Quantization is a widely used compression method that effectively reduces redundancies in over-parameterized neural networks. However, existing quantization techniques for deep neural networks often lack a comprehensive error analysis due to the presence of non-convex loss functions and nonlinear activations. In this paper, we propose a fast stochastic algorithm for quantizing the weights of fully trained neural networks. Our approach leverages a greedy path-following mechanism in combination with a stochastic quantizer. Its computational complexity scales only linearly with the number of weights in the network, thereby enabling the efficient quantization of large networks. Importantly, we establish, for the first time, full-network error bounds, under an infinite alphabet condition and minimal assumptions on the weights and input data. As an application of this result, we prove that when quantizing a multi-layer network having Gaussian weights, the relative square quantization error exhibits a linear decay as the degree of over-parametrization increases. Furthermore, we demonstrate that it is possible to achieve error bounds equivalent to those obtained in the infinite alphabet case, using on the order of a mere $\log\log N$ bits per weight, where $N$ represents the largest number of neurons in a layer.

本文提出了一种快速随机算法来对完全训练的神经网络权重进行量化，建立了全网络错误边界，证明了过参数化程度增加时相对平方量化误差呈线性衰减，同时展示了在每个权重上仅使用O(log(log N))位可以达到等同于无穷字母表情况下的错误边界。

SPFQ：一种用于神经网络量化的随机算法及其误差分析