WaterKron and FlipFlop Hessian: Information-Theoretically Grounded Quantization with Kronecker-factored Hessians

Abstract

How should a Kronecker-factored Hessian approximation be chosen for post-training quantization? We address this question through WaterKron, which combines two-sided GPTQ with row- and column-dependent waterfilling scales and entropy coding. We derive its high-rate distortion with respect to the full Hessian using an explicit Kronecker-Hessian mismatch factor $Φ$. This factor quantifies the asymptotic distortion penalty due to the Kronecker Hessian approximation and provides a criterion for selecting the factors optimally. Minimizing $Φ$ leads to a Gaussian covariance-fitting problem with classical ``flip-flop'' updates. We thus give a rate-distortion justification for using the resulting FlipFlop Hessian in quantization. We evaluate it empirically, finding that the FlipFlop Hessian consistently improves KL divergence and perplexity over input-only, marginal, and Frobenius-based Hessian choices.