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Test-Time Adaptation of Quantized ViTs via Single-Pass Quantizer-Aligned Recalibration

Hyeongheon Cha, Young D. Kwon, Sung-Ju Lee

arXiv:2610.08358v1cs.CVcs.AI

TL;DR

Quantized ViTs lose accuracy under distribution shift, while existing test-time adaptation incurs costs poorly suited to quantized inference or misses frozen-quantizer code distortion. QuAR recalibrates activations at quantizer inputs in one pass without parameter updates. It leads backprop-free TTA on ImageNet-C across evaluated bit-widths and carries its lead across other settings.

  • Problem

    Under distribution shift, per-channel activations occupy frozen quantizers’ calibrated ranges differently, distorting code distributions, while existing TTA methods incur costly adaptation or act away from quantizer inputs.

    Source: §1 · pp. 1–2

  • Method

    QuAR applies parameter-free affine recalibration at frozen quantizer inputs, mapping running per-channel test statistics toward source calibration in one pass without updating model parameters.

    Source: §1 · p. 2

  • Results

    QuAR leads backprop-free TTA at W3A3, W4A4, W6A6, and W8A8 on ImageNet-C ViT-B, beating the strongest baseline by +2.28 at W8A8 and +4.00 at W3A3.

    Source: §1 · p. 2

  • Takeaways & Limitations

    One fixed configuration retains its lead across continual streams, non-i.i.d. label shift, seven OOD benchmarks, and three other backbones.

    Source: §1 · p. 2

    QuAR has not been implemented as an integer-kernel deployment, leaving latency, memory, and energy on such a stack to future work.

    Source: §5 · p. 10

Abstract

from arXiv · show

Post-training quantization is a standard route to fitting vision transformers (ViTs) into edge compute and memory budgets, yet quantized models become especially brittle under distribution shift. Test-time adaptation (TTA) addresses such shifts without labels, but most existing approaches are poorly aligned with the constraints of quantized inference. Prevailing TTA methods recover accuracy through backpropagation, while backprop-free methods often still incur overhead from extra forward passes or parameter updates, and lightweight feature- or logit-level methods recover only part of the loss. Across these approaches, a quantization-specific failure mode that amplifies the drop is not directly targeted: under shift, activations occupy frozen quantizers' calibrated ranges differently, distorting their code distribution. We propose Quantizer-Aligned Recalibration (QuAR), a single-pass TTA method tailored to quantized ViTs that neither backpropagates nor updates any model parameters. QuAR recalibrates activations at the input to a frozen quantizer, mapping the test stream's running per-channel statistics back toward the source calibration. On ImageNet-C with ViT-B, QuAR achieves the highest mean accuracy among state-of-the-art backprop-free TTA methods at 3-, 4-, 6- and 8-bit weight/activation precision, outperforming the strongest baseline by 2.28 points at 8 bits and 4.00 at 3 bits, with 46% lower latency and a memory overhead of only 0.17 MB (0.01% of peak inference memory). Analysis and diagnostics trace the gain to a reduced per-channel mismatch at these quantizers, which restores the code distribution the baselines leave unchanged or distort further. A single fixed configuration remains ahead across continual streams, non-i.i.d. label shift, seven out-of-distribution suites, and three other backbones.

1 INTRODUCTION

Quantized ViTs become brittle under distribution shift because frozen quantizer ranges no longer match test activations, while existing TTA misses this quantizer-level mismatch or incurs costly adaptation. QuAR targets the mismatch without backpropagation or parameter updates.

1 INTRODUCTION

  • Under shift, per-channel activation statistics drift from calibration, distorting frozen quantizers’ code distributions and compounding prediction errors along the path to the logits.The introduction reports that quantization amplifies the accuracy drop 1.7× at W3A3 and that a fifth of channels forfeit at least half a bit of code resolution.Source: §1 · p. 1
  • Existing TTA methods either backpropagate and update parameters or, when backprop-free, often add forward passes and parameter updates while acting away from quantizer inputs.The paper reports that compared methods leave the code distribution unchanged or move it farther from source calibration.Source: §1 · p. 2
  • QuAR applies parameter-free affine recalibrations at frozen quantizer inputs, mapping running per-channel test statistics toward source calibration in late blocks and at the classifier input.The method adds one multiply–add per recalibrated activation within a single forward pass.Source: §1 · p. 2
  • QuAR leads backprop-free TTA at W3A3, W4A4, W6A6, and W8A8 on ImageNet-C ViT-B, beating the strongest baseline by +2.28 at W8A8 and +4.00 at W3A3.It uses one forward pass and no parameter updates, with 46% less latency than ZOA and FOZO; one fixed configuration also leads across continual streams, label shift, seven OOD benchmarks, and three other backbones.Source: §1 · p. 2
  • The paper’s contributions include diagnostics linking gains to per-channel quantizer-input mismatch and a first-order account connecting activation drift, frozen-grid code resolution, and margin-sensitive predictions.The diagnostics include a mismatch measurement, precision sweep, and placement ablation.Source: §1 · p. 2

2 RELATED WORK

Prior work covers quantized ViT deployment and label-free test-time adaptation, but existing methods generally do not combine low-cost adaptation with correction at frozen quantizer inputs. QuAR instead matches per-channel moments at those interfaces in closed form.

