Radio: Rate-Distortion Optimization for Large Language Model Compression

Sean I. Young

REVIEW 4 major objections 5 minor 40 references

Radio: Rate-Distortion Optimization for Large Language Model Compression

T0 review · 4 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read Quantizing an LLM's weights is rate–distortion optimal when each weight group contributes equal marginal output distortion per bit, and the paper's Radio algorithm realizes this rule, beating GPTQ, AWQ, OWQ, QuIP, SqueezeLLM, and…

desk verdict A real empirical contribution in LLM quantization, packaged with a rate-distortion theory that overclaims its assumptions; worth a serious referee. read the letter →

arxiv 2505.03031 v1 pith:I35SPZ26 submitted 2025-05-05 cs.LG cs.CL

Sean I. Young This is my paper · ORCID

classification cs.LGcs.CL

keywords rate-distortiontheoryLLMquantizationpost-trainingcompressionmixed-precisionbitallocationcompandingdualascentperplexity

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

The paper establishes a rate–distortion foundation for post-training LLM weight quantization. It argues that, at a fixed average bit budget, the bit depth assigned to each group of weights should equalize the marginal decrease in the model's output distortion per extra bit across groups, so that groups whose quantization errors propagate more strongly to the output receive more bits. From this rule the paper derives a dual-ascent algorithm, Radio, that alternates between estimating gradient variances by backpropagating through the quantized model and updating bit depths. Quantizing with a companding transform and bias correction, Radio matches or beats GPTQ, AWQ, OWQ, QuIP, SqueezeLLM, and OmniQuant in perplexity at 3–4 bits on the OPT and Llama-2 families, and substantially outperforms OWQ at 2.1–2.8 bits. If the theory holds, it provides a principled way to compress any LLM post-training to a user-specified model size or accuracy.

What carries the argument

The load-bearing object is the per-group distortion function $d_n(B_n) = P_n H_n G_n^2 S_n^2 2^{-2B_n}$, which decomposes the model's expected squared output error into additive contributions from each weight group. The identity that carries the argument is the high-rate quantization law $\mathbb{E}[\Delta_n^2(B_n)] = H_n S_n^2 2^{-2B_n}$ combined with a Jacobian-linearized model, which together imply that the optimal bit allocation equalizes marginal distortion per bit across groups, $-d_n'(B_n)/P_n = V$. This converts a combinatorial mixed-precision search into a smooth convex resource-allocation problem solved by dual ascent, with the gradient variances $G_n^2$ re-estimated by backpropagation at every iteration. The quantizer itself uses companding, a sigmoid transform derived from the cube root of the weight density, so that uniform quantization of the transformed weights approximates the optimal non-uniform quantizer, and bias correction $\mathbf{b}_n^q \leftarrow \mathbf{b}_n + (\boldsymbol{\Theta}_n^q - \boldsymbol{\Theta}_n)\bar{\mathbf{X}}_n$ compensates for the empirically non-zero mean of quantization errors.

What would settle it

Quantize OPT-1.3B at 3 bits with Radio, then, keeping the average bit rate fixed, move one bit from a low-$G_n^2 S_n^2$ group to a high-$G_n^2 S_n^2$ group and measure calibration-set output MSE; if the equal-marginal-distortion allocation is truly optimal, the perturbed allocation must have equal or higher MSE. As a second check, compute the correlation of quantization error vectors across groups: significant correlations would violate the uncorrelatedness assumption in Appendix B, step (b).

