arxiv: 2602.05902 · v2 · submitted 2026-02-05 · 💻 cs.LG · cs.AI

Recognition: 1 theorem link

· Lean Theorem

CoreQ: Learning-Free Mismatch Correction and Successive Rounding for Quantization

Seohyeon Cha , Huancheng Chen , Dongjun Kim , Haoran Zhang , Kevin Chan , Gustavo de Veciana , Haris Vikalo

Authors on Pith no claims yet

Pith reviewed 2026-05-16 06:49 UTC · model grok-4.3

classification 💻 cs.LG cs.AI

keywords post-training quantizationlarge language modelsmismatch correctionsuccessive roundingquantization erroractivation mismatchtriangular least squares

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The pith

A closed-form geometric coefficient corrects quantization mismatch across layers without retraining or tuning

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

Post-training quantization maps pretrained LLM weights to low bits using a small calibration set, but sequential layer processing creates a mismatch: errors from early quantized layers change the inputs seen by later layers. CoreQ derives a per-layer scaling coefficient directly from a geometric breakdown of this mismatch and applies it to adjust each layer's calibration target. The adjusted targets are then quantized by solving a triangular least-squares problem with a greedy successive-rounding algorithm, plus an optional bounded beam-search variant. This removes the need for learned hyperparameters or fixed scaling factors while still compensating for error propagation.

Core claim

CoreQ derives a closed-form coefficient from the geometric decomposition of the activation mismatch and uses it to adaptively correct each layer's calibration target, after which the induced triangular least-squares objective is minimized by an efficient greedy successive-rounding solver.

What carries the argument

Closed-form mismatch correction coefficient obtained from geometric decomposition of the activation error

If this is right

The method improves perplexity and downstream accuracy over strong PTQ baselines across LLM families, scales, bit-widths, and quantization settings.
No hyperparameter tuning is required because the coefficient adapts automatically to each layer.
The K-CoreQ beam-search extension trades modest extra compute for further gains while remaining learning-free.
The triangular least-squares formulation is solved exactly by the greedy successive-rounding procedure.

Where Pith is reading between the lines

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

Similar closed-form geometric corrections could be tested on other sequential compression operations such as pruning or low-rank adaptation.
If the geometric assumption holds for activation mismatches, the same coefficient derivation might apply to calibration objectives in other layer-wise optimization settings.
The absence of learned parameters suggests the approach may maintain performance when calibration data is extremely limited or drawn from a different distribution than the test set.

Load-bearing premise

The geometric decomposition of the mismatch produces a coefficient that compensates for error propagation without introducing bias or overfitting to the finite calibration set.

What would settle it

If applying the derived coefficient to adjust calibration targets produces no improvement or degrades perplexity and accuracy relative to uncorrected PTQ baselines on the same models and calibration data, the central claim is false.

Figures

Figures reproduced from arXiv: 2602.05902 by Dongjun Kim, Gustavo de Veciana, Haoran Zhang, Haris Vikalo, Huancheng Chen, Kevin Chan, Seohyeon Cha.

**Figure 1.** Figure 1: Mean activation error (MAE) |Xf − Xq| across transformer layers of L2-7B on C4, evaluated on the calibration set (left) and a validation set (right) under greedy layer-wise rounding. Although α = 1 achieves the lowest calibration error, intermediate values of α reduce error on the validation set. the corresponding full-precision activations Xl f , and this representation mismatch may propagate to later la… view at source ↗

**Figure 3.** Figure 3: Mean Wiki2 perplexity (± std. over 5 seeds) for a fixed α = E[α] vs. sampled α. Sampling reduces variability across runs and achieves equal or lower mean perplexity across all models. 3.2.1. CLOSED-FORM UPDATE For a fixed quantized candidate Wˆ , L(W , α ˆ ) depends on α only through the α-weighted activations Xα, and admits a closed-form minimizer. Let U := W(Xf − Xq). Then the minimizer of α7→L(W , α ˆ )… view at source ↗

**Figure 4.** Figure 4: Overview of K-SNRQ beam search. Columns of the weight matrix are quantized sequentially from j = n to 1, maintaining a beam of size K. At each step, candidate extensions are scored using the incremental loss and pruned to the top-K beams. 4.2.1. GREEDY ROUNDING: SNRQ At level j, given the suffix qj+1:n, the greedy rounding is the exact minimizer of the conditional discrete subproblem induced by (19), q ⋆ … view at source ↗

