Power-of-Two (PoT) Weights in Large Language Models (LLMs)

Mahmoud Elgenedy

REVIEW 4 major objections 5 minor 14 references

Power-of-Two (PoT) Weights in Large Language Models (LLMs)

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A 4-bit power-of-two weight representation keeps GPT-2 text coherent, cuts memory 8x, and turns multiplication into bit shifts.

desk verdict Plausible accuracy data for known PoT quantization on GPT-2, but the speedup claim is a back-of-envelope estimate, not a result. read the letter →

arxiv 2506.00315 v1 pith:BVGSGDBO submitted 2025-05-31 eess.SP cs.LG

Mahmoud Elgenedy This is my paper

classification eess.SPcs.LG

keywords power-of-twoquantizationlargelanguagemodelpost-trainingbit-shiftarithmeticnon-uniformGPT-2edgeinferencecross-entropyloss

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper attempts to establish that power-of-two quantization is a practical, post-training compression scheme for large language models: restrict every weight to a power of two, store only the exponent, and a GPT-2-scale model keeps producing coherent text while its weight memory shrinks by a factor of 8. The supporting experiments quantize a character-level transformer and a 124-million-parameter GPT-2 model, reporting cross-entropy loss degradation of roughly 0.9 to 1.3 over the floating-point baseline when 4 to 6 bits are used for the power levels. At the same 4-bit budget, PoT (about 4.5 cross-entropy) dramatically outperforms uniform quantization (about 7.7), and generated text remains readable. The broader reason to care is that the same restriction converts multiplication into bit-shifting, which the paper estimates as a 4 to 5 times reduction in processing cost for linear layers and transformer tables, pointing toward language models that can run on edge devices with limited memory and power.

What carries the argument

The formula that carries the argument is the power-of-two quantizer $y = 2^{\mathrm{clip}(\mathrm{round}(\log_2(x/\mathrm{scale})))}$, which maps a floating-point value to the nearest power of two with an exponent restricted to a small integer range. Storing only the exponent is what delivers the memory compression, while the shift-by-exponent interpretation is what replaces multiplication with bit shifting. The paper also treats an epsilon parameter inside the quantize/dequantize simulation as a stabilization control: too large an epsilon destabilizes results, too small one limits resolution, and its choice matters more for PoT than for uniform quantization.

What would settle it

Run the PoT-quantized GPT-2 graph on a cycle-accurate simulator or an actual shift-based kernel and count the executed multiply, shift, and dequantize operations; if the end-to-end latency and energy do not approach the predicted 4–5x improvement over the floating-point model after all dequantizers are removed, the central speedup claim is false.

Watch

Extended reading notes

Core claim

The central discovery is that a logarithmic, nonuniform grid of the form $2^k$ for integer $k$ fits trained transformer weights better than a uniform integer grid at low bit-widths. With 15 power levels, i.e., exponents in $[-7,7]$ requiring only 4 bits, the quantized GPT-2 model has cross-entropy loss around 4.5 compared with 3.167 for floating point, and the generated text is still of good quality; the same 4-bit budget with uniform quantization jumps to about 7.7 and produces broken text. The paper applies PoT to both linear-layer weights and transformer tables, and reports that the memory requirement drops by a factor of $32/4=8$. On the compute side, it argues that because every quantized weight is an exponent, multiplication by that weight becomes a bit shift, giving an estimated processing reduction of roughly 4 to 5 times.

Load-bearing premise

The claim of a 4–5x processing speedup rests on the assumption that the simulated quantized graph, which still contains dequantize operations and computes in floating point, can be compiled into real bit-shift arithmetic without significant overhead; if dequantization, table lookup, or scaling costs remain, the speedup is not established.

