Quantization Effects

Post-training quantization compresses large language models (LLMs) by storing their weights at reduced precision, and each quantized weight introduces an error into the hidden states. Naively, these errors should accumulate with depth and corrupt next-token prediction; randomly initialized models accumulate these discrepancies rapidly, whereas quantized pretrained models accumulate much less hidden-state error and largely maintain downstream task performance, even though they were never trained with quantization noise. This raises the question we address: why does post-training quantization work? Comparing full-precision and quantized forward passes, we identify two mechanisms that characterize pretrained quantization robustness. First, the error a layer newly introduces tends to oppose the error it inherits from the layer's input. The two cancel partially such that the discrepancy between full-precision and quantized passes grows slowly. This counteracting residual interaction develops during pretraining. Our quantitative analysis identifies it as a major factor slowing hidden-error growth. Second, LM-head geometry preferentially preserves the scores and probabilities of high-ranked tokens, which typically represent the model's most confident predictions. Together, these mechanisms explain why quantization error that passes through numerous layers can still produce only small output changes, and we verify the findings across models and quantization settings.

Quantization schemes based on randomized rotations have recently received renewed attention, including the roles of MMSE and unbiased reconstruction scalings. In this note, we point out the connection to classical results in statistical signal processing and communication theory. Specifically, the two reconstruction scales used in the EDEN line of work admit a natural interpretation as finite-dimensional, realization-dependent counterparts of the Wiener and unbiased coefficients in the classical CDEF formulation. At finite blocklength, the CDEF +1 relation holds pointwise for each rotation realization as an exact geometric (Pythagorean) identity, but does not hold after averaging the distortions over the rotation. The classical SNR relation $\sf{SNR}_{\rm MMSE}=\sf{SNR}_{\rm MMSE,U}+1$ is recovered as $d\to\infty$: once the overall scale is handled separately, the empirical coordinate statistics of a randomly rotated vector approach their i.i.d. Gaussian counterparts, and the rotation-dependent quantities concentrate. Importantly, EDEN goes beyond this classical correspondence: for every finite $d$, its Haar-rotation formulation guarantees exact conditional unbiasedness, a stronger property than the second-order notion of unbiasedness in CDEF. We further comment on two distinct roles random rotations play in quantization: one is approximate Gaussianization of the coordinates; the other is decorrelation of reconstruction errors across quantization branches.

Weight quantization largely determines the economics of serving open-weight LLMs. Its costs are usually assessed with capability benchmarks, on which 4-bit quantization of mid-sized models is often considered "nearly free." We examine a different question: when several answers are valid, does quantization change what a model chooses to say? We serve three checkpoints (Qwen3-8B/14B/32B) at three weight precisions (W4A16 AWQ, W8A16 FP8-Marlin, and bf16), holding the hardware, software, and sampling configuration constant, and collect approximately 71,000 completions paired by prompt and seed across two custom, leak-checked prompt batteries. We pre-specified the analyses in three waves in version control. At 8B, int4 reduces output diversity: the probability that two samples for the same scenario recommend the same brand increases by 5.1 percentage points (prompt-paired sign-flip test, Holm p = .023; reproduced at +4.4pp on a full regeneration of the arm), and lexical diversity falls substantially (TTR -0.011, standardized effect -0.51; robust to a length-controlled measure). At 14B and 32B, no content-concentration measure reaches significance; instead, stylistic drift emerges (em-dash rate +0.46/1k words at 14B and +0.61/1k at 32B, both Holm p <= .0024). Pre-specified tests of stereotype direction are null at every scale: outputs concentrate on the modal answer for each prompt rather than on stereotypical answers. Mechanistically, the token-level distribution becomes flatter (decision-token entropy +0.091 bits, p = .015) while the semantic distribution, measured directly from first-token log probabilities, becomes more concentrated (collision +2.6pp, p = .023): individual tokens become less predictable even as meanings become more repetitive. At 8B, the smallest size tested, AWQ-int4 serving measurably narrows the range of suggestions; audits should assess concentration as well as bias.