**Figure 5.** Figure 5: Performance-efficiency trade-offs of K-SNRQ for 3-bit L2-7B. Wiki2 and C4 perplexity versus quantization time (left) and peak GPU memory (right) as beam width K increases. 1000 2000 3000 4000 5000 6000 Search time (ms) 7.84 7.86 7.88 C4 Perplexity ( ) K=1 K=2 K=4 K=6 p=0 p=1 p=2 p=3 p=4 Beam vs cyclic CD K-Beam CD p-pass (K=1) Increase beam Add CD pass 0.0 0.1 0.2 0.3 0.4 PPL / second 0.372 0.004 0.074 0.0… view at source ↗

**Figure 6.** Figure 6: (Left) C4 perplexity vs. search time for increasing beam width K and additional CD passes, with ±1 standard deviation shaded. (Right) Mean marginal perplexity improvement per second for beam expansion versus additional CD passes. memory grows due to the expanded beam state. Overall, beam search closes a substantial fraction of the remaining quality gap, suggesting that residual error is often caused by dec… view at source ↗

**Figure 7.** Figure 7: Robustness to sampling parameter λ in SNRQ-S [PITH_FULL_IMAGE:figures/full_fig_p021_7.png] view at source ↗

**Figure 8.** Figure 8: Mean activation error (MAE) measured on (top) calibration and (bottom) validation set. competitive with strong Hessian-based baselines. In particular, under 3-bit quantization, SNRQ-S achieves the best perplexity on both Wiki and C4, and under 4-bit quantization both SNRQ variants match or slightly improve upon GPTQ/LDLQ while maintaining comparable quantization time. These results suggest that the propose… view at source ↗

**Figure 9.** Figure 9: Performance–efficiency trade-offs of beam search for 3-bit quantized L2-13B. Wiki2 (top) and C4 (bottom) perplexity are shown as functions of quantization time (left) and peak GPU memory (right). 24 [PITH_FULL_IMAGE:figures/full_fig_p024_9.png] view at source ↗

read the original abstract

Post-training quantization (PTQ) enables efficient deployment of large language models by mapping pretrained weights to low-bit formats without retraining, typically using a small calibration set to minimize a layer-wise calibration objective. However, this sequential procedure induces a mismatch: errors from earlier quantized layers alter the inputs received by later layers, causing the activations to deviate from those of the full-precision model. Recent approaches introduce mismatch-aware calibration objectives to compensate for this effect, but leave open how much of the observed mismatch should shift each layer's calibration target. Fully applying this correction can overfit limited calibration data, while scaling the mismatch correction with a fixed coefficient ignores varying reliability of mismatch estimates across layers. To address these limitations, we propose CoreQ, a learning-free PTQ framework that applies a closed-form coefficient for mismatch correction derived from a geometric decomposition of the mismatch. The resulting coefficient adapts the correction across layers, reduces overfitting to finite calibration data, and requires no hyperparameter tuning. Given the corrected target, CoreQ minimizes the induced triangular least-squares objective with an efficient greedy successive-rounding solver and a bounded beam-search extension, K-CoreQ, that trades modest additional compute for improved performance. Across multiple LLM families, scales, bit-widths, and quantization settings, CoreQ improves perplexity and downstream accuracy over strong PTQ baselines.

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.

Desk Editor's Note private letter to a colleague

CoreQ gives a closed-form geometric coefficient for PTQ mismatch correction plus a successive-rounding solver, with reported gains over baselines but open questions on how well the decomposition handles non-linear layers.

read the letter

CoreQ derives a closed-form coefficient from geometric decomposition to adjust the calibration target for mismatch between quantized layers, then minimizes the resulting objective with a greedy successive-rounding procedure and a bounded beam-search variant called K-CoreQ. The coefficient is meant to adapt per layer without extra parameters or tuning, which directly addresses the overfitting risk of full mismatch-aware methods and the under-correction of fixed scaling approaches. The experiments claim consistent improvements in perplexity and downstream accuracy across multiple LLM families, scales, bit-widths, and settings, which is the main supporting evidence so far. The learning-free framing and the efficient solver are the clearest technical steps forward relative to prior PTQ work. The geometric derivation itself looks clean on paper and avoids fitting to the calibration set, which keeps the circularity burden low. The soft spot is the assumption that the decomposition produces an unbiased correction even though mismatch propagates through ReLU, softmax, and LayerNorm. Those operations break the constant-Jacobian premise, so the closed-form coefficient could under-correct in deeper blocks; the paper's empirical results may still hold, but the derivation needs explicit checking against that reality rather than relying on the linear case. This work is aimed at people doing practical post-training quantization for LLM deployment. A reader already running PTQ baselines would get immediate value from the method and the reported numbers. It is coherent enough on its own terms to deserve a serious referee, even if the non-linearity concern requires attention in revision.

