Task-Stratified Knowledge Scaling Laws for Post-Training Quantized Large Language Models

Bohan Yu; Chenxi Zhou; Jiang Li; Jinyu Ye; Jun Zhao; Kang Liu; Pengfei Cao

arxiv: 2508.18609 · v4 · submitted 2025-08-26 · 💻 cs.CL · cs.AI· cs.LG

Task-Stratified Knowledge Scaling Laws for Post-Training Quantized Large Language Models

Chenxi Zhou , Pengfei Cao , Jiang Li , Bohan Yu , Jinyu Ye , Jun Zhao , Kang Liu This is my paper

Pith reviewed 2026-05-18 21:55 UTC · model grok-4.3

classification 💻 cs.CL cs.AIcs.LG

keywords post-training quantizationscaling lawslarge language modelsknowledge capabilitiesmemorizationreasoningquantization sensitivitybit-width

0 comments

The pith

Dividing LLM capabilities into memorization, application, and reasoning reveals that post-training quantization affects each in distinct ways.

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

The paper develops scaling laws for quantized large language models that separate capabilities into memorization, application, and reasoning. It combines model size, bit-width, group size, and calibration set size into one framework and tests it on hundreds of configurations. A reader would care because the results show reasoning demands high precision, application tracks model scale, and memorization depends on calibration quality. This separation explains why uniform quantization often fails and points to targeted fixes that keep useful knowledge intact while cutting compute costs.

Core claim

The authors establish Task-Stratified Knowledge Scaling Laws that unify model size, bit-width, group size, and calibration set size. Validated on 293 PTQ configurations, the laws exhibit strong fit and cross-architecture consistency. They show reasoning is precision-critical, application is scale-responsive, and memorization is calibration-sensitive. In low-bit settings, optimizing the fine-grained factors prevents performance collapse.

What carries the argument

Task-Stratified Knowledge Scaling Laws, which separate capabilities into memorization, application, and reasoning to capture their differing responses to quantization parameters.

If this is right

In low-bit regimes, increasing calibration set size can protect memorization performance without raising bit-width.
Reasoning tasks require higher precision to avoid sharp drops, while application tasks tolerate lower bits if model scale is increased.
Designers can choose group size and calibration settings per capability rather than applying one setting to the whole model.
Uniform low-bit quantization risks collapse mainly in reasoning, so selective precision allocation becomes viable.

Where Pith is reading between the lines

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

The same capability split could be tested on pruning or distillation to see if sensitivities remain consistent across compression methods.
Hardware schedulers might allocate higher precision only to reasoning-heavy layers based on these sensitivities.
The framework suggests general scaling laws for LLMs may need similar stratification when efficiency techniques are involved.

Load-bearing premise

That splitting capabilities into memorization, application, and reasoning accurately captures how quantization changes performance across tasks.

What would settle it

Applying the derived scaling laws to a new collection of PTQ configurations on unseen model architectures and finding that the predicted performance curves deviate substantially from measured results.

read the original abstract

Post-Training Quantization (PTQ) is a critical strategy for efficient Large Language Models (LLMs) deployment. However, existing scaling laws primarily focus on general performance, overlooking crucial fine-grained factors and how quantization differentially impacts diverse knowledge capabilities. To address this, we establish Task-Stratified Knowledge Scaling Laws. By stratifying capabilities into memorization, application, and reasoning, we develop a framework that unifies model size, bit-width, and fine-grained factors: group size and calibration set size. Validated on 293 diverse PTQ configurations, our framework demonstrates strong fit and cross-architecture consistency. It reveals distinct sensitivities across knowledge capabilities: reasoning is precision-critical, application is scale-responsive, and memorization is calibration-sensitive. We highlight that in low-bit scenarios, optimizing these fine-grained factors is essential for preventing performance collapse. These findings provide an empirically-backed foundation for designing knowledge-aware quantization strategies.

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

The paper delivers a solid set of empirical scaling laws that split quantization effects by capability type, but the three-way split into memorization/application/reasoning needs checks against other groupings.

read the letter

The main thing to know is that this work fits scaling laws for post-training quantization that treat memorization, application, and reasoning as separate regimes, then folds in group size and calibration set size alongside model size and bit width. They back it with 293 PTQ runs that hold up across architectures and show reasoning dropping hardest on low precision, application tracking scale more, and memorization reacting to calibration data. That gives a practical handle for low-bit deployment that earlier general scaling laws did not separate out this way. The volume of configurations and the cross-architecture consistency are the strongest parts; they make the reported patterns worth taking seriously for anyone tuning quantization knobs. The advice that fine-grained factors matter most in low-bit regimes follows directly from the fits and is easy to test in practice. The soft spot is the capability split itself. The paper assigns tasks to the three buckets but does not show what happens under different groupings or control for obvious confounds such as output length and dataset size. Without those checks the distinct sensitivities could partly reflect benchmark surface features rather than deeper capability differences. The laws are also empirical fits to the same data used to demonstrate the patterns, which is normal for this style of work but limits how strongly one can claim they reveal intrinsic sensitivities. This is aimed at people who already work on efficient LLM inference and want concrete guidance on where to spend effort when quantizing. A reader who runs quantization experiments or builds scaling-law style models would find the numbers and the unified form useful to build on. It has enough experimental reach and a clear enough question to deserve peer review; referees can ask for the grouping ablations and still get value from the rest.