Quantized vision transformers

  • PTQ fixes low-bit weights and activation quantizers using a calibration set; PTQ4ViT uses twin uniform quantizers and AdaLog uses adaptive logarithmic quantizers for post-Softmax and post-GELU activations.The remaining linear-layer activations stay uniformly quantized under AdaLog, and the calibrated per-tensor ranges remain frozen at test time.Source: §2 · p. 3

Test-time adaptation

  • TENT and related methods adapt online, while ActMAD [Mirza et al., 2023] matches activation moments through backpropagation and LAME [Boudiaf et al., 2022] adjusts output label assignments.PEA uses full-covariance whitening–coloring and requires a second forward pass and a dense matrix per block.Source: §2 · p. 3

Backprop-free TTA for quantized models

  • FOA and FOZO [Wang & Wang, 2026] avoid backpropagation but use extra forward passes, while ZOA [Deng et al., 2025] updates LayerNorm affine parameters without changing quantizers.NEO [Murphy et al., 2026] is single-forward but recenters features only at the classifier.Source: §2 · pp. 3–4
  • QuAR matches per-channel mean and variance in closed form at frozen quantizer inputs, unlike methods that act at block outputs, prompts, LayerNorm parameters, or classifier features.Applying the same affine correction at block outputs recovers only about half the gain; at quantizer inputs it restores code distribution and adds 1.8 to 3.1 points.Source: §2 · p. 4

3 METHOD

QuAR corrects per-channel activation-moment drift before frozen quantizers, using local late-block and global classifier-input recalibration with online target estimates. Its analysis connects this correction to frozen-grid code occupancy, downstream propagation, and accuracy under shift.

3.1 PRELIMINARIES

  • Per-channel activation drift is invisible to per-tensor quantizer parameters, and LayerNorm does not control it because it normalizes across channels rather than across tokens.The mismatch affects scale mainly at interfaces without a preceding normalization, while mean drift occurs everywhere; QuAR aims to correct it online without changing model or quantizer parameters.Source: §3.1 · p. 4

3.2 RECALIBRATION AT THE QUANTIZER INPUTS

  • QuAR recalibrates inputs to every quantized linear in the last four transformer blocks, using a partial local match and a full-strength match at the terminal CLS input to the classification head.The local branch uses λ = 0.5 because full matching can over-correct scale; the terminal head input can be fully matched because no downstream blocks distort it.Source: §3.2 · pp. 4–5, Figure 3 · p. 5

3.3 RUNNING TARGET MOMENTS

  • Target moments are source-initialized and updated online; the default cumulative estimate weighs batches equally, while continually evolving domains use a fixed exponential-moving-average rate of 0.1.Local recalibration uses moments from preceding batches before updating, whereas the global branch updates first to adapt from the first batch of a domain.Source: §3.3 · pp. 5–6

3.4 TEST-TIME COST

  • QuAR needs one forward pass per batch, updates no model or quantizer parameters, stores no past samples, and retains about 170 KB of running moments.That state is reported as 0.01% of peak memory; recalibration adds a multiply and an add per activation.Source: §3.4 · p. 6

3.5 ANALYSIS OF THE QUANTIZATION-SPECIFIC LOSS

  • Under the stated moment-shape and uniform-quantizer assumptions, matching moments returns activations toward the quantizer’s calibrated range and code occupancy.The analysis assumes standardized activation shape is approximately shift-invariant and uses a source-calibrated uniform affine quantizer.Source: §3.5 · p. 6
  • A fifth of channels forfeit at least half a bit of resolution under shift; QuAR reduces that share to 7%, while no compared method removes more than 1%.The gain from scale matching is +2.72 points at interfaces forfeiting resolution versus +1.04 at those forfeiting none; late-block downstream gain is measured ∼4.4× smaller than early-block gain, and accuracy recovery grows as precision falls.Source: §3.5 · p. 7
  • Full recalibration can move code distributions closer to source without improving accuracy, so QuAR uses a partial match to damp finite-sample scale-estimation noise.The deployed configuration uses λ = 0.5 with cumulative target estimates.Source: §3.5 · p. 7

4 EXPERIMENTS

Experiments compare mean top-1 across 15 ImageNet-C corruptions using a quantized ImageNet-trained ViT-B. QuAR leads the compared backprop-free methods at every tested bit-width while requiring one forward pass and no parameter updates.