Quantization is widely used to reduce the computational and memory demands of neural-network inference. In recurrent networks, however, the quantized state is stored and returned at the next time step, so the rule used to store that state can alter subsequent computations. Here, we introduce recurrent-state write-back to denote this rule and isolate its effect in a compact GRU encoder--decoder for fluorescence lifetime imaging, a molecular imaging modality used in quantitative biological imaging. A central task is estimating two lifetime parameters, the short-lived component {\tau}1 and the long-lived component {\tau}2, from high-noise time-resolved fluorescence signals. Holding the trained model fixed, replacing continuous state propagation with deterministic 4-bit state storage increases estimation errors for {\tau}1 and {\tau}2 by approximately 70x and 300x, respectively. Failure occurs when repeated small updates remain below the write threshold, leaving the stored state nearly fixed while the network continues to propose change. Error feedback, residual memory, and direction memory carry information from these suppressed updates across time and recover accuracy without retraining. Precision sweeps show that increasing state precision can worsen a fixed recurrent solution, while matched training shows that compatibility with the state interface can be learned. To test whether this behavior extends beyond the GRU, we repeat the post-training intervention in an independently trained LSTM, where coarse write-back reproduces the failure, error feedback restores accuracy, and state-specific interventions reveal greater sensitivity of the cell state than the hidden state. Our results establish recurrent-state write-back as a key determinant of low-precision recurrent dynamics and identify the state-storage interface as a central design consideration for quantized recurrent inference.

Aggressive quantization disproportionately harms multilingual capability: in the sub-4B INT3 GPTQ regime, we measure 2-4x larger perplexity degradation on non-English languages than on English. We propose Language-Conditional Dequantization (LCD), a post-hoc method that attaches per-language rank-2 LoRA corrections to the linear layers of an already-quantized model, adding 0.12% parameters per language and training in under 20 minutes on a single GPU. Across Qwen2.5-3B and Llama-3.2-3B, LCD recovers 70-83% of the perplexity gap for non-Latin script languages and 17-28% of the GlobalMMLU accuracy gap, outperforming a language-agnostic correction of equal capacity by 3-9 points on typologically distant languages and a data-free low-rank baseline (LQER) by an order of magnitude. We further identify a perplexity-accuracy disconnect and trace it to where quantization concentrates damage: early-depth errors (Llama) propagate downstream and resist local correction, while late-depth errors (Qwen) do not. A layer-restricted variant of LCD validates this mechanism directly.

Post-training quantization reduces the deployment cost of large language models, yet how severely a quantized model degrades is not determined by bit-width alone. We systematically study weight-only post-training quantization across bit-widths, quantization methods, model scales and downstream tasks on multiple model families. We observe that such degradation varies substantially across these factors: 4-bit quantization usually preserves performance, 2-bit often causes broad degradation, and at 3-bit, degradation becomes apparent but varies markedly with task type, quantization method and model scale. To explain this variability, we use the signal-to-noise ratio (SNR) to measure how strongly quantization perturbs full-precision representations. We trace degradation back to two linked processes: how quantization errors arise within individual modules, and how they accumulate across layers. First, a source SNR decomposition shows that newly introduced errors depend on three factors: the magnitude of the weight error, the strength of the task-specific signal, and how strongly the quantization error aligns with task-specific activations. Different factors affect these components in distinct ways. Second, a cross-layer propagation analysis shows that these errors can be attenuated, preserved, or amplified as they pass across layers, and that larger models benefit from weaker error amplification. Together, these results establish that quantization degradation is governed by how errors are introduced at the source and how they accumulate across the network.

Quantization is known to hurt below four bits, but nobody can say which of a model's decisions will change at a given bit-width. This matters most where a model acts rather than answers: a compressed agent stops calling its tools and, one bit lower, loses roughly half its safety refusals, while benchmark scores barely move. Prior work assumes the added noise has a roughly fixed size, which would make confident decisions safe. We measure the decision instead: the margin, the picked option's score minus its best alternative's, tracked before and after quantization across 16 models from 8 families under round-to-nearest, seven under AWQ, two under GPTQ and one under GGUF, at 8 down to 2 bits. The damage is proportional, not fixed in size: the margin is multiplied by a factor that collapses with bit-width (median 0.86 at 4 bits, 0.33 at 3, 0.00 at 2), which we call margin shrinkage. Contraction removes the protection a large margin affords; the model's own biases pick the direction: at 3 bits the decision to call a tool collapses toward inaction while the choice of which tool is untouched. No additive account, including one whose noise grows with the margin, wins a single damaged whether-to-call or safety cell (378 of 378). Given a condition's own constants the relation predicts held-out flip rates to a median 1.7 points, calibrated per decision (error 0.004 over 161,744 predictions), no flip used in the fit. Borrowed constants are wrong by 18-33 points at 3 bits, so the paired margin set has to be measured per model and bit-width: it locates breaking decisions without replacing measurement. At 4 bits the measurement is anchored to behaviour (the most likely token over the whole vocabulary is one of the item's two options in 85% of tool items); we treat the 2-bit floor as where the instrument stops measuring. No label-free repair we tested recovers more than one more bit does.

