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Predicting Quantization Price for Selecting PTQ Configurations Before Deployment

Junbin Qiu, Jian Mu, Weitong Zhang, Yao Shu

arXiv:2609.28270v1cs.CLcs.LG

TL;DR

Weight-space PTQ must rank configurations before a completed quantized model reveals its output drift. The paper derives a covariance-and-curvature price from forward KL and uses it with deployment cost to select configurations. In controlled perturbations, the realized and trace-reduced prices closely align with measured KL drift.

  • Problem

    Weight-space PTQ must choose formats, granularities, quantizer families, transformations, and bits before the completed model reveals output drift, while existing approaches score pieces of degradation separately.

    Source: §1 · p. 1

  • Method

    The method derives a quadratic price for candidate-induced layer-output error covariance from forward KL, using reference-model curvature and deployment cost to guide a budgeted configuration search.

    Source: §1 · p. 2, §3.3 · p. 7

  • Results

    Realized price tracks measured KL drift with ρ = 0.9470 on OPT-125M and ρ = 0.9249 on Qwen3-0.6B; trace-reduced price aligns with ρ = 0.9483 on OPT-125M and ρ = 0.9438 on Qwen3-0.6B.

    Source: §4.1 · pp. 7–8

  • Takeaways & Limitations

    Finite formats, codebooks, granularities, transformations, and fixed-geometry bit allocation can be compared as deployable configurations through their induced covariances and costs.

    Source: §1 · p. 2, §6 · p. 9

    The primary selector is scoped to weight-space PTQ; other quantization settings enter its price language only when exposed as pre-deployment output-error covariance and deployment cost.

    Source: §5 · p. 9

Abstract

from arXiv · show

Weight-space post-training quantization (PTQ) must choose finite formats, granularities, quantizer families, transformations, and bits before the completed quantized model reveals its output-distribution drift. Existing PTQ methods predict important pieces of this degradation, including reconstruction error, Hessian sensitivity, transformation effects, and downstream loss, but these pieces are usually scored after fixing the quantization geometry or inside separate configuration families. We formulate weight-space PTQ as pre-deployment configuration selection using priced layer-output error. Each admissible layer configuration is treated as an error generator with a deployment cost, which induces a layer-output error covariance $\boldsymbol{\Sigma}_l(\alpha_l)$, and the full-precision model prices that covariance by downstream curvature, $\widehat{\rho}_l(\alpha_l)=\frac{1}{2}\operatorname{Tr}\left(\widehat{\mathbf{H}}_l\,\widehat{\boldsymbol{\Sigma}}_l(\alpha_l)\right)$. The price follows from full-precision-to-quantized forward KL, whose first-order term cancels at the reference model. It turns reconstruction and diagonal scores into reduced proxies that drop price factors, while finite formats, codebooks, granularities, and equivalent transformations become comparable candidates through the covariances they induce and the costs they pay. A trace reduction then yields a calibration-time price table and a budgeted price-guided selector, making fixed-geometry bit allocation a special case rather than the organizing problem.

1 Introduction

Weight-space PTQ must rank deployable layer configurations before the completed model reveals its output drift. The paper makes a curvature-priced layer-output error covariance the common basis for comparing those choices under deployment costs.

1 Introduction

  • PTQ must choose formats, granularities, quantizer families, transformations, and bitwidths before the completed model reveals its output-distribution drift.Exhaustively building every candidate is usually impossible, while fixing geometry first misses a central configuration choice.Source: §1 · p. 1
  • Existing methods predict pieces of PTQ degradation, but a common pre-deployment price comparing configurations across quantizers, transformations, and hardware costs is missing.Prior approaches include reconstruction and second-order methods, sensitivity and allocation signals, transformation methods, and downstream-loss objectives.Source: §1 · p. 1
  • The paper prices each candidate-induced layer-output error covariance using downstream curvature from the full-precision reference, deriving the score from forward KL.At the reference model, score cancellation removes the linear term; reconstruction and diagonal scores are reduced proxies that preserve rankings only when dropped factors are effectively constant.Source: §1 · p. 2
  • Finite formats, bits, granularities, codebooks, and equivalent transformations become candidates in a budgeted search through their induced covariances and deployment costs.Fixed-geometry bit allocation is a special case after transformation, quantizer family, and granularity have been held fixed.Source: §1 · p. 2

