Large language models (LLMs) achieve strong performance but incur high deployment costs, motivating extremely low-bit but lossy quantization. Existing quantization algorithms mainly focus on improving the numerical accuracy of forward computation to eliminate performance degradation. In this paper, we show that extremely quantized LLMs suffer from systematic smoothness degradation beyond numerical precision loss. Through a smoothness proxy, we observe that such degradation becomes increasingly severe as the quantization bit-width decreases. Furthermore, based on sequence neighborhood modeling, we find that quantized models exhibit a rapid reduction of effective token candidates within the prediction neighborhood, which directly leads to a sparser decoding tree and degraded generation quality. To validate it, we introduce a simple smoothness-preserving principle in both post-training quantization and quantization-aware training, and demonstrate that preserving smoothness brings additional gains beyond numerical accuracy. The core goal of this paper is to highlight smoothness preservation as an important design consideration for future extreme quantization methods. Code is available at https://github.com/xuyuzhuang11/FINE.
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.
Post-training quantization (PTQ) is one of the most practical ways to reduce the serving cost of Large Language Models (LLMs), but activation quantization remains difficult because outlier-dominated channels lead to large quantization errors. This paper investigates whether part of this degradation is caused by over-migration in scaling-based equivalent transformations. We introduce a quantile-robust scaling policy for SmoothRot-style transforms by replacing max-based activation statistics with high quantiles, and we complement it with constrained gradient-based optimization of channel scales. On LLaMA-3.2-1B under W4A4 quantization, quantile-only policy search improves selected-layer error by 11.1% over the SmoothRot baseline, joint (alpha, q) search improves it by 12%, and training reaches 18.5%. Replaying the best selected-layer policy on all decoder-block down-projection layers reduces the corresponding full-layer mean error from 97.51 to 78.08 (19.9%). The results show that robust migration control and lightweight scale learning provide consistent gains over max-based fixed policies while preserving the equivalent-transform framework.
Model quantization has become essential for efficient large language model deployment, yet existing approaches involve clear trade-offs: methods such as GPTQ and AWQ achieve practical compression but are lossy, while lossless techniques preserve fidelity but typically do not accelerate inference. This paper explores the middle ground of statistically-lossless compression through three complementary notions of losslessness for quantized LLMs. First, task-lossless compression preserves zero-shot benchmark accuracy within natural sampling variance and remains achievable at aggressive bitwidths. Second, we formalize the stricter notion of distribution-lossless compression, requiring the quantized model's next-token distribution to be practically indistinguishable from the original, and propose the Expected Acceptance Rate (EAR), the maximum token-agreement probability under optimal coupling, as a directly interpretable fidelity metric (for example, EAR >= 0.99 indicates 99% agreement). Third, we prove a gamma-squared variance law showing that symmetric quantization inflates noise variance by gamma squared relative to asymmetric quantization, making asymmetry necessary for distribution-lossless fidelity but not for task-level preservation. Using SLQ, a layer-wise non-uniform method with asymmetric quantization and wide bitwidth search, we achieve task-lossless compression at well below 4 bits per parameter (as low as 3.3 bits depending on the model), distribution-lossless compression at 5 to 6 bits per parameter on average, and inference speedups of 1.7 to 3.6x relative to FP16 with optimized kernels. Source code is available at https://github.com/IST-DASLab/SLQ.