Mean Field Analysis of Deep Neural Networks
This provides a theoretical foundation for understanding deep neural network training dynamics, though it is incremental as it builds on existing mean field analysis methods.
The authors tackled the problem of analyzing multi-layer neural networks in the asymptotic regime of large sizes and training iterations, rigorously establishing that the network output converges to a deterministic system of integro-differential equations and recovers a global minimum with zero loss under certain assumptions.
We analyze multi-layer neural networks in the asymptotic regime of simultaneously (A) large network sizes and (B) large numbers of stochastic gradient descent training iterations. We rigorously establish the limiting behavior of the multi-layer neural network output. The limit procedure is valid for any number of hidden layers and it naturally also describes the limiting behavior of the training loss. The ideas that we explore are to (a) take the limits of each hidden layer sequentially and (b) characterize the evolution of parameters in terms of their initialization. The limit satisfies a system of deterministic integro-differential equations. The proof uses methods from weak convergence and stochastic analysis. We show that, under suitable assumptions on the activation functions and the behavior for large times, the limit neural network recovers a global minimum (with zero loss for the objective function).