Conditioning of Random Feature Matrices: Double Descent and Generalization Error

Abstract

We provide (high probability) bounds on the condition number of random feature matrices. In particular, we show that if the complexity ratio $\frac{N}{m}$ where $N$ is the number of neurons and $m$ is the number of data samples scales like ${\log }^{-1}(N)$ or $\log (m)$, then the random feature matrix is well-conditioned. This result holds without the need of regularization and relies on establishing various concentration bounds between dependent components of the random feature matrix. Additionally, we derive bounds on the restricted isometry constant of the random feature matrix. We prove that the risk associated with regression problems using a random feature matrix exhibits the double descent phenomenon and that this is an effect of the double descent behavior of the condition number. The risk bounds include the underparameterized setting using the least squares problem and the overparameterized setting where using either the minimum norm interpolation problem or a sparse regression problem. For the least squares or sparse regression cases, we show that the risk decreases as $m$ and $N$ increase, even in the presence of bounded or random noise. The risk bound matches the optimal scaling in the literature and the constants in our results are explicit and independent of the dimension of the data.

References in corpus (11)

• Reconciling modern machine learning practice and the bias-variance trade-off
• The Lottery Ticket Hypothesis: Finding Sparse, Trainable Neural Networks
• Generalization Properties of Learning with Random Features
• Benign overfitting in ridge regression
• Towards a Mathematical Understanding of Neural Network-Based Machine Learning: what we know and what we don't
• Generalization Bounds for Sparse Random Feature Expansions
• On Random Matrices Arising in Deep Neural Networks. Gaussian Case
• A Random Matrix Analysis of Random Fourier Features: Beyond the Gaussian Kernel, a Precise Phase Transition, and the Corresponding Double Descent
• Generalization error of random features and kernel methods: hypercontractivity and kernel matrix concentration
• Double descent in the condition number
• Avoiding The Double Descent Phenomenon of Random Feature Models Using Hybrid Regularization

Cited by in corpus (1)

• On the Double Descent of Random Features Models Trained with SGD