Editorial extensions

If this is right

At 4 bits, PoT rescues a regime where uniform quantization collapses: on the same GPT-2 evaluation, 4-bit uniform gives about 7.7 cross-entropy while 4-bit PoT gives about 4.5.
Because the scheme is post-training and does not need retraining, it can be applied to an already-trained model, which is the cheapest deployment path for edge hardware.
If the bit-shift conversion is realized in hardware, every linear layer, attention projection, and MLP block gets cheaper, so the savings compound across the whole transformer.
An 8x reduction in weight memory plus the estimated compute reduction would directly lower the energy and memory floor for running language models on small devices.
The PoT range can be swept independently of bit-width, giving a simple knob to trade quality against representable dynamic range.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The accuracy result is separable from the speedup claim: even if real hardware pays overhead for dequantization or scaling, a 4-bit exponent representation with only about 1.3 extra cross-entropy loss remains a viable memory-compression result worth testing on larger models.
The paper's own warning that cross-entropy can be misleading for unstable models implies the decisive next comparison should be generation-oriented, using ROUGE, BERTScore, or human evaluation of samples from the 4-bit PoT model.
Because PoT concentrates levels near zero, it may compose naturally with sparsity and pruning: zero weights can be encoded explicitly, and the same exponent grid could serve both dense and sparse kernels.
A natural testable extension is applying the same post-training exponent grid to an instruction-tuned or reasoning model, since base-model cross-entropy on GPT-2 may not predict chat or agentic task quality.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: this is a class project that applies a known quantization scheme (power-of-two weights) to GPT-2, reports plausible cross-entropy numbers, and then oversells the speedup with a clock-cycle estimate. The accuracy part is worth a glance; the speedup claim is not established.

What's actually new: not much. PoT quantization is well covered in refs 4 and 5 for DNNs and CNNs. This paper runs post-training quantization on GPT-2 (124M) and reports that 4–6 bit PoT gives cross-entropy degradation of roughly 1.3–0.88 over a float baseline. That is a legitimate data point, and the observation that 4-bit PoT beats 4-bit uniform quantization (4.5 vs 7.7 CE) is interesting if it holds up. The paper is honest that this is preliminary and that the generated graph is only a simulation.

Soft spots: the processing speedup is the big one. Section 5.3 says 'multiplication requires approximately 5 clock cycles while bit shifting is just one clock cycle' and Section 6 reports a 5x processing reduction. That is a per-operation estimate, not an end-to-end measurement. The paper's own Section 4.2 says the exported graph 'uses de-quantize operations' and is 'not the final quantized model.' Footnote 3 confirms calculations are done in floating point after dequantization. So no integer-shift model was actually built or timed. Until dequantizers are removed and layer norms, softmax, and accumulations are handled, the 5x claim is unsupported. That is the main flaw.

Second soft spot is the empirical rigor of the CE numbers: one model, no error bars, calibration on 0.05% of OpenWebText, and no code or data. The numbers are internally consistent, so I don't doubt them, but they are not reproducible as presented.

What the paper does well: it clearly explains the quantization setup and points out a real phenomenon—non-uniform PoT has more resolution for small values, which helps low-bit PTQ. That is consistent with prior work and worth keeping in mind.

Who this is for: someone tracking low-bit PTQ for LLMs might skim the table. But as a paper, it lacks the artifacts and measurements needed to be a citable result. I would not bring it to reading group, and I would not accept it for serious peer review in its current form. It is a preliminary class project. If the author adds code, error bars, and an actual integer kernel benchmark, the accuracy observation could become a useful note.

Referee Report

4 major / 5 minor

Summary. The manuscript proposes post-training power-of-two (PoT) quantization of weight matrices and embedding tables in LLMs, with a 124M-parameter GPT-2 as the main testbed. The authors use PyTorch 2 export-mode static quantization to simulate quantization and then restrict weights to powers of two. They report cross-entropy and perplexity for uniform and PoT quantization at several bit widths, and claim memory compression by a factor of 8 and processing speedup by a factor of 4-5 because multiplications become bit shifts. The main evidence is Table 3, where PoT 4-6 bit losses range from roughly 4.08 to 4.5 against a floating-point baseline of about 3.167, together with a sample of generated text.