4.1 MAIN RESULTS

  • At W8A8 and W3A3, QuAR leads the strongest baseline by +2.28 and +4.00 mean-accuracy points, respectively, and wins at all four evaluated bit-widths.The comparison uses mean top-1 on ImageNet-C.Source: §4.1 · p. 7, Table 1 · p. 8
  • Across ten continual rounds, QuAR stays at 35.6–35.8 at W3A3 and 53.9–54.0 at W6A6, beating ZOA by +3.73/+1.19 and NEO-Continual by +4.00/+2.77.The stream carries state across 10 rounds of 15 ImageNet-C corruptions.Source: Table 2 · p. 8, §4.2 · p. 8
  • At one forward and zero parameter writes, QuAR is at least 46% lower latency than two-forward backprop-free baselines; its combined branches add +6.67 at W6A6 over No Adapt.The local branch alone contributes +5.74 and the global branch +3.52 at W6A6.Source: §4.1 · p. 8

4.3 GENERALIZATION ACROSS SHIFTS AND BACKBONES

  • QuAR leads on Sketch, 3DCC, DomainNet and CIFAR-10/100-C across bit-widths, and wins on all three tested transformer backbones at W6A6 with one configuration.On ImageNet-R it is within 0.7 points of the best; on ResNet-50 it scores 31.40 against NEO’s 30.79 and No Adapt’s 30.12.Source: Table 3 · p. 9, Table 4 · p. 9, §4.3 · p. 9

4.1 MAIN RESULTS

  • Across imbalance ratios 1 to 5×10^5, QuAR leads by 1.7–2.7 points while staying stable at 54.55→54.73.The setting uses non-i.i.d. online label shift.Source: §4.2 · p. 9

4.4 ANALYSIS AT THE INTERMEDIATE QUANTIZER INPUTS

  • At 16 late-block interfaces, local recalibration cuts mismatch from 0.347 to 0.153 (−56%); early or middle placement falls below No Adapt, while post-quantizer correction recovers about half the gain.Saturation remains negligible at ≤0.1%.Source: §4.4 · pp. 9–10

4.6 COMPARISON TO BACKPROP-BASED TTA

  • At W6A6, SPA’s higher top-1 requires 5.2× latency and 14.6× inference memory, whereas QuAR adds 0.17 MB at 1.10× latency and runs on Orin Nano.The lightweight backprop-based methods need 9.2 GB and 1.9 to 3.7× latency.Source: Table 7 · p. 10, §4.6 · p. 10

4.5 HYPERPARAMETER SENSITIVITY

  • With λ=0.5 fixed across bit-widths, backbones and datasets, accuracy stays within 0.2 points across batch sizes 1 to 128; every λ from 0.1 to 1.0 beats the best baseline.Eight clean calibration batches come within 1.5 points of the default.Source: §4.5 · p. 10, Table 6 · p. 10

5 LIMITATIONS AND CONCLUSION

QuAR is reported as the most accurate backprop-free method across W3A3–W8A8 on ImageNet-C, with one configuration also carrying across several shifted settings and backbones. The paper notes limits in integer-kernel deployment and a strength parameter chosen once across backbones.

Limitations

  • The authors leave latency, memory, and energy on integer-kernel deployment for future work because QuAR has not been implemented on that stack.The deployable form is an explicit Mul/Add wrapper, which an ONNX Runtime check confirms is representable in standard operators.Source: §5 · p. 10
  • Choosing strength λ once rather than per backbone costs up to 0.2 points on the transformers.On clean source data, top-1 changes by at most 0.42 points across W3A3–W8A8, so the correction is described as safe but inert when adaptation is unnecessary.Source: §5 · p. 10

Conclusion

  • Across W3A3–W8A8 on ImageNet-C, QuAR is the most accurate backprop-free method at low test-time cost and its fixed configuration carries to other streams, suites, and backbones.The conclusion describes the two branches as complementary.Source: §5 · p. 10
  • The analysis identifies activation-moment correction at frozen-quantizer inputs as increasingly valuable as precision falls and the quantization grid loses code resolution.The conclusion says much of the accessible test-time accuracy is recoverable without gradients, extra forward passes, parameter updates, or stored data.Source: §5 · p. 10

AI USE STATEMENT

The authors used generative AI tools for interpretation, editing, figure-script drafting, and code cleanup, then checked the resulting content and take responsibility for the final work.

