vage Egiazarian, Denis Kuznedelev, Elias Frantar, Saleh Ashkboos, Alexander Borzunov, Torsten Hoefler, and Dan Alistarh. Spqr: A sparse-quantized representation for near-lossless llm weight compression.arXiv preprint arXiv:2306.03078,

    work page arXiv

  7. [7]

    The Llama 3 Herd of Models

    Abhimanyu Dubey, Abhinav Jauhri, Abhinav Pandey, Abhishek Kadian, Ahmad Al-Dahle, Aiesha Letman, Akhil Mathur, Alan Schelten, Amy Yang, Angela Fan, et al. The llama 3 herd of models.arXiv preprint arXiv:2407.21783,

    work page internal anchor Pith review Pith/arXiv arXiv

  8. [8]

    GPTQ: Accurate Post-Training Quantization for Generative Pre-trained Transformers

    EliasFrantar, SalehAshkboos, TorstenHoefler, andDanAlistarh. Gptq: Accuratepost-trainingquantization for generative pre-trained transformers.arXiv preprint arXiv:2210.17323,

    work page internal anchor Pith review Pith/arXiv arXiv

  9. [9]

    Gemini: A Family of Highly Capable Multimodal Models

    doi: 10.5281/zenodo.10256836. 11 Gemini Team Google. Gemini: a family of highly capable multimodal models.arXiv preprint arXiv:2312.11805,

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.5281/zenodo.10256836

  10. [10]

    Deep Compression: Compressing Deep Neural Networks with Pruning, Trained Quantization and Huffman Coding

    Song Han, Huizi Mao, and William J Dally. Deep compression: Compressing deep neural networks with pruning, trained quantization and huffman coding.arXiv preprint arXiv:1510.00149,

    work page internal anchor Pith review Pith/arXiv arXiv

  11. [11]

    A survey on recognizing textual entailment as an nlp evaluation.arXiv preprint arXiv:2010.03061,

    Adam Poliak. A survey on recognizing textual entailment as an nlp evaluation.arXiv preprint arXiv:2010.03061,

    work page arXiv 2010

  12. [12]

    Code Llama: Open Foundation Models for Code

    URLhttps://doi.org/10.1038/s41586-023-06924-6. Baptiste Roziere, Jonas Gehring, Fabian Gloeckle, Sten Sootla, Itai Gat, Xiaoqing Ellen Tan, Yossi Adi, Jingyu Liu, Romain Sauvestre, Tal Remez, et al. Code llama: Open foundation models for code.arXiv preprint arXiv:2308.12950,

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1038/s41586-023-06924-6

  13. [13]

    SocialIQA: Commonsense Reasoning about Social Interactions

    URLhttps://arxiv.org/abs/1904.09728. Mingjie Sun, Zhuang Liu, Anna Bair, and J. Zico Kolter. A simple and effective pruning approach for large language models,

    work page internal anchor Pith review arXiv 1904

  14. [14]

    arXiv preprint arXiv:2306.11695 , year=

    URLhttps://arxiv.org/abs/2306.11695. Guangxuan Xiao, Ji Lin, Mickael Seznec, Hao Wu, Julien Demouth, and Song Han. SmoothQuant: Accurate and efficient post-training quantization for large language models. InInternational Conference on Machine Learning,

    work page arXiv

  15. [15]

    Towards large reasoning models: A survey of reinforced reasoning with large language models

    Fengli Xu, Qianyue Hao, Zefang Zong, Jingwei Wang, Yunke Zhang, Jingyi Wang, Xiaochong Lan, Jiahui Gong, Tianjian Ouyang, Fanjin Meng, et al. Towards large reasoning models: A survey of reinforced reasoning with large language models.arXiv preprint arXiv:2501.09686,

    work page arXiv

  16. [16]

    HellaSwag: Can a Machine Really Finish Your Sentence?

    Association for Computational Linguistics. doi: 10.18653/v1/P19-1472. URLhttps: //aclanthology.org/P19-1472/. 13 Appendix A Integration ofADMM-Qwith equivalent transformations We spell out in more detail how equivalent transformations modify the local reconstruction problem solved byADMM-Q. In each case, the transformation rewrites the layer in an equival...

    work page doi:10.18653/v1/p19-1472 2023

  17. [17]

    We conclude that both{D(t)}∞ t=0 and{W (t)}∞ t=0 converge to a shared limit¯D, which completes the proof

    transfers to our setting without modification. We conclude that both{D(t)}∞ t=0 and{W (t)}∞ t=0 converge to a shared limit¯D, which completes the proof. D Additional Experimental Details Computing environments.All experiments were conducted on a computing cluster. Unless otherwise specified, we utilized an Intel Xeon Gold 6248 machine with 16 CPU cores an...

    work page arXiv 2017