SPFQ: A Stochastic Algorithm and Its Error Analysis for Neural Network Quantization

TL;DR

This paper introduces a fast stochastic neural network quantization algorithm with comprehensive error bounds, demonstrating efficient large-scale network compression and improved error decay with over-parameterization.

Contribution

It presents a novel stochastic quantization algorithm with linear complexity and establishes the first full-network error bounds under minimal assumptions.

Findings

Error bounds decay linearly with over-parameterization for Gaussian weights

Quantization achieves similar bounds with only logarithmic bits per weight

Algorithm scales linearly with the number of weights

Abstract

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.