AI Use Statement

  • Generative AI tools helped interpret results, edit text for grammar and readability, draft figure scripts, and clean up code.The authors say they checked every interpretation and figure against the underlying data and reviewed every edited passage.Source: AI USE STATEMENT · p. 11
  • The authors verified that AI-edited code reproduces the reported numbers.They take responsibility for the final content, including AI-assisted text, claims, and artifacts.Source: AI USE STATEMENT · p. 11
  • The authors take responsibility for the final content of the work, including text, claims, and artifacts produced with AI assistance.Source: AI USE STATEMENT · p. 11

ETHICS STATEMENT

The work uses public benchmarks and released checkpoints, adds no dataset or human-subject data, and describes its contribution as methodological.

Ethics Statement

  • The study uses publicly available benchmarks and publicly released model checkpoints, with no human subjects, newly collected data, or introduced dataset.Source: ETHICS STATEMENT · p. 11
  • The methodological contribution inherits the known biases and limitations of the models and datasets it uses, while neither adding to nor removing them.Source: ETHICS STATEMENT · p. 11
  • The authors identify no application-specific risk beyond those of the underlying classifiers and no ethical concern beyond those general to machine-learning research on public benchmarks.They state that test-time adaptation of a quantized model lowers the cost of running vision classifiers on edge hardware.Source: ETHICS STATEMENT · p. 11

REPRODUCIBILITY STATEMENT

The paper specifies a fixed QuAR configuration and evaluation protocols, reports public code and artifacts, and describes the hardware and cost-measurement setup.

Reproducibility Statement

  • QuAR uses one configuration across datasets, bit-widths, and backbones, with cumulative moment estimates except for an α=0.1 EMA in the continual setting.The fixed settings include blocks 8–11, λ=0.5, clip ratio ρ=3, and ϵ=10−6; QuAR is deterministic given the test stream.Source: REPRODUCIBILITY STATEMENT · p. 11
  • Source moments come from one pass over 12,800 unlabeled clean source images, with no target data, labels, or gradients.PTQ4ViT is used for W6A6/W8A8 and AdaLog for W4A4/W3A3; fine-tuned CIFAR and DomainNet models use their own training splits for calibration and source moments.Source: REPRODUCIBILITY STATEMENT · p. 11
  • ImageNet-C uses the RobustBench 5k subset, while online label shift uses the full 50k per corruption; CIFAR-10-C/100-C and DomainNet-126 are evaluated in full.Source: REPRODUCIBILITY STATEMENT · p. 11
  • Code, launch scripts for every table, and source calibration statistics are available in the project repository.The paper identifies PyTorch 2.8, timm 1.0, and CUDA 12.8 as the experiment software stack.Source: REPRODUCIBILITY STATEMENT · p. 11
  • Compute, peak memory, and per-image latency are measured on an NVIDIA TITAN RTX, with on-device measurements on a Jetson Orin Nano.The paper states that optimization-based baselines use their cheapest published two-forward setting for fair cost comparison.Source: REPRODUCIBILITY STATEMENT · p. 11

A EXPERIMENTAL AND REPRODUCIBILITY DETAILS

The experiments use frozen, post-training-quantized ViT-B checkpoints and evaluate adaptation under controlled corruption and shift protocols. The full ImageNet-C comparison favors QuAR at every tested bit-width, with supplementary tables detailing per-corruption results and ablations.

A EXPERIMENTAL AND REPRODUCIBILITY DETAILS

  • Experiments use ViT-B/16 with frozen activation quantizers: PTQ4ViT at W6A6/W8A8 and AdaLog checkpoints at W3A3/W4A4.DeiT-B/16 and ViT-S/16 are freshly quantized at W6A6 for the backbone study.Source: A EXPERIMENTAL AND REPRODUCIBILITY… · p. 15
  • ImageNet-C evaluation uses severity 5 across 15 corruptions, with episodic resets except for continual-stream experiments.The three-seed protocol varies online stream order; its standard deviations reflect online-adaptation variance, not differences in the fixed source checkpoint.Source: A EXPERIMENTAL AND REPRODUCIBILITY… · p. 15
  • QuAR uses one fixed configuration across datasets and bit-widths, while optimization-based baselines use their published, minimum-forward configurations.Source moments are accumulated offline from clean source images, and no corrupted or labeled data are used.Source: A EXPERIMENTAL AND REPRODUCIBILITY… · p. 15

B PER-BIT-WIDTH AND PER-CORRUPTION TABLES

  • QuAR is the best method at every bit-width on the full ImageNet-C test set, with rankings matching the 5k-image comparison.Table 8 reports mean top-1 accuracy over all 15 corruptions and includes a component ablation.Source: Table 8 · p. 16
  • The supplementary tables provide full per-corruption ImageNet-C results, the complete accuracy comparison, and component and sensitivity ablations.The full comparison covers 50,000 images per corruption; the per-corruption table reports seed-0 top-1 accuracy by bit-width.Source: B PER-BIT-WIDTH AND PER-CORRUPTION… · p. 16, Table 8 · p. 16, Table 9 · p. 16

C SENSITIVITY TO THE RECALIBRATION STRENGTH, BATCH SIZE, SOURCE-MOMENT ESTIMATION, AND CLAMP RATIO

The section tests how QuAR’s moment estimates, clamp, branch timing, and recalibration scope affect performance. It also reports transfer to other transformer backbones and convolutional networks.

