SPFQ: a stochastic algorithm and its error analysis for neural network quantization

Abstract Quantization is a widely used compression method that effectively reduces redundancies in overparameterized 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 overparametrization increases. Furthermore, we demonstrate that it is possible to achieve error bounds equivalent to those obtained in the infinite alphabet case, using of the order of a mere $\log \log N$ bits per weight, where $N$ represents the largest number of neurons in a layer.