Referee Report

2 major / 2 minor

Summary. The paper proposes Task-Stratified Knowledge Scaling Laws for post-training quantization (PTQ) of LLMs. It stratifies capabilities into memorization, application, and reasoning to create a unified framework incorporating model size, bit-width, and fine-grained factors (group size and calibration set size). Validated on 293 PTQ configurations with reported cross-architecture consistency, the work claims distinct sensitivities: reasoning is precision-critical, application is scale-responsive, and memorization is calibration-sensitive. It emphasizes optimizing fine-grained factors in low-bit regimes to avoid performance collapse.

Significance. If the central claims hold, the work offers a practical, empirically-backed foundation for knowledge-aware PTQ strategies that go beyond aggregate performance metrics. The scale of validation (293 configurations) and cross-architecture consistency are notable strengths that support generalization. The framework could inform deployment decisions where different capabilities matter differently.

major comments (2)

[Task stratification / capability partitioning] Task stratification section: the assignment of benchmarks to memorization (e.g., MMLU subsets), application, and reasoning (e.g., GSM8K) is presented without inter-rater reliability checks, ablation on alternative groupings, or controls for confounds such as output length and dataset size. This partitioning is load-bearing for the central claim of distinct sensitivities, as the reported differential impacts could reflect surface features of the benchmarks rather than genuine capability differences under quantization.
[Scaling law equations / fitting procedure] Unified scaling law formulation (likely §4 or Eq. defining the stratified law): the exponents and coefficients are free parameters fitted to the same 293 configurations used to validate the sensitivities and cross-architecture consistency. This introduces circularity that weakens the claim that the framework 'unifies' and 'reveals' the sensitivities independently of the fitting data.

minor comments (2)

[Experimental details / results tables] The manuscript should report exact fitting procedures, error bars or confidence intervals on the scaling law parameters, and any post-hoc selection criteria for the 293 configurations to allow full assessment of quantitative robustness.
[Figures] Figure clarity: ensure that plots separating the three capability strata clearly label the axes and include the fitted curves with uncertainty bands for direct visual evaluation of the claimed sensitivities.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed feedback on our manuscript. We address each major comment point by point below, indicating the revisions we will make to strengthen the work.

read point-by-point responses

Referee: [Task stratification / capability partitioning] Task stratification section: the assignment of benchmarks to memorization (e.g., MMLU subsets), application, and reasoning (e.g., GSM8K) is presented without inter-rater reliability checks, ablation on alternative groupings, or controls for confounds such as output length and dataset size. This partitioning is load-bearing for the central claim of distinct sensitivities, as the reported differential impacts could reflect surface features of the benchmarks rather than genuine capability differences under quantization.

Authors: We appreciate the referee's concern regarding the robustness of the task stratification. The benchmark assignments follow conventions established in prior LLM evaluation literature, with MMLU subsets typically linked to knowledge recall and GSM8K to step-by-step reasoning. We acknowledge the absence of formal inter-rater checks, ablations on alternative groupings, and explicit controls for confounds such as output length or dataset size. In the revised manuscript, we will expand the task stratification section to include a detailed rationale for the groupings, an ablation study on alternative partitions, and a discussion of potential surface-feature confounds. These additions will help substantiate that the observed differential sensitivities reflect capability differences rather than benchmark artifacts. revision: yes
Referee: [Scaling law equations / fitting procedure] Unified scaling law formulation (likely §4 or Eq. defining the stratified law): the exponents and coefficients are free parameters fitted to the same 293 configurations used to validate the sensitivities and cross-architecture consistency. This introduces circularity that weakens the claim that the framework 'unifies' and 'reveals' the sensitivities independently of the fitting data.

Authors: We thank the referee for identifying this potential circularity in the fitting procedure. As is standard in scaling-law research, the functional form is fitted to the full set of 293 configurations to derive exponents and coefficients, which are then interpreted to reveal capability-specific sensitivities; the same data also supports fit-quality and cross-architecture analyses. We agree that greater clarity is needed to separate derivation from validation. In the revision, we will add explicit discussion of the fitting methodology, report separate goodness-of-fit statistics, and clarify that the stratified law unifies the factors through its functional form while the sensitivities emerge from the fitted parameters. This addresses the independence concern without altering the empirical approach. revision: partial

Circularity Check

0 steps flagged

No significant circularity in empirical task-stratified scaling laws

full rationale

The paper performs experiments across 293 PTQ configurations, stratifies tasks into memorization/application/reasoning, fits scaling laws to the resulting performance data as functions of model size, bit-width, group size and calibration set size, and reports observed sensitivities from those fits. This constitutes a standard data-driven empirical derivation with no quoted self-definitional loops, fitted parameters renamed as independent predictions, or load-bearing self-citations that reduce the central claims to unverified inputs. The framework is validated against the collected data rather than presupposing its own outputs, making the derivation self-contained.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claims rest on empirical power-law style relationships fitted to PTQ performance data; the stratification into three capability types is introduced without independent justification beyond the observed patterns.

free parameters (1)

exponents and coefficients in the unified scaling law
Fitted to match observed performance across model sizes, bit-widths, group sizes, and calibration sizes.

axioms (1)

domain assumption Power-law relationships adequately describe the interaction of model size, bit-width, and fine-grained quantization factors.
Invoked when constructing the Task-Stratified Knowledge Scaling Laws.

pith-pipeline@v0.9.0 · 5705 in / 1271 out tokens · 34022 ms · 2026-05-18T21:55:52.781171+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Acctask ≈ Ctask × N^αtask × [log2(Cb)]^βtask × G^γtask × [log2(B_eff)]^δtask

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.