# Rethinking Parameter Counting in Deep Models: Effective Dimensionality Revisited

arXiv: https://arxiv.org/abs/2003.02139 · HTML: https://mlsanity.com/paper/2003.02139/

## Key facts

- Cited by 51 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Mar 2020.

## Built on by (most cited first, 30 of 51)

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- [The No Free Lunch Theorem, Kolmogorov Complexity, and the Role of Inductive Biases in Machine Learning](https://mlsanity.com/paper/2304.05366/) (2023)
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- [Deep Learning is Not So Mysterious or Different](https://mlsanity.com/paper/2503.02113/) (2025)
- [On Uncertainty, Tempering, and Data Augmentation in Bayesian Classification](https://mlsanity.com/paper/2203.16481/) (2022)
- [Why neural networks find simple solutions: the many regularizers of geometric complexity](https://mlsanity.com/paper/2209.13083/) (2022)
- [Merging by Matching Models in Task Parameter Subspaces](https://mlsanity.com/paper/2312.04339/) (2023)
- [A U-turn on Double Descent: Rethinking Parameter Counting in Statistical Learning](https://mlsanity.com/paper/2310.18988/) (2023)
- [Adapting the Linearised Laplace Model Evidence for Modern Deep Learning](https://mlsanity.com/paper/2206.08900/) (2022)
- [DCI-ES: An Extended Disentanglement Framework with Connections to Identifiability](https://mlsanity.com/paper/2210.00364/) (2022)
- [Feature Learning and Signal Propagation in Deep Neural Networks](https://mlsanity.com/paper/2110.11749/) (2021)
- [Sampling-based inference for large linear models, with application to linearised Laplace](https://mlsanity.com/paper/2210.04994/) (2022)
- [Learning Dynamics and Generalization in Reinforcement Learning](https://mlsanity.com/paper/2206.02126/) (2022)
- [Why flatness does and does not correlate with generalization for deep neural networks](https://mlsanity.com/paper/2103.06219/) (2021)
- [Should We Learn Most Likely Functions or Parameters?](https://mlsanity.com/paper/2311.15990/) (2023)
- [Adversarial robustness of sparse local Lipschitz predictors](https://mlsanity.com/paper/2202.13216/) (2022)
- [Selectivity Drives Productivity: Efficient Dataset Pruning for Enhanced Transfer Learning](https://mlsanity.com/paper/2310.08782/) (2023)
- [Robustness to Pruning Predicts Generalization in Deep Neural Networks](https://mlsanity.com/paper/2103.06002/) (2021)
- [Low-Precision Arithmetic for Fast Gaussian Processes](https://mlsanity.com/paper/2207.06856/) (2022)
- [Multistage Collaborative Knowledge Distillation from a Large Language Model for Semi-Supervised Sequence Generation](https://mlsanity.com/paper/2311.08640/) (2023)
- [CR-SAM: Curvature Regularized Sharpness-Aware Minimization](https://mlsanity.com/paper/2312.13555/) (2023)
- [Learning Active Subspaces for Effective and Scalable Uncertainty Quantification in Deep Neural Networks](https://mlsanity.com/paper/2309.03061/) (2023)
- [Deep Learning Through A Telescoping Lens: A Simple Model Provides Empirical Insights On Grokking, Gradient Boosting & Beyond](https://mlsanity.com/paper/2411.00247/) (2024)
- [Classical Statistical (In-Sample) Intuitions Don't Generalize Well: A Note on Bias-Variance Tradeoffs, Overfitting and Moving from Fixed to Random Designs](https://mlsanity.com/paper/2409.18842/) (2024)
- [Compressibility Measures Complexity: Minimum Description Length Meets Singular Learning Theory](https://mlsanity.com/paper/2510.12077/) (2025)
- [Optimization Variance: Exploring Generalization Properties of DNNs](https://mlsanity.com/paper/2106.01714/) (2021)
- [Perspectives on the State and Future of Deep Learning - 2023](https://mlsanity.com/paper/2312.09323/) (2023)
- [GRAWA: Gradient-based Weighted Averaging for Distributed Training of Deep Learning Models](https://mlsanity.com/paper/2403.04206/) (2024)

## Abstract (arXiv)

Neural networks appear to have mysterious generalization properties when using parameter counting as a proxy for complexity. Indeed, neural networks often have many more parameters than there are data points, yet still provide good generalization performance. Moreover, when we measure generalization as a function of parameters, we see double descent behaviour, where the test error decreases, increases, and then again decreases. We show that many of these properties become understandable when viewed through the lens of effective dimensionality, which measures the dimensionality of the parameter space determined by the data. We relate effective dimensionality to posterior contraction in Bayesian deep learning, model selection, width-depth tradeoffs, double descent, and functional diversity in loss surfaces, leading to a richer understanding of the interplay between parameters and functions in deep models. We also show that effective dimensionality compares favourably to alternative norm- and flatness- based generalization measures.