Referee Report

1 major / 2 minor

Summary. The manuscript proposes CoreQ, a learning-free post-training quantization (PTQ) framework for large language models. It derives a closed-form coefficient for mismatch correction from a geometric decomposition of the quantization-induced input deviation, uses this to adaptively adjust layer-wise calibration targets, and minimizes the resulting triangular least-squares objective via a greedy successive-rounding solver (with an optional bounded beam-search extension K-CoreQ). Experiments report improved perplexity and downstream accuracy over strong PTQ baselines across LLM families, scales, bit-widths, and settings.

Significance. If the geometric decomposition produces an unbiased closed-form coefficient that reliably compensates mismatch propagation without overfitting the calibration set, CoreQ would represent a meaningful advance in parameter-free PTQ by eliminating hyperparameter tuning and providing an efficient solver for the induced objective. The approach directly targets a core limitation of sequential layer-wise PTQ.

major comments (1)

[§3 (geometric decomposition and coefficient derivation)] The central derivation assumes that geometric decomposition of the mismatch yields a closed-form coefficient that compensates error propagation. However, quantization mismatch propagates through non-linear activations (ReLU, softmax, LayerNorm) whose Jacobians are state-dependent and non-constant, violating the linear projection assumption used to obtain the coefficient. This risks under-correction in deeper blocks and residual input deviation that successive rounding cannot fully recover. The manuscript must explicitly state the linearity assumption and provide either a proof of robustness or empirical validation that the coefficient remains effective under realistic non-linear propagation.

minor comments (2)

[Abstract and §4] Clarify the precise definition of the 'triangular least-squares objective' and how the successive-rounding solver exploits its structure; the abstract introduces the term without sufficient context for readers unfamiliar with the formulation.
[§3.3] The abstract states that the coefficient 'adapts the correction across layers' and 'requires no hyperparameter tuning'; confirm that no implicit scaling or layer-specific thresholds are introduced in the implementation details.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. We address the major comment regarding the linearity assumption in the geometric decomposition below, and we are prepared to revise the manuscript accordingly.

read point-by-point responses

Referee: [§3 (geometric decomposition and coefficient derivation)] The central derivation assumes that geometric decomposition of the mismatch yields a closed-form coefficient that compensates error propagation. However, quantization mismatch propagates through non-linear activations (ReLU, softmax, LayerNorm) whose Jacobians are state-dependent and non-constant, violating the linear projection assumption used to obtain the coefficient. This risks under-correction in deeper blocks and residual input deviation that successive rounding cannot fully recover. The manuscript must explicitly state the linearity assumption and provide either a proof of robustness or empirical validation that the coefficient remains effective under realistic non-linear propagation.

Authors: We agree that the derivation in §3 employs a first-order linear approximation by modeling the effective mismatch propagation as a geometric projection onto the calibration direction. This yields the closed-form coefficient without requiring learned parameters. While non-linear activations introduce state-dependent effects that are not captured exactly, the coefficient is intended to compensate the dominant linear component of the input deviation. Our experiments across diverse LLM families, scales, and bit-widths show consistent gains in perplexity and downstream accuracy, indicating that the approximation remains effective in practice and that residual deviations are further mitigated by the successive-rounding solver. In the revised manuscript we will explicitly state the linearity assumption in §3, add a short discussion of its scope, and include additional empirical validation (e.g., layer-wise mismatch reduction plots) demonstrating robustness under realistic non-linear propagation. A full analytic proof is intractable given the composition of non-linearities, but the provided empirical evidence supports the coefficient’s utility. revision: partial

Circularity Check

0 steps flagged

No significant circularity: closed-form geometric coefficient and independent solver

full rationale

The paper's central derivation applies a closed-form coefficient obtained from geometric decomposition of the mismatch, presented as learning-free and without reference to fitted parameters or self-citation chains. The successive-rounding solver then minimizes the resulting triangular least-squares objective independently of the coefficient derivation. No equation or claim reduces by construction to its own inputs; the coefficient adapts across layers via the decomposition rather than being defined in terms of the final output or calibrated to the target result. This satisfies the self-contained benchmark with external falsifiability via perplexity and accuracy metrics.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that mismatch can be decomposed geometrically into a usable closed-form per-layer coefficient without introducing free parameters or new entities.