When decoder language models are used as classifiers, predicted class probabilities depend on implementation choices, including the prompt template, verbalizer (label-to-token mapping), and scoring rule, that are rarely treated as experimental variables. We present a controlled evaluation of three Mistral-7B variants (Base, BioMistral, and Instruct) on PubMed RCT sentence classification (n=2000) under FP16, INT8, and INT4 precision using four answer-text prompt templates. Our primary finding is that the probability extraction protocol dominates apparent calibration. Switching from summed to mean token log-likelihood scoring reverses the calibration ranking between models: BioMistral average expected calibration error increases from 0.097 to 0.289, whereas Instruct decreases from 0.237 to 0.096, while accuracy changes by less than 1 percentage point for the specialized models but 4-6 percentage points for the base model. Prompt template choice produces accuracy differences of 7-24 percentage points, comparable to or larger than model-level effects. On one template, BioMistral outperforms Instruct although the overall mean favors Instruct by only 1.3 percentage points. For BioMistral and Instruct, INT8 quantization changes accuracy and F1 by only 1-2 percentage points relative to FP16, whereas the base model shows larger INT8 effects on some templates (up to +4.2 percentage points). INT4 produces heterogeneous but non-catastrophic effects. Temperature scaling reduces expected calibration error under summed scoring for both models but only for that scoring rule. A fine-tuned PubMedBERT reference achieves 82.7% accuracy but uses about 176000 labeled training examples, precluding direct comparison. These results demonstrate that prompt template design and scoring normalization are first-order experimental decisions when evaluating decoder language model calibration.

Deploying large language models in realistic server environments poses challenges, as the system needs to provide high-quality responses with low latency. Quantization is a common approach to reduce the memory footprint and improve inference efficiency, yet its impact on latency and throughput is rarely evaluated under controlled, orchestration-level workloads. In this work we study the quantization trade-offs of two translation model families, EuroLLM \citep{martins2025eurollm} and Hy-MT2 \citep{zheng2026hy} across five models ranging from 1.7B to 22B for efficient deployment on a single A100 or H100 GPU. We demonstrate that combining a document-chunking strategy with W4A8 or W8A8 quantization improves the latency-throughput Pareto-curve under a wide range of workloads. Furthermore, since standard machine translation (MT) benchmarks rely on isolated sentences and fail to capture long-context dynamics, we introduce a document-level evaluation from WMT24++ to assess how text chunking strategies affect translation quality under quantization. Our results reveal that standard segment-level evaluation can fail to predict the interaction between quantization and long-context document translation. While Hy-MT2 remains robust under quantization, EuroLLM shows strong sensitivity and translation quality collapses rapidly for all considered quantization formats. Overall, our experiments show that the trade-off between inference efficiency and translation quality depends not only on the quantization format, but also on the choice of text chunking strategy.

Parameter-efficient fine-tuning (PEFT) and low-bit quantization are now standard tools for adapting language models under tight compute budgets, yet their interaction is most often studied on billion-parameter models where the design space is expensive to explore. We ask a complementary question: on a specific, fully reproducible 60M-parameter encoder-decoder model (T5-small) and a single-table text-to-SQL benchmark (WikiSQL), how much task accuracy does each efficiency knob actually cost? We run a controlled, single-variable study over (i) LoRA rank r in {2, 4, 8, 16, 32}, (ii) the set of adapted modules, and (iii) numerical precision. We report task accuracy alongside system-level metrics including trainable parameters, peak training memory, inference latency, and throughput, and frame adaptation as a constrained trade-off rather than an accuracy-only objective. Our results show that LoRA with r=16 recovers within 11.6 percentage points of full fine-tuning accuracy (59.6% vs. 71.2% exact-match) while training fewer than 1% of parameters and consuming 31% less peak GPU memory. Within this setting, rank beyond r=16 yields no measurable accuracy gain. QLoRA with INT8 and NF4 quantization achieves comparable accuracy (52.8% and 53.2%) at dramatically lower memory cost (0.60 GB each), demonstrating a compelling trade-off for memory-constrained deployments. All code, configurations, and logs are released for full reproducibility.