Significance. If the accuracy numbers and the speedup claim held, converting matrix multiplications into bit shifts at 4-6 bits would be an interesting contribution to edge deployment of LLMs. The paper has clear strengths: it evaluates against an externally pretrained GPT-2 checkpoint rather than fitting its own baseline; the reported cross-entropy values are internally consistent with Table 3; and the problem it addresses is practically important. However, the experimentally supported contribution is currently limited to a single model, a single run per configuration, a very small calibration set, and an unmeasured hardware claim. The core idea is not flawed, but the evidence is preliminary.

major comments (4)

[§5.3 and §6] The central claim of a 4-5x processing reduction is not established. The only evidence is the clock-cycle statement that multiplication takes about 5 cycles and bit shifting about 1 cycle. The paper's own §4.2 says the generated quantized graph "is not the final quantized model" and "uses de-quantize operations," and footnote 3 confirms that "calculations will be done in floating point after dequantization operation." No timing of an end-to-end integer-shift inference graph is reported. The speedup claim should be backed by a kernel-level or end-to-end measurement on a real integer-shift graph, or explicitly relabeled as a per-operation theoretical bound.
[Table 3 and §5.3] The comparative claim that at the same 4-bit budget uniform quantization "shows a big failure (cross entropy ≈ 7.7)" while PoT gives ≈4.5 is not supported by the table. The 7.7 value is listed for Normal quantization with integer range [-25, 25] and labeled 6 bits; no 4-bit uniform quantization row is reported. The text also refers to this as "5-bits" on the preceding line. The bit-width labels need to be corrected and a true 4-bit uniform baseline should be reported before making the same-bit comparison.
[§5.1 and §5.3] The accuracy results rest on a single pretrained GPT-2 model, a single quantization run per configuration, and evaluation/calibration on 0.05% of OpenWebText. No error bars or repeated runs are provided, so the reported differences among PoT cross-entropy values (4.08, 4.23, 4.3, 4.5) and the claimed degradation range of about 0.88-1.3 may be within measurement noise. The paper should state that these are single-run estimates and, ideally, provide variance estimates or re-evaluation on a larger held-out subset.
[§4.2 and Table 2] The PoT quantization procedure is underspecified. It is not described how per-tensor scale factors are determined for each linear layer or embedding table, how signed PoT levels such as [-7, 7] are encoded in 4 bits, what epsilon values were used, or how the "customized quantizer configuration function" maps the standard quantization flow to the log2-based PoT mapping. Without these details, the reported cross-entropy numbers cannot be reproduced or compared with prior PoT work.

minor comments (5)

[§3.2] The statement that the evaluation dataset is 0.05% of the total OpenWebText dataset should specify whether this refers to tokens, documents, or files, and how many evaluation batches were used.
[§5.2 and §1] There are typos: "wights" should be "weights" in §5.2, and "Al progress" in the Section 1 quote should be "AI progress."
[Table 3] The table layout is confusing because the two 4-bit PoT rows have very different cross-entropy values (9.563 and 4.5) and the text does not map the rows to the exponent ranges mentioned in the bullets, such as [-7, 7].
[Figure 3 and §5.3] The claim that the generated text has "very good quality" is subjective; no automatic text-quality metric is reported, even though §5.1 lists ROUGE and BERTScore as future work.
[References] References [7] and [8] are informal repository/dataset citations without version numbers or access dates; please provide complete bibliographic information.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the accuracy results are measured against an external GPT-2 baseline and the speedup estimate is an unverified hardware assumption, not a circular derivation.

full rationale

The paper's central quantitative claims are not circular. Cross-entropy values in Table 3 are obtained by measuring a post-training-quantized GPT-2 model against a fixed pretrained checkpoint (OpenAI GPT-2, Ref [9]) and a held-out test split, so the reported degradation is externally anchored rather than defined by the method. The PoT mapping in Table 2, y = 2^clip(round(log2(x/scale))), is a standard quantization equation adopted from prior work (Refs [4,5]) and is not constructed to force the reported loss values. The epsilon hyperparameter is tuned for numerical stability, but it is not fitted to the target cross-entropy metric in a way that makes the result true by construction. The only materially weak step is the claimed 'factor of 4-5' processing reduction in Sections 5.3 and 6, which is inferred from an assumed per-operation clock-cycle ratio (multiplication ~5 cycles vs bit shift ~1 cycle) rather than measured end-to-end; the paper itself notes in Section 4.2 and footnote 3 that the generated graph is not the final quantized model and that dequantize operations remain. That is an unsupported assumption or limitation, not circularity, because it does not use the target claim as an input. There are no load-bearing self-citations, no imported uniqueness theorems, and no renaming of a known result as a new derivation. Therefore the circularity score is 0.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