axioms (1)

domain assumption Mismatch between quantized and full-precision activations admits a geometric decomposition that produces an adaptive closed-form correction coefficient.
This assumption enables the learning-free property and is invoked to justify the coefficient derivation.

pith-pipeline@v0.9.0 · 5557 in / 1197 out tokens · 36339 ms · 2026-05-16T06:49:20.266163+00:00 · methodology

discussion (0)

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the interpolated proxy L(Ŵ;α) admits the decomposition L(Ŵ;α) ≡ α Lasym(Ŵ) + (1-α) Lsym(Ŵ)

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Saliency-Aware Regularized Quantization Calibration for Large Language Models
cs.AI 2026-05 unverdicted novelty 6.0

SARQC augments standard PTQ calibration with a saliency-aware regularizer to keep quantized weights closer to original floating-point values, yielding improved perplexity and zero-shot accuracy on dense and MoE LLMs.
Saliency-Aware Regularized Quantization Calibration for Large Language Models
cs.AI 2026-05 unverdicted novelty 4.0

SARQC augments standard PTQ calibration with a saliency-aware regularization term that reduces generalization risk and yields better perplexity and zero-shot accuracy on dense and MoE LLMs.

Reference graph

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11 Regularized Calibration with Successive Rounding for Post-Training Quantization A. Algorithms We provide pseudocode for SNRQ and K-SNRQ, together with practical implementation details that bridge the objective reformulation and search procedures used in our codebase. Specifically, Algorithm 1 presents the base SNRQ algorithm, while Algorithm 2 introduc...

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Remarks.Because the tail minimization (error feedback) solution to a least-squares problem depends on the target through correlations with Xq:,:, replacing Tq by rq changes the continuous optimum ∆W ⋆ :,q: in general, and therefore can change the discrete greedy choice for the quantized column ˆW:,q as well. If GPTAQ didnotapply the surrogate replacement ...

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Additional Results C.3.1

L2-7B L2-13B L3-8B α Wiki2 (↓) C4 (↓) Avg.Acc (↑) Wiki2 (↓) C4 (↓) Avg.Acc (↑) Wiki2 (↓) C4 (↓) Avg.Acc (↑) 0.0 6.62 8.13 66.48 5.537.1371.14 12.85 13.33 65.78 0.25 6.38 7.89 66.93 5.567.1270.33 9.49 11.96 69.53 0.5 6.37 7.86 66.96 5.62 7.16 70.73 8.48 11.87 70.53 0.75 6.43 7.96 66.87 5.79 7.29 69.30 8.94 12.65 68.26 1.0 6.52 8.05 66.55 5.72 7.23 70.45 11...

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We bold the best results, as well as those whose value falls within the top score±standard deviation

3-bit quantization results with incoherence processing (Chee et al., 2023; Tseng et al., 2024). We bold the best results, as well as those whose value falls within the top score±standard deviation. Model Method Wiki2 (↓) C4 (↓) Avg.Acc (↑)Q.Time (s) L3-8B QuIP 8.63 11.80 67.60 763.8 GPTAQ 9.07 12.15 65.77 715.0 SNRQ-C 8.74 11.87 67.60 594.4 SNRQ-S 8.55 11...

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We follow the experimental protocol of Li et al

with 22M and 86M parameters, respectively. We follow the experimental protocol of Li et al. (2025): 128 images are sampled from the ImageNet training set and used for calibration. We apply weight clipping based on mean squared error and adopt asymmetric quantization grids for both weights and activations. Quantization time is measured on a single NVIDIA R...

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[27]

Average extra GPU time (ms) per module in L2-7B from (i) obtaining closed-form α in SNRQ-C and (ii) sampling in SNRQ-S. Method attn.k-proj attn.v-proj attn.q-proj attn.o-proj mlp.up-proj mlp.gate-proj mlp.down-projAvg SNRQ-C 17.22 10.46 10.46 10.32 27.07 27.02 85.21 26.82 SNRQ-S 10.04 0 0 9.99 10.15 0 36.75 16.73 Table 13.Beam width vs. coordinate-descent...

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[28]

2 3 4 5 6 7 8 Beam size K 7.09 GPTAQ Performance-Time Comparison 1500 2000 2500 Search Time (ms) 7.05 7.06 7.07C4 Perplexity SNRQ Least Square Fit ±1 band SNRQ (K =

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