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Extended reading notes

Core claim

The paper's central discovery is that mixed-precision LLM weight quantization has a closed-form rate–distortion solution. Measuring output distortion as the expected squared deviation of the model's predictions, the distortion contributed by quantizing weight group $n$ at $B_n$ bits is approximately $d_n(B_n) = P_n H_n G_n^2 S_n^2 2^{-2B_n}$, where $P_n$ is the group size, $G_n^2$ is the average squared norm of the output Jacobian with respect to that group's weights, $S_n^2$ is the weight variance, and $H_n$ is a distribution-dependent constant. Because this family is monotone and convex in $B_n$, the constrained optimum equalizes the marginal distortion per bit, $-d_n'(B_n)/P_n = V$ for all $n$, which is equivalent to the allocation rule $B_n = \tfrac{1}{2}\log_2(2\ln 2\, G_n^2 S_n^2 / V)$ with the dual variable $V$ tuned to meet the bit budget. The paper turns this identity into an algorithm: estimate $G_n^2$ by backpropagating a PCA-projected calibration loss through the quantized model, update bit depths and $V$ by dual ascent, group weights into rows or columns to exploit variance heterogeneity, quantize with a companding transform matched to the weight distribution, and correct the induced output bias by updating each layer's bias vector. Empirically, Radio achieves lower WikiText2 perplexity than GPTQ, AWQ, OWQ, QuIP, SqueezeLLM, and OmniQuant at 3–4 bits on OPT and Llama-2, and lower perplexity than OWQ at 2.1–2.8 bits.

Load-bearing premise

The derivation assumes that the high-rate quantization law (error variance halves per added bit) stays accurate at the 2–4 bit depths used in the experiments and that quantization errors in different weight groups are uncorrelated; if either assumption fails, the additive distortion that Radio optimizes no longer equals the true output distortion.

Editorial extensions

If this is right

At any user-specified average bit rate, Radio's bit-depth assignment is the exact minimizer of the least-squares output distortion up to integer rounding, removing the combinatorial search that restricts mixed-precision methods to coarse bit-depth options.
The method quantizes post-training to any target model size or fractional bit rate and scales to hundreds of billions of parameters, quantizing a 7B model in about 47 minutes.
Radio achieves the lowest WikiText2 perplexity among GPTQ, QuIP, OWQ, AWQ, SqueezeLLM, and OmniQuant at 3–4 bits on the OPT and Llama-2 families, with the largest gains on small models (up to 4.55 perplexity on 3-bit OPT-125M).
At 2.1–2.8 bits, Radio-quantized Llama-2 models are substantially more accurate than OWQ at the same bit rates, with perplexity gaps that grow as bit depth decreases.
Quantizing low-variance weights to zero yields a mild pruning effect that slightly improves generalization, and a custom kernel dequantizes mixed-precision weights on the fly to accelerate matrix-vector multiplication by 1.4–3.3× for 3-bit models.

Reading between the lines

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

The equal-marginal-distortion rule generalizes beyond squared output error: replacing the output Jacobian with gradients of a task loss (e.g., accuracy on GSM8K) would produce a task-aware bit allocation; the paper does not test this, but the derivation does not depend on the loss being squared.
Equation (9) predicts that the bit saving from grouping grows with the heterogeneity of $G_n^2 S_n^2$ across groups, which suggests a testable extension: searching over permutations or orthogonal rotations of each weight matrix to maximize that heterogeneity, beyond the row/column grouping Radio uses, should yield further compression.
The paper's silence on whether quantization errors are uncorrelated across groups is testable: computing the empirical cross-group error correlation matrix on a Radio-quantized model would show whether the additive distortion decomposition (Appendix B, step b) holds, and if not, a covariance-aware bit allocation would be needed.
Table 1 shows that large models (66B, 70B) are compressed almost as well by simple round-to-nearest as by Radio, which suggests a practical scaling rule: for very large models, companding plus bias correction may capture most of the benefit, while the full dual-ascent allocation matters most for small- and medium-size models.

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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

The genuinely new thing here is applying classical rate-distortion bit allocation to LLM weight compression: per-group bit depths are chosen by dual ascent on gradient variances estimated with autograd, then combined with companded quantization and bias correction. The empirical payoff is real, especially at 2.x bits on Llama-2 (e.g., 7B: 7.05 vs 9.98 PPL at 2.4 bits versus OWQ/256). At 3–4 bits, Radio is competitive and occasionally better than GPTQ, AWQ, and OmniQuant. The empirical claim is the contribution, not the theoretical framing.

