Limitations of Lazy Training of Two-layers Neural Networks

Abstract

We study the supervised learning problem under either of the following two models: (1) Feature vectors ${\mathbf{x}}_i$ are $d$-dimensional Gaussians and responses are $y_i=f_{\ast }({\mathbf{x}}_i)$ for $f_{\ast }$ an unknown quadratic function; (2) Feature vectors ${\mathbf{x}}_i$ are distributed as a mixture of two $d$-dimensional centered Gaussians, and $y_i$'s are the corresponding class labels. We use two-layers neural networks with quadratic activations, and compare three different learning regimes: the random features (RF) regime in which we only train the second-layer weights; the neural tangent (NT) regime in which we train a linearization of the neural network around its initialization; the fully trained neural network (NN) regime in which we train all the weights in the network. We prove that, even for the simple quadratic model of point (1), there is a potentially unbounded gap between the prediction risk achieved in these three training regimes, when the number of neurons is smaller than the ambient dimension. When the number of neurons is larger than the number of dimensions, the problem is significantly easier and both NT and NN learning achieve zero risk.

arXiv comment39 pages; 2 pdf figures

References in corpus (9)

• A Convergence Theory for Deep Learning via Over-Parameterization
• On Exact Computation with an Infinitely Wide Neural Net
• Gradient Descent Provably Optimizes Over-parameterized Neural Networks
• No Spurious Local Minima in Nonconvex Low Rank Problems: A Unified Geometric Analysis
• Stochastic Gradient Descent Optimizes Over-parameterized Deep ReLU Networks
• Gradient Descent Finds Global Minima of Deep Neural Networks
• Generalization Properties of Learning with Random Features
• Learning One-hidden-layer Neural Networks with Landscape Design
• On the Power and Limitations of Random Features for Understanding Neural Networks

Cited by in corpus (7)

• Proving the Lottery Ticket Hypothesis: Pruning is All You Need
• Asymptotics of Wide Networks from Feynman Diagrams
• A Mean-field Analysis of Deep ResNet and Beyond: Towards Provable Optimization Via Overparameterization From Depth
• Beyond Linearization: On Quadratic and Higher-Order Approximation of Wide Neural Networks
• When Does Preconditioning Help or Hurt Generalization?
• Neural Tangent Kernel Maximum Mean Discrepancy
• Deep Networks Provably Classify Data on Curves