A Note on the Global Convergence of Multilayer Neural Networks in the Mean Field Regime
This work provides theoretical convergence guarantees for deep neural networks, addressing a foundational problem in machine learning, though it is incremental as it builds on prior results.
The authors extended their previous framework to prove global convergence guarantees for multilayer neural networks of any depth, using a correlated initialization that enables universal approximation property propagation through network depth.
In a recent work, we introduced a rigorous framework to describe the mean field limit of the gradient-based learning dynamics of multilayer neural networks, based on the idea of a neuronal embedding. There we also proved a global convergence guarantee for three-layer (as well as two-layer) networks using this framework. In this companion note, we point out that the insights in our previous work can be readily extended to prove a global convergence guarantee for multilayer networks of any depths. Unlike our previous three-layer global convergence guarantee that assumes i.i.d. initializations, our present result applies to a type of correlated initialization. This initialization allows to, at any finite training time, propagate a certain universal approximation property through the depth of the neural network. To achieve this effect, we introduce a bidirectional diversity condition.