C SENSITIVITY TO THE RECALIBRATION STRENGTH, BATCH SIZE, SOURCE-MOMENT ESTIMATION, AND CLAMP RATIO

  • The variance-ratio clamp makes cumulative-moment adaptation nearly batch-size invariant: accuracy stays within 0.2 points from batch sizes 1 to 128.Without the clamp, batch-1 accuracy falls from 54.31 to 53.29; the clamp is most useful when small batches or class concentration drive target variance toward zero.Source: C SENSITIVITY TO THE RECALIBRATION… · p. 17, Table 12 · p. 18
  • The deployed running-moment estimator and exact pooled Welford estimates differ by at most about 0.15 points, with the deployed estimator best at batch size 1.The local branch’s per-channel moments pool B·N tokens; the clamp bounds the CLS branch when its single-token estimate can degenerate.Source: C SENSITIVITY TO THE RECALIBRATION… · pp. 17–18, Table 13 · p. 18
  • Using the current batch’s own moments for local correction costs 2.0 points at batch size 64 and 16.2 at batch size 1, and performs poorly under label shift.Pooling across batches keeps the correction from moving with batch composition; a full extra pass to obtain converged moments stays within 0.3 points of the deployed estimate.Source: C SENSITIVITY TO THE RECALIBRATION… · p. 18, Table 14 · p. 19
  • The global branch updates before correcting, while the local branch uses preceding-batch moments and updates afterward; reversing these choices costs accuracy early or under label shift.Only the global branch acts on the first batch, and accuracy is within about one point of the full-stream value after five batches.Source: C SENSITIVITY TO THE RECALIBRATION… · pp. 18–19, Table 15 · p. 19
  • At W6A6, local mean-only recalibration adds +0.85 over the global branch alone, and local scale matching adds a further +2.30.Full QuAR is best at both tested batch sizes; dropping local variance matching or either branch costs accuracy.Source: C SENSITIVITY TO THE RECALIBRATION… · p. 19, Table 16 · p. 20

D GENERALIZATION TO OTHER BACKBONES

  • At W6A6, QuAR is best on all three tested backbones with λ=0.5; ViT-S improves from 35.55 to 38.29, ahead of NEO by +2.2.The full match λ=1 over-corrects per-channel scale on the additional backbones, while λ=0.5 transfers as a moderated setting.Source: D GENERALIZATION TO OTHER BACKBONES · pp. 19–20

E APPLYING QUAR TO A CONVOLUTIONAL BACKBONE (CNN)

  • On ImageNet-C, the unchanged λ=0.5 configuration raises GroupNorm ResNet-50 accuracy from 30.12 to 31.40 ±0.05, leading NEO and T3A.For BatchNorm ResNet-50, it raises accuracy from 17.32 to 24.57 ±0.06.Source: E APPLYING QUAR TO A CONVOLUTIONAL… · p. 21

F RECALIBRATION SCOPE

  • Recalibrating blocks 8–11 at all four interfaces gives the lowest-overhead scope on the accuracy plateau; recalibrating the full network collapses accuracy.At W6A6, the full-network scope reaches 31.93 versus 53.42 for the late-block default; adding post-GELU mlp.fc2 helps at every tested bit-width.Source: F RECALIBRATION SCOPE · pp. 21–22

G GENERALIZATION BEYOND IMAGENET-C: IMAGENET-R, SKETCH, C, AND 3DCC

QuAR is evaluated on rendition, sketch, and held-out corruption shifts in addition to the standard ImageNet-C setting. Its results vary by shift type, while held-out corruption suites show gains across all tested bit-widths.

ImageNet-R and ImageNet-Sketch

  • QuAR is best on ImageNet-Sketch at every bit-width and improves over No Adapt on ImageNet-R, but recentering baselines lead it there by 0.11 to 0.65 points.The λ=0.5 partial recalibration keeps ImageNet-R above No Adapt, whereas λ=1 pushes it below that baseline.Source: G GENERALIZATION BEYOND IMAGENET-C:… · pp. 22–23

Held-out corruption suites: ImageNet-C and ImageNet-3DCC

Held-out ImageNet-C corruptions

H GENERALIZATION BEYOND IMAGENET: CIFAR-10-C, CIFAR-100-C AND DOMAINNET-126

The paper tests a fixed QuAR configuration on CIFAR-C and DomainNet-126, then compares it with FOA at FOA’s recommended, higher-forward setting. QuAR leads broadly, though the runner-up depends on the dataset and ImageNet-R is an exception in the FOA comparison.