2 Pre-Deployment Configuration Selection in Weight-Space PTQ

The formulation treats each layer choice as an admissible configuration applied to a full-precision reference and calibration distribution. It spans quantization and transformation choices while recording deployment cost separately from accuracy degradation.

2.1 Reference Model, Candidate Coordinates, and Target Drift

  • Each layer configuration specifies pre-deployment choices such as bitwidth, granularity, quantizer family, format, or transformation against a full-precision reference and calibration inputs.The target is the completed model’s output-distribution drift, but evaluating it requires building that model, motivating a calibration-time ranking rule.Source: §2.1 · pp. 2–3

2.2 Admissible PTQ Configuration Families and Costs

  • Candidate families change the finite-format replacement or coordinate while preserving the floating-point layer; granularities, codebooks, and transformations alter the perturbation being considered.Equivalent transformations move quantization difficulty between weight and input coordinates.Source: §2.2 · p. 3
  • Deployment cost combines bit, metadata, and kernel costs, and the backend-supported candidates must fit the deployment budget.The cost records deployability rather than accuracy degradation; groups and codebooks can add metadata or lookup overhead, while transformations may require online kernels.Source: §2.2 · pp. 3–4

3 Priced Layer-Output Error for Configuration Selection

The section derives a calibration-time price for layer-output error from forward KL, then uses candidate-induced covariances and deployment costs to select PTQ configurations before completed models are built.

3.1 Forward KL Induces the Layer-Output Error Price

  • At the full-precision reference, forward KL’s linear term vanishes, leaving a quadratic price that combines candidate-induced layer-output error covariance with output-side curvature.The price approximates model-level drift using layer statistics; a PSD Gauss-Newton surrogate may replace the curvature in implementation.Source: §3.1 · p. 4
  • Reconstruction drops downstream output curvature, while diagonal input-statistic scores also drop cross-channel covariance; rankings persist only when omitted factors are effectively constant across candidates.Otherwise, reliable selection keeps the full price factors explicit.Source: §3.1 · p. 5

3.2 From Candidate Families to Price Factors

  • Finite formats, bits, granularities, and codebooks determine perturbation moments, while equivalent transformations change their generating coordinates and input covariance; candidates also carry deployment costs.The reference-side curvature remains fixed across these choices.Source: §3.2 · p. 5
  • Under determinant normalization, whitening minimizes worst-direction input variance, but practical transformations balance input-metric flattening against the transformed-weight quantization residual.The transformation changes both the input covariance and the finite-format residual.Source: §3.2 · pp. 5–6
  • When candidate-side covariance factors are nearly isotropic, the price reduces to two candidate-side scalars and a cached reference-side curvature trace, without replacing curvature by the identity.The reduction is a scoring shortcut, not a configuration class.Source: §3.2 · p. 6

3.3 Budgeted Price-Guided Configuration Search

  • The selector estimates candidate scales and costs, caches a reference-curvature trace, and minimizes predicted configuration price subject to the deployment budget.This calibration-time table replaces exhaustive completed-model trials.Source: §3.3 · pp. 6–7

4 Experiments

The experiments test whether calibration-time prices track full-precision-to-quantized KL drift and whether price-guided selection improves configuration decisions. They report alignment across controlled perturbations and competitive outcomes across bit, transformation, and granularity choices.

