A Rigorous, Tractable Measure of Model Complexity
Provides a unified, principled complexity measure for parametric and kernel-based models, benefiting researchers and practitioners in model selection and interpretation.
The authors propose a new, mathematically rigorous and computationally tractable measure of model complexity based on gradient similarities across inputs, which generalizes several existing model-specific complexity measures. They use this measure to gain new insights into the double descent phenomenon across various models.
An accurate assessment of a model's complexity is crucial for topics such as interpretation, generalization, and model selection. However, most existing complexity measures either rely on heuristic assumptions or are computationally prohibitive. In this paper, we present a mathematically rigorous yet easy-to-compute measure of model complexity that is based on the similarities between the model gradients across inputs. It is thus well-defined for any parametric model, but also for kernel-based non-parametric models. We prove that our measure of complexity generalizes model-specific complexity measures such as polynomial degree (for polynomial regression), kernel length scale (for Matérn kernels), number of neighbors (for k-nearest neighbors), number of splits (for decision trees), and number of trees (for random forests). We also use our measure to obtain new insights into the double descent phenomenon for random Fourier features, random forests, neural networks, and gradient boosting.