The paper does several things well. The formulation is clean and implementable, the companding step is standard but sensibly applied, and bias correction addresses a real failure of the zero-mean assumption. The experiments are unusually broad: OPT 125M–66B and Llama-2 7B–70B, perplexity plus GSM8K and five common-sense QA benchmarks, plus a custom CUDA kernel. Ablations in Table 3 are useful, and Appendix F's recasting of OBS is a nice pedagogical addition. The reference list covers the relevant quantization and rate-distortion literature; the self-citations to earlier transform-quantization work are relevant, not turf-building.

The soft spots are real but not fatal. The derivation in Appendix B assumes quantization errors across groups are uncorrelated and that the high-rate formula E[Δ^2] = H S^2 2^{-2B} holds at 2–4 bits. The paper explicitly admits zero-mean error is false and patches it, but never tests uncorrelatedness. The stress-test point is legitimate: for a fixed model, the errors are deterministic, not random, so the additive distortion decomposition is at best a modeling assumption. That undercuts the "foundations" language and the exact-optimality claim in Eq. (4), but it does not undercut the empirical gains, which are heuristics that clearly work. The reproducibility section has two small holes: the smoothing factor α in Algorithm 1 is never given a value, and the text says code is on GitHub but no URL appears. The WikiText2 model-selection issue is minor; the paper says it selects on validation and reports test, which is normal.

Take-home: I trust the numbers and would cite this as a competitive post-training quantization method. The theory needs an honest caveat paragraph before I would call it foundational. This deserves peer review, with a request to measure off-diagonal Hessian terms on a small model and fill the reproducibility gaps.

Referee Report

4 major / 5 minor

Summary. The paper proposes Radio, a post-training LLM weight quantization method based on a rate-distortion formulation. The central idea is to allocate bit depths to weight groups by minimizing an approximate expected-output-distortion objective subject to an average-bit-rate constraint. The authors derive a high-rate approximation in which the per-group distortion is d_n(B_n) = P_n H_n G_n^2 S_n^2 2^(-2B_n), leading to the optimality condition that the marginal distortion per bit be equalized across groups. They solve the relaxed problem by dual ascent, using backpropagated gradient variances to set bit depths, and combine the allocation with companded scalar quantization, bias correction, and row-column grouping. Experiments on OPT and Llama-2 at 3-4 bits and 2.x bits report perplexity and downstream-task results comparing favorably with GPTQ, AWQ, OWQ, QuIP, SqueezeLLM, and OmniQuant.

Significance. If the theoretical derivation were fully justified, Radio would offer a principled, per-group mixed-precision allocation with a clean rate-distortion interpretation, going beyond the coarser mixed-precision schemes in much of the prior literature. The empirical study is broad: it covers multiple model families and scales, includes 2.x-bit results where Radio shows large perplexity improvements over OWQ, provides ablations separating step-size, bit-depth, and companding effects, and reports runtime and kernel-level speedups. The appendices give detailed derivations, algorithm parameters, and a CUDA kernel, and the paper states that code is available. The main caveat is that the optimality claim rests on high-rate and zero-mean/uncorrelated quantization-error assumptions that are acknowledged to be only partially valid at the operating bit depths; the empirical results are promising, but the theoretical 'foundations' claim is not yet established as stated.

major comments (4)

