Generalization behavior of OPTQ and the role of regularization

Post-training quantization (PTQ) compresses large language models by mapping weights to low-bit representations. The scaling factor that defines the quantization grid is typically chosen using simple, data-free heuristics. In this work, we present PiSO (Piecewise Scale Optimization), an algorithm that leverages calibration data to compute the optimal channel-wise weight scales exactly and efficiently under round-to-nearest quantization. PiSO partitions the scale search space into finitely many intervals on which the objective admits a closed-form minimizer. We extend PiSO to group-wise quantization via principled heuristics and propose effective strategies for interleaving scale optimization with error correction. Experiments on Llama and Qwen models across multiple model sizes and target weight bit-widths demonstrate consistent improvements in perplexity and downstream zero-shot accuracy, both standalone and combined with error correction. In particular, we observe increased benefits as the target bit-width narrows and quantization becomes more challenging.

One of the fundamental challenges in the deep learning community is to theoretically understand how well a deep neural network generalizes to unseen data. However, current approaches often yield generalization bounds that are either too loose to be informative of the true generalization error or only valid to the compressed nets. In this study, we present a simple yet non-vacuous generalization bound from the optimization perspective. We achieve this goal by leveraging that the hypothesis set accessed by stochastic gradient algorithms is essentially fractal-like and thus can derive a tighter bound over the algorithm-dependent Rademacher complexity. The main argument rests on modeling the discrete-time recursion process via a continuous-time stochastic differential equation driven by fractional Brownian motion. Numerical studies demonstrate that our approach is able to yield plausible generalization guarantees for modern neural networks such as ResNet and Vision Transformer, even when they are trained on a large-scale dataset (e.g. ImageNet-1K).

Post-training quantization (PTQ) methods in the GPTQ family minimize a layer-wise reconstruction error on a uniform grid whose scale must be chosen; the common max-based choice degrades sharply at low bit-widths. We study how sensitive this objective is to the scale. For a layer with i.i.d. Gaussian weights and calibration activations of sufficiently large effective rank, we prove that, as the width grows, the normalized round-to-nearest loss converges with high probability, uniformly over all scales, to the mean-squared error of a uniform quantizer applied to a standard Gaussian; we verify the effective-rank condition for wide, randomly initialized MLPs with odd Lipschitz activations and isotropic Gaussian calibration data. The limiting objective has a unique nondegenerate minimizer, whose scale decreases strictly with the number of levels and whose curvature with respect to relative scale errors decays approximately exponentially with the bit-width. GPTQ experiments on five LLMs show the same trend: the scale rule changes perplexity substantially at 2--3 bits and negligibly from 6 bits on, and a local measure of GPTQ scale sensitivity decreases with bit-width in line with the Gaussian curvature. The Gaussian-optimal scale fails on raw weights; after Hadamard incoherence processing it matches the best searched rule at 3 bits and above without any search, but remains clearly worse at 2 bits.