SandwichQuant: Which Parameters Matter Before and After Quantization?

Low-bit quantization suffers severe accuracy degradation on compact networks, rooted in the dominant full-parameter coupled training paradigm that ignores parameter subspace heterogeneity. Their limited feature redundancy leaves little room to absorb quantization errors. Conventional pipelines adopt monolithic optimization: PTQ reconstructs fixed pretrained models without improving inherent quantization friendliness; QAT updates all parameters jointly, suffering from gradient coupling between backbone weights and calibration parameters. In this paper, we identify normalization affine parameters as a low-dimensional high-leverage subspace dominating quantization robustness, and propose Normalization Affine Preconditioning (NAP) for targeted subspace optimization. For PTQ, NAP freezes backbone weights and fine-tunes only affine parameters under the target fake-quantization graph on full-precision models, proactively boosting quantization friendliness before downstream reconstruction. For QAT, we introduce an alternating QAT-NAP schema that decouples feature learning and numerical calibration, breaking the performance ceiling of saturated joint training. Theoretical analysis confirms BN affine parameters fully cancel the channel-wise affine component of quantization distortion, while nonlinear rounding and clipping residuals form the irreducible error boundary; distillation-guided NAP acts as directional flatness optimization, projecting teacher-student logit mismatch onto the restricted subspace. Experiments on ImageNet and CIFAR-100 show NAP recovers severely collapsed low-bit quantization, consistently boosts reconstruction-based PTQ, and outperforms saturated full-parameter QAT with negligible tuning cost. This work reveals the principle of targeted low-dimensional subspace optimization, offering a new perspective beyond full-parameter coupled training for efficient deep learning.

Post-training quantization (PTQ) compresses large language models by mapping weights to low-bit representations. While much work has focused on how weights are rounded onto the quantization grid, the scaling factor that defines the grid is typically chosen with simple, data-free heuristics. We observe that, under round-to-nearest (RTN) quantization, the grid assignments are piecewise constant in the scale, making the output reconstruction error piecewise quadratic over finitely many intervals, each admitting a closed-form optimization. Building on this insight, we present PiSO (Piecewise Scale Optimization), an algorithm that leverages calibration data to efficiently select channel-wise scales to achieve globally optimal output reconstruction error to arbitrary accuracy. We extend PiSO to group-wise quantization via tractable objectives. We also propose strategies for integrating it with error-correction methods such as GPTQ and Qronos, which greedily adjust weight assignments to compensate for the rounding errors of previously quantized weights. Experiments on Llama and Qwen models across multiple sizes and bit-widths show consistent improvements in perplexity and zero-shot accuracy, both standalone and combined with error correction, with gains increasing as the bit-width narrows. For instance, on Llama-3, PiSO reduces the 2-bit perplexity of Qronos by up to 7x, and at 3 bits, PiSO without error correction outperforms GPTQ with absmax scales.

Low-rank quantization error compensation (LQEC) recovers the accuracy lost under aggressive weight quantization by attaching a closed-form rank-$r$ adapter beside each frozen quantized weight, without any training. We show that existing compensators are limited by two shared simplifications. They calibrate symmetrically, evaluating the full-precision and compensated weights on the same activation, which yields a compensation target that is inherently high-rank -- so a fixed rank budget captures only a small fraction of it. And they minimize only the second-order term of the loss, although the compensated model is not stationary: a first-order descent direction larger than the applied compensation itself remains in every layer, and no reconstruction objective can absorb it. We propose a two-stage closed-form framework that removes both simplifications. Stage 1 aligns each layer's output with the full-precision model under a Fisher-weighted asymmetric objective, concentrating the rank budget on a rank-compressible target. Stage 2 re-measures statistics on the compensated model and applies a rank-constrained natural-gradient step that absorbs the remaining first-order signal. Every adapter is the result of a single truncated SVD; backward passes serve only to collect statistics. At 2 bits under QuIP#, our method reduces WikiText-2 perplexity from 12.43 to 10.26 on Qwen3-8B and from 21.11 to 13.22 on Qwen3-4B. On the held-out C4 corpus, it recovers 51% and 84% of the gap to FP16, versus 31% and 63% for the strongest baseline, with consistent gains in the seven-task zero-shot average, at higher bit-widths, and under a distinct quantizer.