Vision-language models (VLMs) deployed on consumer hardware must decide when to answer and when to defer, and that decision depends on having a confidence signal that tracks correctness. A practitioner with a fixed memory budget faces a choice between a small model at full precision, the same small model quantized, and a larger model quantized into the same footprint -- three configurations that push the confidence signal in opposing directions. We measure, on identical inputs, how model scale and 4-bit quantization affect two confidence signals in the Qwen2-VL family: the confidence a model states in natural language, and its own mean token probability over the answer it generates. Across 5,700 predictions spanning six realistic photographic degradations at three severities, we find that scale sharply improves the model's internal uncertainty signal (mean error-detection AUROC 0.80 to 0.98 from 2B to 7B) while its verbalized confidence stays weak and often at chance (mean 0.61 to 0.69): the gap between what the model knows and what it says widens rather than closes with size. We find that 4-bit quantization is nearly free for accuracy (-1.6 points) but expensive for the confidence signal (internal AUROC 0.95 to 0.80, and the verbalized-confidence parse rate collapses from 99% to 64%). For a fixed memory budget the recommendation is therefore to prefer a larger quantized model over a smaller full-precision one: 7B-4bit gives both the best accuracy and the best uncertainty signal (internal AUROC 0.98) of the three configurations that fit. We frame the results as selective-prediction operating points so they translate directly into a deployment recommendation, and we argue that error-detection AUROC, not calibration error, is the metric that exposes the difference between the two signals.

Post-training quantization (PTQ) has become a practical solution for deploying deep learning models on resource-constrained edge devices by compressing high-precision floating-point weights into low-precision representations without requiring retraining. Past research has demonstrated that quantization largely preserves classification accuracy; however, whether it also preserves the model's internal reasoning remains an open question. This study presents a systematic evaluation on how static PTQ affects the interpretability / explainability of five widely used CNN architectures: VGG19, ResNet18, EfficientNet-B0, DenseNet161, and MobileNetV2 at INT8 and INT4 precision. We employ a dual interpretability framework that combines Grad-CAM for spatial attention analysis with LIME for input-level feature attribution, and systematically compare full-precision and quantized models on two binary classification datasets. Interpretability is evaluated using three complementary metrics: the Pearson correlation coefficient, structural similarity index, and top-20% IoU to capture distributional and structural variations in model explanations, supplemented by deletion/insertion faithfulness analysis. The results show that classification accuracy is not a reliable indicator of interpretability stability under reduced precision. DenseNet161 maintains strong feature consistency across both precision levels, whereas EfficientNet-B0, despite achieving competitive spatial attention and classification accuracy at INT8 precision, exhibits a substantial degradation in input-level feature attribution. These findings have direct implications for the trustworthy deployment of quantized models in applications with high interpretability requirements, demonstrating that architecture selection is as important as the quantization strategy.

The growing adoption of local inference frameworks such as Ollama has made it increasingly common for developers to run large code models on laptops and other resource-constrained hardware. In these settings, post-training quantization is essential for reducing memory footprint and enabling practical deployment, yet its impact on generated code remains insufficiently understood. We empirically evaluate six state-of-the-art quantization methods (GPTQ, AWQ, QuIP#, AQLM, BitsAndBytes, and GGUF) on two representative large code model families, Qwen2.5-Coder and CodeLlama, using the multilingual McEval and CoderEval benchmarks for Python and Java. We assess functional correctness (pass@1) together with maintainability, reliability, security, and structural complexity. We also introduce a novel analysis of robustness under varying prompt complexity, characterized by Shannon entropy and token length. Our results show that quantization techniques differ meaningfully in their impact on correctness and code quality. AQLM consistently matches or exceeds the full-precision baseline, whereas QuIP# exhibits the largest correctness degradation, particularly on complex prompts. Security attributes remain stable across models, benchmarks, and programming languages, while robustness to prompt complexity varies across techniques. These findings provide practical guidance for selecting quantization strategies for deploying large code models on resource-constrained hardware and highlight the importance of evaluating quantized models beyond functional correctness.

Reported serving speedups from quantized kernels typically bundle the weight format, the kernel, and the inference runtime into one number. We present an attribution study on four NVIDIA RTX A5000 GPUs, 24 GiB each, on a single host with NVLink-bridged pairs. A matched intermediate stack that keeps the faster runtime without the quantized kernel splits the full speedup into a runtime part and a kernel and quantization part. Under matched greedy decoding the full stack reaches $2.58\times$ end to end, with the runtime change accounting for about two thirds of that gain on a logarithmic scale; across three similar model families the kernel and quantization part moves by at most 1.5%. Sharding one instance across all four cards falls well below doubling: a profiler trace attributes about 80% of the per token shortfall to coordination, and an NVLink versus PCIe control on the same hardware shows similar realized bandwidth on both links, pointing away from link bandwidth as the cause. Whether to run one sharded instance or several independent ones depends on the workload and the model, with the ranking reversing on the larger model: the smaller model splits between sharding and multiple instances by workload, while the larger model favors two paired instances on every workload. Quantization extends sustainable concurrent users roughly four times past a reproducible half precision memory cliff. Differences in sampling mode and prompt pool between the two stacks are documented as threats to validity.