H GENERALIZATION BEYOND IMAGENET: CIFAR-10-C, CIFAR-100-C AND DOMAINNET-126

  • QuAR is most accurate in all twelve CIFAR-10-C, CIFAR-100-C, and DomainNet-126 dataset-by-bit-width cells, ahead of the runner-up by 0.6 to 3.7 points.The evaluation uses ViT-B/16 models fine-tuned on their source data and quantized under the main protocol.Source: H GENERALIZATION BEYOND IMAGENET:… · p. 24, Table 21 · p. 25
  • Against No Adapt, QuAR’s gain grows as precision falls on CIFAR-10-C, CIFAR-100-C, and DomainNet-126.The reported ranges are +2.9 to +5.7 on CIFAR-10, +3.7 to +6.5 on CIFAR-100, +2.2 to +4.4 under PTQ4ViT on DomainNet, and +3.6 to +5.3 under AdaLog.Source: H GENERALIZATION BEYOND IMAGENET:… · p. 25
  • The runner-up varies by dataset: FOZO leads the CIFAR comparisons in several settings, while T3A is runner-up on DomainNet-126 at every bit-width.FOA and FOZO also fall below No Adapt at some DomainNet-126 bit-widths.Source: H GENERALIZATION BEYOND IMAGENET:… · p. 25

I FOA AT ITS RECOMMENDED CONFIGURATION (K=28)

  • At FOA’s recommended K=28 setting, it remains below one-forward QuAR at every bit-width on ImageNet-C, ImageNet-C (Mintun), Sketch, and 3DCC; ImageNet-R is the exception.K=28 uses about 29 forwards per batch and runs at about 162 ms per image, roughly 25× QuAR’s 6.6 ms.Source: I FOA AT ITS RECOMMENDED CONFIGURATION… · p. 26, Table 22 · p. 26

J COMPARISON WITH ADAPTIVE QUANTILE RECALIBRATION (AQR)

QuAR outperforms AQR on quantized ViT-B, including at batch size 1, and remains the most accurate method across online label-shift ratios. Its deployed recalibration changes clean-source accuracy by at most 0.42 points.

J COMPARISON WITH ADAPTIVE QUANTILE RECALIBRATION (AQR)

  • At batch sizes 64–512, QuAR is +2.0 to +3.0 points higher than AQR; at batch size 1, AQR scores 18.73 versus QuAR’s 54.27.A post-hoc tuned AQR placement and scope still reaches only 52.41 at batch 64, below QuAR.Source: J COMPARISON WITH ADAPTIVE QUANTILE… · pp. 26–27, Table 23 · p. 27
  • QuAR aligns source-calibrated moments at frozen quantizer inputs, whereas AQR remaps full activation distributions after LayerNorm using per-batch percentiles.The paper attributes QuAR’s advantage to this quantizer-aligned operating point and to more stable moment estimates as batches shrink; the default AQR configuration was checked on full-precision ViT-B.Source: J COMPARISON WITH ADAPTIVE QUANTILE… · pp. 26–27

K ONLINE LABEL SHIFT

L SPA UNDER ONLINE LABEL SHIFT

  • SPA varies by more than eight points across seeds on the same stream, with collapses concentrated in blur corruptions and occurring within individual corruptions.Bare SPA lacks an entropy threshold and redundancy filter to reject near-single-class batches; adaptation resets per corruption rather than accumulating along the stream.Source: L SPA UNDER ONLINE LABEL SHIFT · p. 28

M CLEAN SOURCE: BEHAVIOR UNDER NO SHIFT

  • On clean ImageNet validation, QuAR changes top-1 accuracy by at most 0.42 points across W3A3–W8A8.With no shift, running target moments coincide with source moments, so recalibration reduces to the identity apart from finite-sample estimation noise.Source: M CLEAN SOURCE: BEHAVIOR UNDER NO SHIFT · p. 28, Table 25 · p. 28

N ROBUSTNESS TO THE QUANTIZER CHOICE

QuAR remains the best method across the tested quantizer–bit-width settings, except for the degenerate PTQ4ViT W3A3 case. In asymmetric deployment, it also leads at W4A8, where activation precision matters more than weight precision in the reported comparisons.