4.1 Realized Price and Implemented-Trace-Price Align with KL Drift

  • Realized price tracks KL drift, with ρ = 0.9470 on OPT-125M and ρ = 0.9249 on Qwen3-0.6B; the two-layer price also tracks drift.The two-layer result supports the local linearized decomposition described in Thm. 3.1.Source: §4.1 · p. 7, Figure 1 · p. 8
  • Implemented trace price retains nearly the same KL-drift alignment, with ρ = 0.9483 on OPT-125M and ρ = 0.9438 on Qwen3-0.6B.This controlled comparison tests the trace reduction as a low-cost pre-deployment statistic.Source: §4.1 · pp. 7–8

4.2 Price-Guided Configuration Selection

  • The price-guided allocation gives the best average downstream task score among the three 3-bit methods, while AMQ has the lowest WikiText2 PPL and much larger allocation time.Compared with HIGGS, price-guided allocation is faster and stronger on both PPL and average task accuracy.Source: Table 1 · p. 8, §4.2 · pp. 8–9
  • Price-guided choices give the best PPL and average task score in both transformation selection and granularity allocation.The comparisons use CALM-CKA and random choice for transformations, and fixed group-128 and random choice for granularity.Source: §4.2 · p. 9

5 Related Work and Scope

The paper positions its selector at the configuration-allocation level, making geometry choices comparable through priced layer-output error while keeping the primary scope to weight-space PTQ.

5 Related Work and Scope

  • Unlike fixed-geometry allocation references, the formulation includes geometry itself in the pre-deployment configuration set and prices transformations, granularities, quantizer families, and bits with one downstream metric.The cited fixed-geometry references include HAWQ variants, HIGGS-style methods, and Q-Palette.Source: §5 · p. 9
  • Transformations become comparable before deployment through the priced layer-output error they induce, while end-loss methods are nearest in metric.The passage identifies GuidedQuant and YAQA as end-loss neighbors.Source: §5 · p. 9
  • The primary selector is weight-space PTQ; other quantization settings fit the same price language only when represented by pre-deployment output-error covariance and deployment cost.The local quadratic price assumes low-bit nonlocality, moment-model mismatch, cross-layer coupling, and curvature-proxy error remain secondary to the predicted price gap.Source: §5 · p. 9

6 Conclusion

The conclusion frames weight-space PTQ as pre-deployment selection over configurations priced by forward-KL curvature and candidate-induced error covariance. The extended related work distinguishes this configuration-level decision from neighboring PTQ ingredients and deployment constraints.

6 Conclusion

  • Forward-KL score cancellation leaves a quadratic quantization price: the reference model supplies downstream curvature and each candidate supplies its error covariance.A trace reduction provides a practical calibration-time score for the budgeted search.Source: §6 · p. 9

A Extended Related Work

  • The paper applies local second-order, reconstruction, and end-loss ingredients at a different decision level: predicting prices for candidate weight-space configurations under deployment constraints.The passage distinguishes this configuration-level use from methods targeting compensation, reconstruction refinements, or output-distribution preservation.Source: A Extended Related Work · p. 13
  • Outliers, rotations, affine transforms, and expanded perturbation sets matter in the pricing view through the weight perturbation distributions they induce; activation quantization remains a neighboring problem.The passage names sparse exceptions, codebooks, vector quantization, and trellis-coded quantization among the expanded choices.Source: A Extended Related Work · p. 13
  • Deployment methods remain discrete constrained selection problems after candidate weight formats receive predicted prices and deployment costs.The passage identifies the allocation contribution as reducing priced perturbations and transformation terms to coefficients such as cl, not claiming continuous KKT allocation is new.Source: A Extended Related Work · p. 13

B.1 Proof of Thm. 3.1

The proof develops the local quadratic price from forward-KL expansion and then gives finite-format and equivalent-transformation consequences for the covariance being priced.

