# Computing the Information Content of Trained Neural Networks

arXiv: https://arxiv.org/abs/2103.01045 · HTML: https://mlsanity.com/paper/2103.01045/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 36 references that are in the map.
- First posted on arXiv 1 Mar 2021.

## Builds on (most cited first, 10 of 10)

- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Deep Image Prior](https://mlsanity.com/paper/1711.10925/) (2017)
- [A Closer Look at Memorization in Deep Networks](https://mlsanity.com/paper/1706.05394/) (2017)
- [Wide Neural Networks of Any Depth Evolve as Linear Models Under Gradient Descent](https://mlsanity.com/paper/1902.06720/) (2019)
- [On Lazy Training in Differentiable Programming](https://mlsanity.com/paper/1812.07956/) (2018)
- [Deep Neural Networks as Gaussian Processes](https://mlsanity.com/paper/1711.00165/) (2017)
- [Computing Nonvacuous Generalization Bounds for Deep (Stochastic) Neural Networks with Many More Parameters than Training Data](https://mlsanity.com/paper/1703.11008/) (2017)
- [Deep learning generalizes because the parameter-function map is biased towards simple functions](https://mlsanity.com/paper/1805.08522/) (2018)
- [Is SGD a Bayesian sampler? Well, almost](https://mlsanity.com/paper/2006.15191/) (2020)
- [Learning by Turning: Neural Architecture Aware Optimisation](https://mlsanity.com/paper/2102.07227/) (2021)

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

- [Wide Bayesian neural networks have a simple weight posterior: theory and accelerated sampling](https://mlsanity.com/paper/2206.07673/) (2022)
- [Steering Information Utility in Key-Value Memory for Language Model Post-Training](https://mlsanity.com/paper/2507.05158/) (2025)
- [Multiclass classification utilising an estimated algorithmic probability prior](https://mlsanity.com/paper/2212.07426/) (2022)

## Abstract (arXiv)

How much information does a learning algorithm extract from the training data and store in a neural network's weights? Too much, and the network would overfit to the training data. Too little, and the network would not fit to anything at all. Na"ively, the amount of information the network stores should scale in proportion to the number of trainable weights. This raises the question: how can neural networks with vastly more weights than training data still generalise? A simple resolution to this conundrum is that the number of weights is usually a bad proxy for the actual amount of information stored. For instance, typical weight vectors may be highly compressible. Then another question occurs: is it possible to compute the actual amount of information stored? This paper derives both a consistent estimator and a closed-form upper bound on the information content of infinitely wide neural networks. The derivation is based on an identification between neural information content and the negative log probability of a Gaussian orthant. This identification yields bounds that analytically control the generalisation behaviour of the entire solution space of infinitely wide networks. The bounds have a simple dependence on both the network architecture and the training data. Corroborating the findings of Valle-P'erez et al. (2019), who conducted a similar analysis using approximate Gaussian integration techniques, the bounds are found to be both non-vacuous and correlated with the empirical generalisation behaviour at finite width.