[Appendix B, Eqs. (12)-(13)] The derivation of Eq. (5), and hence the optimality conditions in Eq. (4), requires the quantization errors Δ_1,...,Δ_N to be zero-mean and mutually uncorrelated. Section 3.2 explicitly notes that zero-mean is empirically false and patches it with bias correction, but no analogous patch or check is provided for uncorrelatedness. For a fixed pretrained model, the Δ_n are deterministic functions of the weights, and the expectation over calibration inputs X does not make the cross terms E_X[(J^T J)_{nm}] Δ_n Δ_m vanish for n≠m. The paper should either provide evidence that the off-diagonal Hessian cross-terms are negligible at 2-4 bits, or reformulate the derivation using a stochastic/dithered quantizer that makes the uncorrelatedness assumption meaningful. Without this, Eq. (4) does not actually characterize the rate-distortion optimal bit allocation.
[Appendix B, step (c), and Sec. 3.2] The identity E[Δ_n^2(B_n)] = H_n S_n^2 2^(-2B_n) is a high-rate asymptotic result from Gersho and Gray, and the companding transform in Eq. (8) is derived as asymptotically optimal as B→∞. The experiments, however, operate at 2-4 bits, where the asymptotic expansion is strained, and the paper provides no validation that the measured per-group quantization error follows the assumed exponential form at these depths. A direct diagnostic, such as plotting measured group distortion versus 2^(-2B) for representative layers, or a statement quantifying the approximation error, would be needed to support the claim that Algorithm 1 optimizes the true output distortion.
[Eq. (6) and Algorithm 1, line 15] The printed dual-ascent update in Sec. 3.1, B_n ← clamp(1/2 log2(G_n^2 S_n^2 V / (2 ln2)), 0, 8), appears to invert the argument of the logarithm relative to Algorithm 1 line 15 and the optimality condition in Eq. (4). The correct form, following from -2 ln2 H G_n^2 S_n^2 2^(-2B_n) = -V, is B_n ← clamp(1/2 log2(2 ln2 H G_n^2 S_n^2 / V), 0, 8). As printed, Eq. (6) would allocate fewer bits as V increases, which is the opposite of the intended dual-ascent behavior. This inconsistency must be corrected.
[Discussion, Sec. 5] The statement that the paper formulates weight quantization 'as a convex optimization problem' is inaccurate for the original objective in Eq. (3), which is a discrete, generally nonconvex least-squares problem. Only the high-rate surrogate with fixed G_n^2 and S_n^2 is convex in the relaxed bit depths B_n. The discussion should be reworded to distinguish the surrogate objective actually optimized from the original problem.

minor comments (5)

[Appendix E] The appendix says 'Table 6 lists output produced by different quantized Llama-2-70b models,' but Table 6 is already used for running times; the example outputs should be renumbered as a separate table.
[Sec. 3.2] The citation to '(Young et al., 2021)' for coarse 1D grid fine-tuning does not match the reference list, which contains Young et al. (2019) on transform quantization for CNN compression; please correct the year or the reference.
[Figure 4] The labels 'C4 (Full)' and 'WT2 (Full)' in Figure 4 are not defined in the text; please clarify what 'Full' refers to.
[Reproducibility Statement] The reproducibility statement mentions a 'GitHub project website' but does not provide the URL; please include the exact repository address.
[Table 1] Some 4-bit Radio perplexity values are below the FP16 baseline (e.g., OPT-125M: 27.23 vs. 27.65), which is plausible as a regularization effect but deserves a one-sentence comment in the text.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: Radio's bit-allocation rule is derived from standard rate-distortion theory and validated on independent test sets; the only self-citation is a non-load-bearing fine-tuning reference.

full rationale

Radio's core allocation rule (Eqs. 4-6) is derived in Appendix B from a linearized output model plus the classical Gersho-Gray high-rate quantization result; neither the optimality conditions nor the per-group distortion d_n(B_n) = P_n H_n G_n^2 S_n^2 2^{-2B_n} is defined in terms of the perplexities it later predicts. The bit depths are optimized against a calibration-set distortion objective (Algorithm 1), and the reported WikiText2, C4, GSM8K, and commonsense-QA results are independent test-set evaluations, so the empirical claims are not forced by construction. The companding function (8) follows from the Laplace density and standard rate-distortion companding, not from the target results. The only self-citation (Young et al. for coarse-grid fine-tuning of (S, mu)) is a post-processing hyperparameter choice and is not load-bearing for the rate-distortion derivation. The unverified uncorrelatedness and zero-mean assumptions in Appendix B are correctness risks, not circularity: they are stated approximations imported from classical rate-distortion theory, and the paper explicitly acknowledges the zero-mean failure and patches it with bias correction. Therefore no circular step is identified.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The method adds no new physical or mathematical entities. Its load-bearing support is a set of classical high-rate quantization results applied at low bit depths, plus hand-chosen hyperparameters; no fitted constant is used to produce the reported perplexities.