N ROBUSTNESS TO THE QUANTIZER CHOICE

  • QuAR is best in all seven non-degenerate quantizer×bit-width settings; under AdaLog, its gain over No Adapt grows from 5.08 at W8A8 to 7.63 at W3A3.PTQ4ViT at W3A3 is excluded because it collapses to near-chance for all methods; under PTQ4ViT the gain grows from 6.08 at W8A8 to 7.47 at W4A4.Source: N ROBUSTNESS TO THE QUANTIZER CHOICE · pp. 28–29, Table 26 · p. 29

O ASYMMETRIC DEPLOYMENT BIT-WIDTHS

  • The reported W4A8 comparison uses AdaLog because PTQ4ViT collapses to 76.62, while AdaLog reaches 85.22 on 10,240 held-out clean images.The paper notes that W4A8 is an asymmetric format used in deployment-realistic settings.Source: O ASYMMETRIC DEPLOYMENT BIT-WIDTHS · p. 29
  • At W4A8, QuAR reaches 60.11±0.07 top-1 versus 58.01 for the strongest baseline.Halving weight precision from 8 to 4 bits moves each of seven rows by at most 0.42, while lowering activation precision from 8 to 4 bits moves them by six to eight points.Source: O ASYMMETRIC DEPLOYMENT BIT-WIDTHS · p. 29, Table 27 · p. 29

P MECHANISM ANALYSIS

The mechanism analysis links quantization’s larger shift degradation to activation mismatch at quantizer inputs and shows that correction placement matters. QuAR’s in-place implementation preserves single-forward operation with negligible added memory.

P MECHANISM ANALYSIS

  • The clean-to-corruption accuracy loss grows from 28.7 points at full precision to 48.4 at W3A3.Under shift, the unquantized margin falls from 3.78 to 2.05 while activation quantization error changes by −3.2% to +2.2%; harmful flips grow by 2.7 points at W8A8 and 19.5 at W3A3.Source: P MECHANISM ANALYSIS · pp. 30–31, Table 28 · p. 30, Table 30 · p. 31
  • Local-only and QuAR reduce combined per-channel mismatch at the late-block interfaces by 56.0%, while Global-only and NEO do not reduce it.The reduction is partial because λ=0.5 applies partial recalibration; the code-distribution diagnostic compares de-quantized values rather than bin indices.Source: P MECHANISM ANALYSIS · pp. 30–31, Table 29 · p. 31
  • The local branch gains +4.5 to +5.9 points over No Adapt at every tested precision, with its marginal gain over Global-only peaking at +3.88 at W3A3.The precision sweep uses source-initialized cumulative moments and λ=0.5.Source: P MECHANISM ANALYSIS · p. 31, Table 31 · p. 32
  • At every tested bit-width, correcting activations before the quantizer is best; correcting after it falls below No Adapt at W3A3 and W4A4.The after-quantizer variant scores 27.98 and 47.01 against No Adapt’s 28.35 and 47.30; block-output correction yields 1.8–3.1 fewer points than quantizer-input placement.Source: Table 32 · p. 32, P MECHANISM ANALYSIS · p. 32

Q MEASURED TEST-TIME COST

  • At W6A6, QuAR runs at 1.10× No Adapt latency, versus about 2× for FOA, FOZO, and ZOA, with 0.17 MB residual peak memory overhead.The in-place implementation avoids a buffered corrected tensor that would cost 148 MB at batch 64 and is bit-identical to the buffered path.Source: Q MEASURED TEST-TIME COST · p. 33, Table 33 · p. 33

R ON-DEVICE MEASUREMENT ON A JETSON ORIN NANO

On a Jetson Orin Nano, QuAR’s relative latency advantage is reproduced, while batch 16 permits comparison with backprop-based methods that do not fit at the protocol batch. The paper also verifies graph representability but leaves full integer-only deployment for future work.

R ON-DEVICE MEASUREMENT ON A JETSON ORIN NANO

S RELATION TO BACKPROP-BASED TEST-TIME ADAPTATION

  • Backprop-based TTA requires gradients and parameter updates, while QuAR’s local branch applies a closed-form partial correction to the tensor read by each frozen quantizer.The paper reports that the correction’s late pre-quantizer placement and partial strength are load-bearing; full matching and placements away from those interfaces perform worse.Source: S RELATION TO BACKPROP-BASED TEST-TIME… · pp. 34–35, Table 36 · p. 35, Table 37 · p. 35

T ONNX REPRESENTABILITY CHECK

  • The W8A8 recalibration sites export as standard ONNX per-channel Mul/Add pairs that reproduce PyTorch logits, but full integer-only deployment with end-to-end accuracy parity remains future work.The per-channel shift does not fold into a per-tensor quantizer, so the explicit wrapper is the intended representation.Source: T ONNX REPRESENTABILITY CHECK · p. 35

U PROOFS AND ASSUMPTIONS

This appendix provides first-order proofs and assumption-based analysis of quantizer recalibration through per-channel moments, explaining code-distribution restoration, correction strength, source-anchored stability, and scale matching for a log-quantized interface. The results are explanatory rather than worst-case guarantees and do not bound errors from changes in activation shape or inter-channel correlations.