Post-training quantization (PTQ) compresses deep neural networks for deployment under limited memory and computational budgets. However, low-bit (i.e., 2-bit or 4-bit) PTQ often suffers from substantial performance degradation. Most existing PTQ methods operate on an unconstrained full-precision (FP) model and primarily address quantization errors through post-hoc reconstruction. We argue that low-bit PTQ accuracy is limited not only by post-quantization error minimization, but also by the quantization-error tolerance of a FP model itself. In this paper, we propose Efficient Tuning Before Quantization (ETBQ), a pre-conditioning tuning stage for Stochastic Gradient Descent (SGD)-optimized models before PTQ. During tuning, the FP model is optimized under perturbations sampled from the error distributions of weight and activation quantization, guiding the model toward a loss-landscape region that is less sensitive to the subsequent PTQ. Unlike QAT, ETBQ does not train a fake-quantized deployment model, which is computationally and memory intensive. Instead, ETBQ outputs a FP model that can be used by any PTQ backend. Experiments on CIFAR-100, Tiny-ImageNet, ImageNet, and Cityscapes provide consistent evidence that ETBQ improves low-bit PTQ across diverse tasks. Under W2A4 settings, e.g., ETBQ improves over naive PTQ by 2.14% top-1 accuracy on Tiny-ImageNet and by 5.80% mIoU on Cityscapes. Code is available at https://github.com/xpxpxp2001xpxpxp/ETBQ.

Low-bit quantization makes small reasoning models inexpensive to deploy but can degrade their chains of thought. This motivates decoder-side monitors that intervene when generation becomes unreliable. We show that a natural candidate, the centered token log-probability increment $\log p(w_t)+H_t$, is the wrong observable for this purpose. Under the model's own sampling law it is a mean-zero martingale by construction, so it measures sampling self-consistency rather than trajectory health and is nearly silent during confident repetition, where both $\log p(w_t)$ and entropy are close to zero. We introduce a training-free decoding controller that combines (i) a degeneration-aware alarm score fusing token uncertainty with explicit verbatim repetition and (ii) a calibrated e-process-inspired sequential detector. The raw product process is Ville-valid under a conditional-mean null, while the deployed CUSUM-floored statistic is treated as an empirical change detector because the score is history-dependent and autocorrelated. On GSM8K with DeepSeek-R1-Distill-Qwen-1.5B in FP16 and INT4, calibration turns a monitor that fires on 93--95% of generations into a selective detector of failing traces ($φ\approx 0.3$, precision $\approx 0.6$ against a 0.38 base rate). In this pilot, the controller reduces measured verbatim-degeneration signals and yields a positive but statistically inconclusive INT4 accuracy change from 63% to 69% (paired McNemar $p=0.18$, $n=100$), at a 28% token-budget cost. We also find that non-termination, rather than looping, is the dominant failure mode on GSM8K. The main contribution is methodological: an explanation of why centered token log-probability is inadequate for decoder monitoring and a calibrated, cautiously evaluated replacement.

Neural networks can learn algorithmic input-output mappings, but trusting a learned executor requires more than a correct final answer because the state transitions that produce it are usually hidden. To make those transitions visible, we introduce a trace-supervised symbolic neural CPU, a factorized learned execution architecture that combines recurrent control, an explicit operation router over a fixed differentiable arithmetic-logic unit bank, destination-masked register writeback, complete trajectory supervision and matched fixed-point replay. The model exposes the selected operation, source and destination registers, register trajectory, memory signals and writeback semantics at every step. On the principal 16-wide benchmark, the non-quantized executor reproduces reference execution exactly, while the eight-bit quantization-simulated executor preserves the symbolic operation path through programs of 1,000 instructions. When the same execution is evaluated against a matched fixed-point replay, the residual numerical drift disappears, showing that it comes from a mismatch between continuous and low-precision reference semantics rather than from execution failure. We compare recurrent, Transformer, temporal-convolution, temporal graph-inspired and state-space controllers, and the ablations show that operation-gate supervision is necessary for an inspectable execution path. Hidden-opcode memory-pressure tasks expose the remaining limits in delayed state use and temporal binding. We also extend the interface with ValueMemory, hybrid adaptive leaky integrate-and-fire controllers, candidate-constrained symbolic control trained through behaviour cloning and actor-critic reinforcement learning, and an RV32I base-integer semantic bridge. Together, these results establish a trace-verifiable framework for interpretable, low-precision and controllable neural execution.