B.1 Proof of Thm. 3.1

  • At the full-precision reference, Taylor expansion has no first-order term; the retained second-order block yields the layerwise price, while omitted terms form a relative diagnostic.The proof uses the reference-side Hessian and layer-output drift to construct the quadratic contribution.Source: §B.1 · p. 14

B.2 Finite-Format Moment Bound

  • For grouped scalar rounding, the finite-format moment bound uses a per-group variance of δ_g(α_l)^2/12, recovering the bit and granularity bound.The bound follows under the independent zero-mean moment model.Source: §B.2 · p. 15

B.3 Equivalent Transformation Covariance

  • An equivalent transformation preserves the floating-point output and output-side curvature, while moving the input-side metric and perturbation covariance.Vectorizing the quadratic form makes the transformed input covariance explicit as the factor P C P^T.Source: §B.3 · p. 15

B.4 Proof of the Affine Whitening Optimum

These diagnostics characterize when whitening and reduced PTQ scores are useful, and where their simplifications stop. They show that diagonal scaling cannot remove normalized input correlations, while curvature and covariance structure determine whether scalarized proxies retain the full price.

B.4 Proof of the Affine Whitening Optimum

  • B.4: Preserving the determinant, affine whitening makes PCP⊤=(det C)^(1/n)I and attains the lower bounds on the maximum eigenvalue and condition number.The stated optimizer form follows from the equality condition that all eigenvalues of PCP⊤ are equal.Source: §B.4 · p. 15

B.5 Whitening Normalizations and Scale Coupling

  • B.5: Whitening leaves a scale degree of freedom; its normalization affects the candidate coefficient, and adaptive scalar-quantizer distortion can partly offset that scale.For homogeneous range statistics, Rq can scale as 1/σ, but this cancellation is quantizer-model-dependent rather than universal.Source: §B.5 · pp. 15–16

B.6 Trace-Score Reduction

  • B.6: The trace reduction leaves a residual term when candidate noise is non-isotropic, so the scalarized score is not exact in that case.The cited bound uses trace duality.Source: §B.6 · p. 16

B.7 Diagonal Scaling Shows What Scaling Cannot Change

  • B.7: Diagonal scaling changes variances but leaves standardized correlations unchanged, so it cannot remove off-diagonal dependence in C.Source: §B.7 · p. 16

B.8 Rotation Coherence Bounds Scalar-Quantizer Distortion

  • B.8: Rotation leaves the input covariance spectrum unchanged; any improvement in the stated bound comes only through the scalar-quantizer distortion factor.Source: §B.8 · pp. 16–17

B.9 Curvature and Covariance Structure Diagnostics

  • B.9: Diagonal-input and imatrix-style scores are reduced proxies; when off-diagonal input covariance matters, the full covariance or transformed covariance must be priced.Diagonal dominance of output curvature does not make it a scalar identity, and imatrix-style weights omit both output curvature and cross-channel input covariance.Source: §B.9 · p. 17

B.10 Weight-Activation Extension and Reduced Allocation Laws

The appendix extends the output-price operator to a neighboring weight-activation branch and derives reduced allocation laws. The joint branch relies on noise assumptions, while the continuous bit-allocation law is only a surrogate for discrete budgeted selection.

B.10 Weight-Activation Extension and Reduced Allocation Laws

  • The weight-activation branch retains weight- and activation-side output-error terms under conditional zero-mean activation noise and an approximately zero cross moment.If the conditional zero-mean assumption fails, omitted first-order bias and cross moment remain validity diagnostics, particularly in low-bit or strongly clipped regimes.Source: §B.10 · pp. 17–18
  • Preprocessing can make the weight-side input metric more isotropic while changing activation-noise covariance and propagation; scaling can change activation price for shared-range or non-equivariant quantizers.For scale-equivariant per-channel activation quantization, the activation-side effect can cancel in the simplified surrogate.Source: §B.10 · p. 18
  • Under the continuous interior bit-budget constraint, the reduced allocation law equalizes optimized retained prices across layers.The law assumes positive retained scalar coefficients and continuous bit variables.Source: §B.10 · p. 18
  • For discrete hardware bitsets, unequal costs, KV-cache constraints, or active bounds, continuous bit allocation is a warm start or diagnostic, not a replacement for the discrete selector.Source: §B.10 · p. 18

C Price-Guided Configuration Selection on 3B and 8B Models

On Llama-3.2-3B and Llama-3.1-8B, price-guided selection performs consistently well for transformation and granularity choices, while bit-allocation results are less uniform. The experiments compare it with task-specific baselines under the stated configuration settings.