free parameters (6)

Dual update step size beta = 2
Set in Algorithm 1 and text; affects convergence speed, not final optimum, but is a hand-chosen constant.
Gradient variance smoothing factor alpha = unspecified
Used in Algorithm 1 lines 11 and 13 to accumulate running means of inputs and gradient variances; value not reported anywhere in the paper.
Group size = 512 for OPT, 256 for Llama-2
Selected per model family after hyperparameter search; Table 2c shows it changes perplexity by up to 0.2-0.3 at 3 bits.
Minibatch size = 16
Chosen as optimal in ablations; Table 2a shows low sensitivity.
Number of tokens per sequence = 17
Chosen as optimal in ablations; Table 2b shows low sensitivity.
Maximum optimization iterations = 64
Algorithm 1; Figure 4 shows about 20 iterations needed, so 64 is a safe early-stopped value.

assumptions (5)

domain assumption High-rate quantization distortion relation E[Delta_n^2] = H_n S_n^2 2^(-2B_n) holds at the 2-4 bit depths used
Invoked in Appendix B step (c) from Gersho and Gray; the experiments quantize at 2-3 bits, outside the asymptotic high-rate regime where this relation is exact.
domain assumption Quantization errors Delta_n across layers are zero-mean and uncorrelated
Used in Appendix B steps (a)-(b) to drop the linear term and cross terms. The paper itself notes zero-mean is false and adds bias correction, but uncorrelatedness remains untested.
domain assumption Weight distributions are similar enough across layers that H_n can be treated as constant
Section 3.1: 'Assuming weights are distributed across layers with H1=...=HN, factors Hn and the constant can be removed'; if H_n varies, the optimal bit allocation changes.
domain assumption Weights in each group follow a Laplace distribution for the companding transform
Section 3.2 derives the sigmoid (8) using a Laplace density; the paper notes weights may be Gaussian or Laplace but does not validate per-layer.
domain assumption Second-order residual terms in the Taylor expansion of the distortion are negligible
Appendix B approximates the Hessian as J^T J, dropping the sum of r_m times the Hessian of r_m.

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Cite this review

Pith. "Pith review of Radio: Rate-Distortion Optimization for Large Language Model Compression." pith.science (2026). https://pith.science/paper/I35SPZ26

@misc{pith2026250503031,
  author       = {Pith},
  title        = {Pith review of: Radio: Rate-Distortion Optimization for Large Language Model Compression},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I35SPZ26}},
  note         = {Machine review of arXiv:2505.03031}
}

read the original abstract

In recent years, the compression of large language models (LLMs) has emerged as a key problem in facilitating LLM deployment on resource-limited devices, reducing compute costs, and mitigating the environmental footprint due to large-scale AI infrastructure. Here, we establish the foundations of LLM quantization from a rate-distortion theory perspective and propose a quantization technique based on simple rate-distortion optimization. Our technique scales to models containing hundreds of billions of weight parameters and offers users the flexibility to compress models, post-training, to a model size or accuracy specified by the user.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

40 extracted references · 30 canonical work pages

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Introduction Large Language Models (LLMs) have become a universal framework for solving a wide range of problems in natural language processing, ranging from text summarization and translation to conversational AI. While LLMs have already surpassed traditional methods in many of these tasks, they involve tens to hundreds of billions of weight parameters (...

work page 2023
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Previous Work Earlier work on neural network model quantization can be traced back to Vanhoucke et al. (2011), who demonstrated Radio: Rate–Distortion Optimization for Large Language Model Compression 2 that 8-bit integer arithmetic is sufficient for neural network training and inference without incurring a significant loss of model accuracy. In general, ...