No new physical or conceptual entities are introduced; PoT quantization, epsilon, and scales are standard quantization constructs. The free parameters are the exponent ranges, the epsilon threshold, and the per-tensor scales, all of which affect the reported results.

free parameters (3)

PoT exponent range limits = [-7,7] for 4-bit; [-15,15] for 5-bit; per-layer values not reported
Chosen by hand and reported in Table 3; the paper notes that ranges below these degrade sharply, so the selected ranges directly shape the headline results.
epsilon floating-point stability threshold = Not reported
Section 5.3 bullet 3 states epsilon is chosen based on the range of quantized data and has more impact on PoT than on normal quantization; no fixed criterion is given.
Per-tensor quantization scale for PoT = Not reported
Post-training quantization calibrates scale from data ranges, and the reported cross-entropy values depend on these scales, but the paper does not report them.

assumptions (4)

standard math Properties of log2 and exponent representation support y = 2^clip(round(log2(x/scale)))
The quantization equation in Table 2 is standard arithmetic; no proof is needed.
domain assumption Calibration on 0.05% of OpenWebText yields quantization scales representative of the full distribution
Section 3.2 states the evaluation dataset is 0.05% of the total; no sensitivity analysis is provided.
domain assumption Pretrained GPT-2 weights can be rounded to powers of two without retraining and still produce coherent text
This is the central premise of post-training quantization; the paper tests only one model and one checkpoint.
domain assumption Multiplication requires about 5 clock cycles and bit-shifting about 1 clock cycle
Section 5.3 bullet 4 uses this assumption to claim a processing reduction factor of 4-5, without hardware measurement.

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Cite this review

Pith. "Pith review of Power-of-Two (PoT) Weights in Large Language Models (LLMs)." pith.science (2026). https://pith.science/paper/BVGSGDBO

@misc{pith2026250600315,
  author       = {Pith},
  title        = {Pith review of: Power-of-Two (PoT) Weights in Large Language Models (LLMs)},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BVGSGDBO}},
  note         = {Machine review of arXiv:2506.00315}
}

abstract

Complexity of Neural Networks is increasing rapidly due to the massive increase in model parameters. Specifically, in Large Language Models (LLMs), the number of model parameters has grown exponentially in the past few years, for example, from 1.5 billion parameters in GPT2 to 175 billion in GPT3. This raises a significant challenge for implementation, especially for Edge devices where memory and processing power are very limited. In this work, we investigate reducing LLM complexity with special type of quantization, power of two (PoT), for linear layers weights and transformer tables. PoT not only provides memory reduction but more importantly provides significant computational reduction through converting multiplication to bit shifting. We obtained preliminary results of PoT quantization on Nano-GPT implementation using Shakespeare dataset. We then extended results to 124-M GPT-2 model. The PoT quantization results are shown to be very promising with cross entropy loss degradation $\approx$[1.3-0.88] with number of bits range [4-6] to represent power levels.

Figures

Figures reproduced from arXiv: 2506.00315 by the authors.

**Figure 2.** shows the PyTorch 2 Export quantization framework. The first step is to export the trained floating-point model into a graph basic tensor operations. Second, “Prepare Pt2e”, where observers are added before and after each operation (used to calibrate input and output range). The third step is to run the prepared model with realistic data so that observers can do calibration. Final step is to convert each observer in… view at source ↗

**Figure 3.** Output generated text with 4 bits PoT Considering PoT quantization, we observe cross entropy loss to degrade to ≈[4.08-4.5] based on the range limit. This slightly increase in loss is expected, in fact, it is very promising given that we did not perform any training, i.e., we only quantize weights after training (post training quantization PTQ not quantization assisted training QAT). In the following, we summarize v… view at source ↗

**Figure 4.** Snapshot of quantized graph of GPT-2 model [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

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Reference graph

Works this paper leans on

14 extracted references · 9 canonical work pages

[1]

