A Mean Field Theory of Quantized Deep Networks: The Quantization-Depth Trade-Off

TL;DR

This paper uses mean-field theory to analyze how quantization affects deep neural networks, deriving formulas for maximum trainable depth based on quantization levels, which informs efficient low-precision model design.

Contribution

It introduces a mean-field framework for quantized networks, deriving a closed-form equation for maximum trainable depth as a function of quantization levels.

Findings

Maximum trainable depth scales as N^{1.82} with quantization levels N.

Proposed initialization schemes improve signal propagation in quantized networks.

Theoretical insights guide the design of resource-efficient deep models.

Abstract

Reducing the precision of weights and activation functions in neural network training, with minimal impact on performance, is essential for the deployment of these models in resource-constrained environments. We apply mean-field techniques to networks with quantized activations in order to evaluate the degree to which quantization degrades signal propagation at initialization. We derive initialization schemes which maximize signal propagation in such networks and suggest why this is helpful for generalization. Building on these results, we obtain a closed form implicit equation for $L_{\max}$, the maximal trainable depth (and hence model capacity), given $N$, the number of quantization levels in the activation function. Solving this equation numerically, we obtain asymptotically: $L_{\max}\propto N^{1.82}$.