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Regression of a Target Function on the Sphere

Code for the paper Learning sparse features can lead to overfitting in neural networks appeared at NeurIPS 2022. Details for running experiments can be found in experiments.md.

Neural Network Training

Run main.py

Architecture. Train a one hidden-layer fully-connected neural network on the mean square error.

Data-set. Data samples x can be drawn from

• the d-dimensional normal distribution, args.dataset = normal;
• uniform distribution inside the sphere, uniform;
• or uniform distribution on the spherical surface, sphere.

The target is either

• the sample norm ||x||, args.target = norm;
• or a Gaussian random field computed through an ~infinite-width teacher network (args.target = teacher) with relu or abs activation function to some power a. Here examples of the Gaussian random field defined on the sphere in d=3:

Algorithm. Full batch gradient descent can be performed with

• the alpha-trick by setting args.alpha larger (lazy) or smaller (feature) than one.
• a regularization args.l on the l2 norm of the parameters (args.reg = 'l2'), on the path norm ||w1|| * |w2| (args.reg = 'l1') or on the l1 norm |w2| by fixing the first layer weights on the unit sphere ||w1|| = 1 (args.reg = 'l1' and args.w1_norm1 = 1).

Additionally, conic gradient descent [Chizat and Bach, 2018] can be performed by setting args.conic_gd = 1.

Kernel Ridge Regression (KRR)

Run main_krr.py

Student kernel. Analytical NTK of an infinite-width one-hidden-layer neural network.

Data-set. see above.

Ridge. The regularization or ridge parameter can be fixed by args.l (default: 0).