URL: " 'urlintro :=

ENTRY address author booktitle chapter edition editor howpublished institution journal key month note number organization pages publisher school series title type volume year eprint doi pubmed url lastchecked label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block STRINGS urlintro eprinturl eprintpr...
[2]

write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize " " * FUNCT...
[3]

Gholami, S

A. Gholami, S. Kim, Z. Dong, Z. Yao, M. W. Mahoney, and K. Keutzer, ``A survey of quantization methods for efficient neural network inference,'' in Low-power computer vision , pp. 291--326, Chapman and Hall/CRC, 2022

work page 2022
[4]

Krishnamoorthi, ``Quantizing deep convolutional networks for efficient inference: A whitepaper,'' arXiv preprint arXiv:1806.08342 , 2018

R. Krishnamoorthi, ``Quantizing deep convolutional networks for efficient inference: A whitepaper,'' arXiv preprint arXiv:1806.08342 , 2018

arXiv 2018
[5]

H. Wu, P. Judd, X. Zhang, M. Isaev, and P. Micikevicius, ``Integer quantization for deep learning inference: Principles and empirical evaluation,'' arXiv preprint arXiv:2004.09602 , 2020

arXiv 2004
[6]

Przewlocka-Rus, S

D. Przewlocka-Rus, S. S. Sarwar, H. E. Sumbul, Y. Li, and B. De Salvo, ``Power-of-two quantization for low bitwidth and hardware compliant neural networks,'' arXiv preprint arXiv:2203.05025 , 2022

arXiv 2022
[7]

McDanel, S

B. McDanel, S. Q. Zhang, H. Kung, and X. Dong, ``Full-stack optimization for accelerating cnns using powers-of-two weights with fpga validation,'' in Proceedings of the ACM International Conference on Supercomputing , pp. 449--460, 2019

work page 2019
[8]

M. C. Neves, ``What are quantized llms?,'' Mar 2025

work page 2025

Show all 14 references

[9]

Karpathy, ``Karpathy-nanogpt: The simplest, fastest repository for training/finetuning medium-sized gpts.''
[10]

openwebtext

A. Gokaslan, ``Dataset card for "openwebtext".''
[11]

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[12]

OpenAI, ``tiktoken.''
[13]

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[14]

``Pytorch 2.0 export post training static quantization.''

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

URL: " 'urlintro :=

ENTRY address author booktitle chapter edition editor howpublished institution journal key month note number organization pages publisher school series title type volume year eprint doi pubmed url lastchecked label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block STRINGS urlintro eprinturl eprintpr...

[2] [2]

write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize " " * FUNCT...

[3] [3]

Gholami, S

A. Gholami, S. Kim, Z. Dong, Z. Yao, M. W. Mahoney, and K. Keutzer, ``A survey of quantization methods for efficient neural network inference,'' in Low-power computer vision , pp. 291--326, Chapman and Hall/CRC, 2022

work page 2022

[4] [4]

Krishnamoorthi, ``Quantizing deep convolutional networks for efficient inference: A whitepaper,'' arXiv preprint arXiv:1806.08342 , 2018

R. Krishnamoorthi, ``Quantizing deep convolutional networks for efficient inference: A whitepaper,'' arXiv preprint arXiv:1806.08342 , 2018

arXiv 2018

[5] [5]

H. Wu, P. Judd, X. Zhang, M. Isaev, and P. Micikevicius, ``Integer quantization for deep learning inference: Principles and empirical evaluation,'' arXiv preprint arXiv:2004.09602 , 2020

arXiv 2004

[6] [6]

Przewlocka-Rus, S

D. Przewlocka-Rus, S. S. Sarwar, H. E. Sumbul, Y. Li, and B. De Salvo, ``Power-of-two quantization for low bitwidth and hardware compliant neural networks,'' arXiv preprint arXiv:2203.05025 , 2022

arXiv 2022

[7] [7]

McDanel, S

B. McDanel, S. Q. Zhang, H. Kung, and X. Dong, ``Full-stack optimization for accelerating cnns using powers-of-two weights with fpga validation,'' in Proceedings of the ACM International Conference on Supercomputing , pp. 449--460, 2019

work page 2019

[8] [8]

M. C. Neves, ``What are quantized llms?,'' Mar 2025

work page 2025