A Functional-Space Mean-Field Theory of Partially-Trained Three-Layer Neural Networks

by   Zhengdao Chen, et al.

To understand the training dynamics of neural networks (NNs), prior studies have considered the infinite-width mean-field (MF) limit of two-layer NN, establishing theoretical guarantees of its convergence under gradient flow training as well as its approximation and generalization capabilities. In this work, we study the infinite-width limit of a type of three-layer NN model whose first layer is random and fixed. To define the limiting model rigorously, we generalize the MF theory of two-layer NNs by treating the neurons as belonging to functional spaces. Then, by writing the MF training dynamics as a kernel gradient flow with a time-varying kernel that remains positive-definite, we prove that its training loss in L_2 regression decays to zero at a linear rate. Furthermore, we define function spaces that include the solutions obtainable through the MF training dynamics and prove Rademacher complexity bounds for these spaces. Our theory accommodates different scaling choices of the model, resulting in two regimes of the MF limit that demonstrate distinctive behaviors while both exhibiting feature learning.


page 1

page 2

page 3

page 4


Neural Hilbert Ladders: Multi-Layer Neural Networks in Function Space

The characterization of the functions spaces explored by neural networks...

On Feature Learning in Neural Networks with Global Convergence Guarantees

We study the optimization of wide neural networks (NNs) via gradient flo...

Phase diagram for two-layer ReLU neural networks at infinite-width limit

How neural network behaves during the training over different choices of...

Towards a General Theory of Infinite-Width Limits of Neural Classifiers

Obtaining theoretical guarantees for neural networks training appears to...

Self-Consistent Dynamical Field Theory of Kernel Evolution in Wide Neural Networks

We analyze feature learning in infinite width neural networks trained wi...

Global Convergence of Second-order Dynamics in Two-layer Neural Networks

Recent results have shown that for two-layer fully connected neural netw...

Please sign up or login with your details

Forgot password? Click here to reset