A Rigorous, Tractable Measure of Model Complexity

TL;DR

This paper introduces a new, mathematically rigorous, and computationally efficient measure of model complexity based on gradient similarities, applicable to a wide range of models.

Contribution

It proposes a novel complexity measure that generalizes existing model-specific measures and provides insights into phenomena like double descent across various models.

Findings

The measure is well-defined for parametric and kernel-based models.

It generalizes polynomial degree, kernel length scale, and other complexity metrics.

Provides new understanding of double descent in different models.

Abstract

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 Mat\'ern 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.