Post-training quantization is widely used to deploy large language models in resource-constrained settings, yet its evaluation relies almost exclusively on accuracy and perplexity. We show that these metrics fail to capture behavioral changes induced by quantization. We introduce correctness agreement, a decision-level metric that measures overlap in correct predictions between a base model and its quantized variants, independent of absolute accuracy. Across multiple models and quantization schemes from 8-bit to 2-bit, we find that behavioral divergence emerges under moderate quantization even when task performance appears preserved. To explain this effect, we analyze quantization as a structural operator on attention weights and quantify layer-wise distortions using statistical and distributional measures. Our results reveal non-linear breakpoints at low bit-widths and show that query and key projections are consistently more sensitive than value and output projections. These findings expose an illusion of equivalence between base and quantized models and motivate behavioral evaluation beyond conventional performance metrics.

We study the spectral perturbation of the empirical Fisher Information Matrix (FIM) of a parametric statistical model under two structured perturbations: departure of the input from a reference (in-distribution) ensemble, and finite-precision (quantized) perturbation of the model's parameters. For the first, under an explicit local curvature-monotonicity hypothesis on the dominant eigenvalue lambda_max of the FIM, we show departure from a reference manifold provably elevates lambda_max relative to a calibration baseline (Proposition 3.2), and discuss why this hypothesis is required, since curvature need not increase monotonically under every perturbation. Our principal result is a directional eigenvalue perturbation bound, via Weyl's inequality, showing lambda_max under a quantization noise perturbation is lower bounded by its unperturbed value up to a third-order remainder, and, under a mild genericity condition, strictly exceeds it at leading order (Theorem 4.3). We give two tractable approximations to lambda_max -- one heuristic, one with a rigorous two-sided bound -- and a completeness result for a threshold-based partition of an augmented state space. These results motivate using sigma_t = lambda_max(F_t)/lambda_base as a runtime monitoring statistic for deployed language models: the quantization result offers a mechanism for an empirical observation of our own, where a calibration threshold for this statistic was approximately 244 times larger than a preliminary full-precision estimate on a 4-bit quantized model, a single measurement rather than a value derived in closed form. We report supporting measurements (twelve models, n=1,080 trajectories) broadly consistent with our predictions, discuss the scope and limitations of every result, and state as an open problem the closed-form prediction of the quantization inflation magnitude our bound does not supply.

In recent years, weight quantization that encodes the learnable parameters of large language models in an $n$-bit format has garnered significant attention due to its potential for model compression and inference acceleration. Many practical techniques have been developed; however, the theoretical understanding of many aspects, especially the approximation and degradation of expressive power as the number of quantization bits decreases, remains unclear. In this paper, we provide a theoretical investigation into the expressive capability of large language models relative to the number of quantization bits. We argue that 1.58-bit is the limiting precision for weight quantization by establishing the universal approximation and expressive collapse properties of weight-quantized models with respect to the number of quantization bits. Additionally, we confirm that weight quantization leads to expressive degradation, in which the expressive capacity of weight-quantized models degrades polynomially as the number of quantization bits decreases. These theoretical findings provide a solid foundation for advancing weight quantization in the context of scaling laws and shed insights for future research in model compression and inference acceleration.

As large language models scale, memory bandwidth for key-value caches and retrieval-augmented generation systems becomes a critical bottleneck. While 1-bit quantization addresses this constraint, recent TurboQuant relies on dense random rotation matrices to condition the vector distribution before quantization. This projection demands millions of floating-point multiplications per embedding, making it difficult to deploy on constrained edge silicon. We introduce Fast-TurboQuant, a multiplier-free projection architecture that replaces the dense matrix with a structured fast Johnson-Lindenstrauss transform. By applying a Rademacher phase inversion followed by a fast Walsh-Hadamard transform (FWHT), the method leverages sub-Gaussian concentration to satisfy the prerequisites of scalar Lloyd-Max quantization without Gaussian projections. This substitution reduces the arithmetic complexity to only additions, eliminating hardware multipliers. Evaluation on DBpedia OpenAI-3 Large embeddings demonstrates a 19.7 times algorithmic speedup under sequential execution. Furthermore, the dimension expansion due to the FWHT zero-padding reduces the mean squared error and improves Recall@10.