C Price-Guided Configuration Selection on 3B and 8B Models

  • The experiments use Llama-3.2-1B as the main-text model and apply the same selection protocol to Llama-3.2-3B and Llama-3.1-8B.The three blocks compare bit allocation with HIGGS and AMQ, transformation selection with CALM-CKA and random choice, and granularity allocation with fixed group-128 and random choice.Source: C Price-Guided Configuration Selection… · p. 18
  • Price-guided transformation selection gives the best PPL and average task score on both models; granularity selection gives the best PPL on both and best average score on Llama-3.1-8B.On Llama-3.2-3B, transformation selection improves over CALM-CKA from 8.81 to 8.58 PPL and from 58.58 to 58.96 average score; on Llama-3.1-8B, PPL changes from 6.97 to 6.86 and average score from 64.18 to 64.24. Random baselines are less reliable, especially in PPL.Source: C Price-Guided Configuration Selection… · p. 18, Table 2 · p. 19, Table 3 · p. 19
  • Bit-allocation results vary by model: price-guided selection has the best average downstream score on Llama-3.2-3B, while AMQ is stronger across the Llama-3.1-8B block.On Llama-3.2-3B, the price-guided average downstream score is 53.64, while AMQ has the best PPL; on Llama-3.1-8B, price-guided selection ranks second in average score among the three methods. AMQ takes much more time because it searches by building and evaluating candidate quantized models.Source: Table 3 · p. 19, C Price-Guided Configuration Selection… · p. 20

D.1 Baselines

  • Baselines span reconstruction- or search-based bit allocation, representation-similarity transformation selection, and fixed or random granularity choices.HIGGS uses a linearized reconstruction objective and AMQ uses discrete search; CALM-CKA selects by representation similarity, while granularity comparisons include fixed group-128 and random choice.Source: §D.1 · p. 20

D.2 Hyperparameters and Implementation Details

Implementation uses WikiText2 calibration data and specifies separate candidate sets and procedures for selection, followed by defined perplexity and downstream-accuracy evaluations.

D.2 Hyperparameters and Implementation Details

  • Calibration statistics for Llama models use 32 WikiText2 validation samples, sequence length 512, and seed 0; KL alignment uses 128 validation samples of length 1024.The Llama models quantize transformer linear projections except embeddings, normalization layers, rotary embeddings, and the language-model head, yielding 112, 196, and 224 quantized layers for the 1B, 3B, and 8B models.Source: §D.2 · p. 20
  • For Llama selection, downstream curvature is estimated with a Gauss–Newton Fisher trace from one forward and backward pass per layer, without Hessian-vector probes.The price-guided bit assignment uses a binary MILP with choices {2, 3, 4} bits, group size 128, and an AMQ-compatible budget of 3.0 ± 0.005; HIGGS and AMQ use the same bit set, group size, budget, and GPTQ backend.Source: §D.2 · pp. 20–21
  • Transformation selection fixes W4 quantization and group size 128, while granularity selection fixes W4, no transformation, uniform integer quantization, and min–max clipping.Transformation candidates include identity, scaling, Hadamard rotation, and normalized diagonal whitening; granularity candidates include per-tensor, per-channel, group sizes 32, 64, and 128, and 64 × 64 blocks. Granularity costs include FP32 scale metadata.Source: §D.2 · p. 21
  • PPL is evaluated on WikiText2 test data, and downstream accuracy is the arithmetic mean across five LightEval tasks.PPL uses full-text concatenation, sequence length 2048, 141 sequences, and 288,627 supervised tokens; downstream tasks use batch size 32 and greedy decoding.Source: §D.2 · p. 21
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