Rethinking Numerical Stability in Hessian-Based PTQ: Accumulation-Order Sensitivity

Post-Training Quantization (PTQ) is widely used to reduce the computational and memory costs of large language models. Hessian-based PTQ methods such as GPTQ estimate empirical second-order statistics from calibration samples to guide weight quantization. Although the sum of these statistics is invariant to sample order in exact arithmetic, floating-point non-associativity can yield different empirical Hessians depending solely on the order in which the same samples are accumulated. While calibration sample selection is commonly treated as an important experimental condition, accumulation order is typically left uncontrolled. To our knowledge, we provide the first systematic evidence that reordering fixed calibration samples alone affects quantization and downstream performance in Hessian-based PTQ, with variability comparable to calibration resampling. We isolate this effect by varying only accumulation order across Qwen and Llama models, multiple Hessian-based PTQ methods, and weight bit-widths. In some settings, order-induced performance variation exceeds inter-method performance gaps and even reverses method rankings. Further analysis shows that order-dependent perturbations in the empirical Hessian are amplified through subsequent Hessian-dependent operations, ultimately altering quantization decisions. As a practical control, numerical stabilization incorporating compensated summation substantially suppresses order-induced variation. These findings highlight the need to account for accumulation order when evaluating and comparing Hessian-based PTQ methods. Our code is available at https://anonymous.4open.science/r/ReSQ-B3EF.