We introduce the Trajectory-based Quantization Sensitivity Score (TQS), a metric that reframes post-training quantization (PTQ) through the lens of dynamical-systems stability. By modeling the network's rollout as a discrete-time dynamical system, TQS characterizes how quantization-induced errors propagate and amplify over the rollout horizon. Unlike conventional PTQ methods, where sensitivity analysis is often coupled to the quantization procedure, TQS enables a priori sensitivity estimation decoupled from quantizer selection and bit-width assignment. This separation allows for quantization budget planning even for black-box or compiled networks with fused operators. Building on this, we present TQS-PTQ, a flexible mixed-precision framework that requires no calibration data or costly second-order approximations. Our experiments show that a dynamical-systems perspective provides a robust, high-performing pathway for low-precision deployment in resource-constrained settings.

Quantized checkpoints are often screened first with quality metrics and only later, if at all, with direct safety tests. This paper audits that shortcut on a matched 51-row matrix spanning 6 models, 4 families, a 7-level GGUF ladder, and AWQ/GPTQ INT4 checkpoints. In this matrix the shortcut fails: all 36 quality-safety pairings split direction across models, and 9 hidden-danger rows plus 1 near-hidden-danger row show quality stable or improved while refusal falls by 12-68 percentage points. Seven of the 11 AWQ/GPTQ rows are hidden-danger. A four-probe mechanistic follow-up over the 17 Hugging Face-backed FP16/AWQ/GPTQ cells does not rescue it: entropy, refusal-direction, and calibration probes are weak or null separators of dangerous rows, and although probe-identified safety-associated neurons absorb 1.39$\times$ more quantization error overall ($p < 5 \times 10^{-7}$), the effect is not regime-specific. Claude Sonnet 4 relabels 11,470 items in a predefined stratified set, agrees with the primary gemma3:12b judge on 89.9% of rows ($κ= 0.873$, 95% CI [0.866, 0.881]), and changes 0/10 hidden-danger cells. A calibrated study-internal behavioral screen -- the Refusal Template Stability Index (RTSI), built from four refusal-template drift features and calibrated on this matrix -- routes 10/10 hidden- or near-hidden-danger rows to direct safety testing (Wilson 95% CI lower bound 0.72) while leaving 23 of 45 non-baseline rows in a low-risk bucket under both in-sample scoring and row-level leave-one-out validation; on the same matrix, the best single-feature baselines (unique-prefix-rate-delta, raw refusal-rate delta) recover 9/10 and 8/10 respectively at matched bucket size, and cross-stack transfer requires recalibration. For the quantized checkpoints, model families, and safety outcomes studied here, retained quality cannot waive direct safety evaluation.

Large-scale transformer training and deployment are increasingly constrained by the transfer of activations, gradients, and optimizer states across accelerators. Low-bit quantization offers a natural remedy, but transformer activations are often heavy-tailed and outlier-dominated, making simple quantization highly lossy. We show that this difficulty is not only a property of the quantizer, but also of the architecture. Specifically, residual connections can drive transformer activations away from Gaussianity during training. Using controlled comparisons between residual and residual-free transformers, we demonstrate that this effect leads to substantially higher quantization error and accuracy degradation at low precision in residual models. We explain the phenomenon through an excess kurtosis analysis, showing that residual mixing can amplify non-Gaussianity, whereas dense mixing in residual-free contracts non-Gaussianity. We then show that residual-free transformers can be made trainable using orthogonal initialization, spectral or second-order optimization, and depth-aware scaling of attention temperature. In language tasks, while there is a small drop in full precision performance, these models retain near-Gaussian activations and exhibit significantly improved robustness to low-bit quantization. Our results identify an accuracy--compressibility trade-off in transformer design and motivate architecture-level approaches to quantization-friendly foundation models.

Existing scaling laws for Large Language Models (LLMs), predominantly monotonic power laws, fail to explain emerging non-monotonic phenomena such as catastrophic overtraining and quantization-induced degradation, where performance deteriorates despite increased compute. We propose the Shannon Scaling Law, a unified theoretical framework that models LLM training as information transmission over a noisy channel, grounded in the Shannon-Hartley theorem. By mapping model parameters to channel bandwidth and training tokens to signal power, our formulation explicitly captures the interaction between learning signal and intrinsic noise. This perspective reveals a fundamental Shannon capacity for LLMs: scaling model size or data without preserving a sufficient signal-to-noise ratio (SNR) inevitably amplifies noise, inducing a transition from monotonic improvement to U-shaped performance degradation. We validate our theory through experiments on Pythia and OLMo2 under perturbations, including Gaussian noise, quantization and supervised fine-tuning on math, QA and code tasks. The Shannon Scaling Law consistently outperforms classical scaling laws and recent perturbation-aware laws, achieving strong $R^2$ scores and accurately capturing loss basins missed by prior approaches. It also extrapolates: fitted on $\leq$6.9B Pythia models with $\leq$180B tokens, it predicts the unseen 12B model up to 307B tokens at pooled $R^2{=}0.847$, while monotonic baselines collapse.

