# Measuring Neural Network Complexity via Effective Degrees of Freedom

arXiv: https://arxiv.org/abs/2602.13442 · HTML: https://mlsanity.com/paper/2602.13442/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 27 references that are in the map.
- First posted on arXiv 13 Feb 2026.

## Builds on (most cited first, 6 of 6)

- [Randomness In Neural Network Training: Characterizing The Impact of Tooling](https://mlsanity.com/paper/2106.11872/) (2021)
- [Interpretable Convolutional Neural Networks via Feedforward Design](https://mlsanity.com/paper/1810.02786/) (2018)
- [Quantifying Inherent Randomness in Machine Learning Algorithms](https://mlsanity.com/paper/2206.12353/) (2022)
- [Simple Feedfoward Neural Networks are Almost All You Need for Time Series Forecasting](https://mlsanity.com/paper/2503.23621/) (2025)
- [Computing AIC for black-box models using Generalised Degrees of Freedom: a comparison with cross-validation](https://mlsanity.com/paper/1603.02743/) (2016)
- [Investigating Statistical Inference and Covariate Effects in Shallow Neural Networks](https://mlsanity.com/paper/2311.08139/) (2023)

## Built on by (most cited first, 1 of 1)

- [A Rigorous, Tractable Measure of Model Complexity](https://mlsanity.com/paper/2605.21167/) (2026)

## Abstract (arXiv)

Quantifying the complexity of feed-forward neural networks (FFNNs) remains challenging due to their nonlinear, hierarchical structure and numerous parameters. We apply generalized degrees of freedom (GDF) to measure model complexity in FFNNs with binary outcomes, adapting the algorithm for discrete responses. We compare GDF with both the effective number of parameters derived via log-likelihood cross-validation and the null degrees of freedom of Landsittel et al. Through simulation studies and a real data analysis, we demonstrate that GDF provides a robust assessment of model complexity for neural network models, as it depends only on the sensitivity of fitted values to perturbations in the observed responses rather than on assumptions about the likelihood. In contrast, cross-validation-based estimates of model complexity and the null degrees of freedom rely on the correctness of the assumed likelihood and may exhibit substantial variability. We find that GDF, cross-validation-based measures, and null degrees of freedom yield similar assessments of model complexity only when the fitted model adequately represents the data-generating mechanism. These findings highlight GDF as a stable and broadly applicable measure of model complexity for neural networks in statistical modeling.