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

arXiv: https://arxiv.org/abs/1906.00771 · HTML: https://mlsanity.com/paper/1906.00771/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Jun 2019.

## Built on by (most cited first, 3 of 3)

- [VC dimension of partially quantized neural networks in the overparametrized regime](https://mlsanity.com/paper/2110.02456/) (2021)
- [Which Decisions Low-Bit Quantization Breaks, and How to Predict Them](https://mlsanity.com/paper/2608.06564/) (2026)
- [Depth Laws for the Precision Floor of Trained Neural Networks: Amplification, Residual Scaling, and a Quantization-Aware Training Paradox](https://mlsanity.com/paper/2609.32060/) (2026)

## Abstract (arXiv)

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 Lmax, 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: Lmaxpropto N1.82.