Restored quantizer operating point

  • Mean deviation (0.235) exceeds scale deviation (0.112) by a factor of 2.1, so recalibration mainly relocates activations on the frozen grid and restores source code occupancy.Under the stated shift-shape assumption, matching source moments restores the source code distribution; the frozen step leaves granular error power unchanged, while recalibration restores the source clipping probability.Source: U PROOFS AND ASSUMPTIONS · pp. 36–37

Where code starvation occurs

  • At unnormalized interfaces, 28.9% of channels lose at least half a bit and the worst loses 2.51 bits; no channel crosses that threshold after LayerNorm.The analysis reports that starvation is confined to the attention-output and post-GELU interfaces without preceding normalization.Source: U PROOFS AND ASSUMPTIONS · p. 37, Table 39 · p. 39

Code-distribution restoration

  • QuAR removes 52% of code-distribution drift overall; its local branch removes 47.2% across 16 intermediate interfaces, while its global branch removes 90.0% at the head input.At the head input, NEO removes 20.6% and FOA 48.0%; only QuAR is high in both interface groups.Source: U PROOFS AND ASSUMPTIONS · p. 39

From code restoration to accuracy

  • A bound on propagated quantization error comprises a frozen-step granular term and a clipping term that grows when activations leave the calibrated range.Returning activations to the calibrated operating point restores the margin toward its source value and can cap quantization-induced label flips, but accuracy also depends on estimation error and signal fidelity.Source: U PROOFS AND ASSUMPTIONS · p. 40
  • Full matching restores 84.2% of code-distribution drift but scores 53.52, below QuAR’s 54.35, so maximizing restoration does not maximize accuracy.The analysis attributes this tradeoff to recalibration estimation error and identifies an intermediate shrinkage strength as optimal.Source: U PROOFS AND ASSUMPTIONS · p. 40

Depth asymmetry

  • A perturbation’s downstream gain falls from 0.26 in early blocks 0–3 to 0.059 in late blocks 8–11, an approximately 4.4× attenuation that makes late recalibration lower-risk.The bound links the risk of an imperfect correction to downstream gain; the accuracy benefit and number of blocks are determined empirically.Source: U PROOFS AND ASSUMPTIONS · p. 41, Figure 6 · p. 41

Optimal recalibration strength

  • The optimal recalibration strength decreases as scale-estimation noise rises and equals 1 only with noiseless estimates; ViT-B prefers λ=0.5–0.7, while ViT-S and distilled DeiT-B prefer λ ≤0.5.The analysis interprets this as shrinkage of variance matching toward the identity, with λ=0.5 serving as a single robust default.Source: U PROOFS AND ASSUMPTIONS · pp. 41–42

Stable partial restoration is optimal

  • With a genuine scale shift and noisy estimates, optimal restoration is interior and partial: λ⋆∈(0, 1), because no correction leaves bias while full matching injects variance.The stationary point is λ⋆ = (1 − r)²/[(1 − r)² + r²δ²] + O(δ²); it depends on the unknown shift and noise, and tends toward zero when estimation noise makes correction unhelpful. The authors describe λ=0.5 only as a robust value in an empirically flat region, not as the optimum.Source: U PROOFS AND ASSUMPTIONS · p. 42

Drift-free continual adaptation

  • A fixed source anchor and clamped gain keep QuAR’s running state bounded over arbitrarily long streams, unlike online optimizer updates that can drift from their initial parameters.The comparison names entropy minimization and ZOA (Deng et al., 2025) as unanchored updates; EATA and SAR use explicit safeguards against long-stream instability.Source: U PROOFS AND ASSUMPTIONS · pp. 42–43

The single log-quantized (post-GELU) interface

  • For AdaLog’s post-GELU mlp.fc2 interface, scale correction shifts the log-magnitude distribution toward its source position, restoring clip/drop probability while leaving scale-invariant granular error unchanged.This is an approximate, first-order argument rather than a direct low-bit code-distribution measurement; the diagnostic is measured at W6A6, where the interface uses uniform quantization.Source: U PROOFS AND ASSUMPTIONS · p. 43

Relation to the sensitivity analysis of ZOA

  • Halving activation bit-width moves the same methods by six to eight points, versus at most 0.42 points for weight bit-width; this motivates targeting activation-grid drift beyond ZOA’s weight-error account (Deng et al., 2025).The analyses agree that quantization sensitivity increases at lower precision, but differ on what is recoverable: QuAR targets per-channel mean drift and scale contraction at frozen activation quantizers.Source: U PROOFS AND ASSUMPTIONS · pp. 43–44

Scope

  • The analysis is first-order and moment-level: it explains mismatch reduction, partial λ, and absence of drift, but does not bound residual error from higher-moment or inter-channel-correlation shifts.Symmetric-code restoration is measured across all 17 recalibration interfaces at W6A6 (52%); QuAR is reported as the best adapter at every bit-width yielding a usable model.Source: U PROOFS AND ASSUMPTIONS · p. 44
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