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Quantization Framework Here, we use the task of next-token prediction in language modeling as a running example. For our purposes, the end-to-end mapping of input token embeddings to its predicted next-token embeddings by a pretrained language model can be expressed in the most general form as 𝐙=𝑓(𝐗)=𝑓(𝐗,𝚯1,...,𝚯𝑁,𝐛1,...,𝐛𝑁), (1) in which 𝐗∈ℝ𝐿×𝐸 denotes a...

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Discussion Formulating weight quantization as a convex optimization problem as we have done here yields two benefits. First, it explicates the objective we seek to optimize (minimizing output distortion in this case) and sets us on a path to solve the right problem using modern automatic differentiation tools e.g. PyTorch’s autograd library. Second, it en...

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using backpropagation to facilitate the flow of gradient through to quantization operators. One shortcoming of QAT methods is that model training needs to be repeated for different quantized model bit depths and accuracy, which can render them less suited for quantizing larger neural network models such as LLMs. More recent quantization methods for langua...

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Loss-aware quantization methods (Hou & Kwok, 2018; Nahshan et al., 2020; Qu et al.,

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as well as orthogonal transform of weight matrices prior to their quantization (Ashkboos et al., 2024). While mixed-precision weight quantization is a promising paradigm for handling weights with different sensitivity, current mixed-precision approaches (Wang et al., 2019; Chen et al., 2021; Lee et al., 2024; Dettmers et al.,

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A small fraction of weights is quantized to zero and pruned away due to low variance, with smaller groups increasing the degree of pruning (a)

41.13 45.05 53.24 73.36 77.95 81.69 54.06 57.83 63.64 75.84 77.26 81.66 68.03 71.67 76.24 Radio/256 (3.0000 bits) (Ours) 41.21 45.73 53.84 72.60 77.95 82.32 53.95 58.55 63.86 77.20 78.51 81.88 69.14 71.11 77.43 (c) Scores for 3-bit Llama-2 models on common sense QA (Arc-Challe...
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We can see from Table 4 (a) that Radio-quantized Llama-2 models are considerably more accurate at these bit depths than their OWQ counterparts. This is expected since Radio assigns bit depths from the range (0,𝐵max) commensurately with gradient variances whereas OWQ opts to pr...

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For simplicity, let us rewrite model (4) as 𝑓( ⋅ ,𝚯1,𝚯2,...,𝚯𝑁)=𝑓( ⋅ ,𝚯), where 𝚯 is a vector of all model weights across different layers of the model

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1992
[2006]

One can think of 𝐺12,𝐺22,...,𝐺𝑁2 as the diagonal elements of a non-diagonal Hessian matrix used in e.g

every time the descent direction is re-computed. One can think of 𝐺12,𝐺22,...,𝐺𝑁2 as the diagonal elements of a non-diagonal Hessian matrix used in e.g. the Gauss–Newton method, but whose off-diagonal elements disappear in the expectation due to multiplication by uncorrelated ...

1991
[2011]

companded

is to alternately update the primal 𝐵1,...,𝐵𝑁, and dual 𝑉 variables, with one set held fixed while updating the other variables. After initializing 𝐵1= ⋅ ⋅ ⋅𝐵𝑁=∞, 𝑉 to some small positive number, and computing 𝐺12,...,𝐺𝑁2, both sets of variables {𝐵𝑛},𝑉 can be updated iterative...

2019
[2017]

Choi et al., 2017; Wang et al., 2019), determining the optimal bit depth (Wang et al., 2019; D

and activations (Y. Choi et al., 2017; Wang et al., 2019), determining the optimal bit depth (Wang et al., 2019; D. Zhang et al.,

2017
[2018]

by using mixed bit depth quantization (Wang et al., 2019; Chen et al.,

2019
[2019]

Zhang et al., 2018; Zhou et al.,

integrates the quantization process into training by allowing the model to adapt to the reduced bit precision in weights (Esser et al., 2019; Jacob et al., 2018; D. Zhang et al., 2018; Zhou et al.,

2019
[2020]

Data-free quantization (Nagel et al., 2019; Xu et al., 2020; K

seek to minimize the accuracy loss in quantized models by calibrating quantization and biases on a set of calibration examples. Data-free quantization (Nagel et al., 2019; Xu et al., 2020; K. Choi et al., 2021; Qian et al.,

2019
[2021]

Next Token Prediction

for question-answering tasks. Next Token Prediction. As our first set of experiments, we quantize the Meta OPT and Llama 2 models to 3 and 4 bits on average and measure the performance of the quantized models via perplexity, a stringent accuracy metric. We use a combined row–c...