Quantization can preserve predictive accuracy under low-bit deployment while silently breaking algorithmic recourse: an actionable change that flips a decision before quantization may fail after quantization, or become substantially more costly. We formalize counterfactual sensitivity under quantization through validity, cost, and direction stability, and introduce two metrics: Validity Drop (VD) and Counterfactual Recourse Gap (CRG) that reveal recourse failures invisible to accuracy. We propose Counterfactual-Faithful Quantization (CFQ), which trains quantizer parameters and mixed-precision bit allocation to preserve counterfactual behavior by enforcing the target outcome at teacher recourse points under a global bit budget. A margin-based analysis gives a sufficient condition for recourse transfer under bounded quantization perturbations. Experiments on Adult, German Credit, and COMPAS show that accuracy-matched baselines can significantly degrade recourse stability, while CFQ maintains accuracy and substantially improves VD and CRG across bit budgets.

The Muon optimizer has emerged as a compelling alternative to Adam for training large language models, achieving remarkable computational savings through gradient orthogonalization. However, Muon's optimizer state is more sensitive to quantization errors: because the orthogonalization discards the magnitudes of singular values and retains only directional information, even small quantization errors in singular vector directions are amplified in the update. In this work, we propose MuonQ, a low-bit Muon training framework built on the principle of directional fidelity optimization. First, we apply a pre-quantization normalization so that each step introduces quantization errors of the same magnitude, preventing the accumulated error from developing a preferred direction. Second, we introduce a structural decomposition that separately quantizes the dominant singular components via power iteration, ensuring that quantization errors perturb only singular value magnitudes rather than rotating singular vector directions. Third, we adopt $μ$-law companding quantization to allocate higher resolution to densely packed momentum values, shifting the quantization objective from outlier preservation to dense-region distinguishability. Together, these techniques enable stable 4-bit quantization of Muon's optimizer states. Pre-training experiments on GPT-style and LLaMA-style models demonstrate that MuonQ at 4-bit precision closely matches full-precision Muon in both training loss and downstream task accuracy, while reducing optimizer state memory by up to 7.3 $\times$. Our code is available at https://github.com/YupengSu/MuonQ.

Deep learning models are the most efficient models in many machine learning tasks. The main disadvantage when using them in IoT, mobile devices, independent autonomous or real-time systems is their complexity and memory size. Therefore, much research has concentrated on compression techniques of deep learning architectures. One of the most popular technique is quantization. In most of the works, the quantization is done based on the nearest neighbour quantization technique. This work focuses on improving the quantization efficiency in pretrained and quantized models. This approach has the potential to improve the final accuracy of quantized models. The main postulate of the work is that final quantization states of the network based on nearest neighbour rounding does not guarantee optimal accuracy. In the presented work, the evolution strategy is used as an optimization approach. The evolution in each iteration changes the values of the small percentage of weights. It shifts theirs values to different quantization states. The work shows that proposed evolution with an appropriate set of operators and parameters can fast improve the accuracy of the quantized models. The results are presented for popular architectures such as VGG and Resnet for image classification and detection. Additionally, simulations were carried out for the autoencoder architecture.

Natural Language Inference (NLI) is a central task in natural language understanding with applications in fact-checking, question answering, and information retrieval. Despite its importance, current NLI systems heavily rely on supervised learning with datasets that often contain annotation artifacts and biases, limiting generalization and real-world applicability. In this work, we apply a reinforcement learning-based approach using Group Relative Policy Optimization (GRPO) for Chain-of-Thought (CoT) learning in NLI, eliminating the need for human-labeled rationales and enabling this type of training on challenging datasets such as ANLI. We fine-tune 7B, 14B, and 32B language models using parameter-efficient techniques (LoRA and QLoRA), demonstrating strong performance across standard and adversarial NLI benchmarks. At the 32B scale, GRPO-trained models generalize better than other supervised baselines in adversarial sets. With AWQ quantization, the 32B model fits within 22GB of CUDA memory. This work provides a scalable and practical framework for building robust NLI systems without sacrificing inference quality.