BaKron: Efficient Quantization with Kronecker-Factored Hessians

Post-training quantization (PTQ) is a widely adopted technique for compressing large language models (LLMs) without retraining. Existing second-order PTQ methods, including GPTQ, construct quantization objectives exclusively from input activation statistics, effectively assuming that all output channels contribute equally to the layer-wise reconstruction objective. We propose KronQ, a PTQ framework that challenges this assumption by introducing the gradient covariance into the quantization pipeline. Under the Kronecker-factored Hessian approximation, the quantization loss depends jointly on both the activation and gradient covariances, and KronQ exploits this at two complementary levels. (1) KronQ introduces bidirectional incoherence processing, extending the existing input-side random rotation to the output dimension using the gradient covariance, reducing weight magnitude variance across both input and output dimensions. (2) KronQ derives a new sensitivity metric for inter-layer mixed-precision allocation, driven by the gradient and activation Hessian traces. Notably, in the case of 2-bit weight-only quantization on LLaMA-3-70B, while GPTQ and GPTAQ diverge or produce degenerate quantizations (>2000 perplexity on WikiText-2), KronQ achieves 7.93 perplexity.

We present Activation Residual Hessian Quantization (ARHQ), a post-training weight splitting method designed to mitigate error propagation in low-bit activation-weight quantization. By constructing an input-side residual Hessian from activation quantization residuals (G_x), ARHQ analytically identifies and isolates error-sensitive weight directions into a high-precision low-rank branch. This is achieved via a closed-form truncated SVD on the scaled weight matrix W G^{1/2}_x . Experimental results on Qwen3-4B-Thinking-2507 demonstrate that ARHQ significantly improves layer-wise SNR and preserves downstream reasoning performance on ZebraLogic even under aggressive quantization. The code is available at https://github.com/BeautMoonQ/ARHQ.

Quantization-aware training (QAT) is widely adopted to quantize language models by training full-precision weights using gradients from the quantized model. The main bottleneck is its slow convergence and early performance plateau, particularly below 4-bit-widths. While this problem has been observed in prior work, its precise cause remains unclear. In this paper, we analyze the convergence of QAT by estimating the spectrum of the loss-surface Hessians. We find that the weights converge to flat regions around saddle points, where a large fraction of the Hessian eigenvalues are both positive and negative. During training, an increasing fraction of Hessian eigenvalues concentrates around zero, whose magnitude decreases. At lower bit-widths, the magnitude of eigenvalues in the Hessian spectrum is significantly smaller. To mitigate these issues, we propose an algorithm called WinQ to accelerate QAT, which involves: (1) periodically resetting weights to the linear interpolation of full-precision and quantized weights, reducing the distance to the quantization grid and increasing eigenvalue magnitude, and (2) computing gradients of noise-injected weights to regularize the Hessian. Extensive experiments show that WinQ accelerates QAT by up to 4 times across various quantization methods and models. Under the same training cost, WinQ improves state-of-the-art sub-4-bit quantization by up to 8.8%. These results are consistent across 16 settings with different language models, quantization methods, and bit widths.