2022
[2022]

More recent extensions (Dettmers et al., 2023; Lee et al.,

has been proposed for compressing 10–100 billion parameter models. More recent extensions (Dettmers et al., 2023; Lee et al.,

2023
[2023]

For calibration data, we source 128 examples from the training split of the C4 dataset (Raffel et al., 2020)

families of language models (obtained from the HuggingFace Hub), comparing the performance of the proposed quantization method with baselines on next token prediction and question answering tasks. For calibration data, we source 128 examples from the training split of the C4 d...

2020
[2024]

to GPTQ incorporate the handling of the more sensitive model weights by re-scaling them or by retaining the original weight values similar to (Lin et al., 2024; Xiao et al., 2023), low-rank decomposition of quantization error matrices (Shao et al.,

2024
[2025]

Copyright 2025 by the author(s). Abstract In recent years, the compression of large language models (LLMs) has emerged as a key problem in facilitating LLM deployment on resource-limited devices, reducing compute costs, and mitigating the environmental footprint due to large-s...

2025

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [1]

Introduction Large Language Models (LLMs) have become a universal framework for solving a wide range of problems in natural language processing, ranging from text summarization and translation to conversational AI. While LLMs have already surpassed traditional methods in many of these tasks, they involve tens to hundreds of billions of weight parameters (...

work page 2023

[2] [2]

Previous Work Earlier work on neural network model quantization can be traced back to Vanhoucke et al. (2011), who demonstrated Radio: Rate–Distortion Optimization for Large Language Model Compression 2 that 8-bit integer arithmetic is sufficient for neural network training and inference without incurring a significant loss of model accuracy. In general, ...

work page 2011

[3] [3]

Quantization Framework Here, we use the task of next-token prediction in language modeling as a running example. For our purposes, the end-to-end mapping of input token embeddings to its predicted next-token embeddings by a pretrained language model can be expressed in the most general form as 𝐙=𝑓(𝐗)=𝑓(𝐗,𝚯1,...,𝚯𝑁,𝐛1,...,𝐛𝑁), (1) in which 𝐗∈ℝ𝐿×𝐸 denotes a...

work page 2022

[4] [5]

Discussion Formulating weight quantization as a convex optimization problem as we have done here yields two benefits. First, it explicates the objective we seek to optimize (minimizing output distortion in this case) and sets us on a path to solve the right problem using modern automatic differentiation tools e.g. PyTorch’s autograd library. Second, it en...

work page 2022

[5] [6]

using backpropagation to facilitate the flow of gradient through to quantization operators. One shortcoming of QAT methods is that model training needs to be repeated for different quantized model bit depths and accuracy, which can render them less suited for quantizing larger neural network models such as LLMs. More recent quantization methods for langua...

work page 2019

[6] [8]

Loss-aware quantization methods (Hou & Kwok, 2018; Nahshan et al., 2020; Qu et al.,

to improve the accuracy of the quantized model. Loss-aware quantization methods (Hou & Kwok, 2018; Nahshan et al., 2020; Qu et al.,

work page 2018

[7] [10]

Choi et al., 2021)

or using synthetic data in place of real calibration data (K. Choi et al., 2021). For the compression of LLMs in particular, an extension to the Optimum Brain Surgeon (OBS) algorithm (Hassibi & Stork,

work page 2021

[8] [13]

as well as orthogonal transform of weight matrices prior to their quantization (Ashkboos et al., 2024). While mixed-precision weight quantization is a promising paradigm for handling weights with different sensitivity, current mixed-precision approaches (Wang et al., 2019; Chen et al., 2021; Lee et al., 2024; Dettmers et al.,

work page 2024