Global Convergence of Three-layer Neural Networks in the Mean Field Regime

• URL: http://arxiv.org/abs/2105.05228v1
• Date: Tue, 11 May 2021 17:45:42 GMT
• Title: Global Convergence of Three-layer Neural Networks in the Mean Field Regime
• Authors: Huy Tuan Pham, Phan-Minh Nguyen
• Abstract summary: In the mean field regime, neural networks are appropriately scaled so that as the width tends to infinity, the learning dynamics tends to a nonlinear and nontrivial dynamical limit, known as the mean field limit. Recent works have successfully applied such analysis to two-layer networks and provided global convergence guarantees. We prove a global convergence result for unregularized feedforward three-layer networks in the mean field regime.
• Score: 3.553493344868413
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In the mean field regime, neural networks are appropriately scaled so that as the width tends to infinity, the learning dynamics tends to a nonlinear and nontrivial dynamical limit, known as the mean field limit. This lends a way to study large-width neural networks via analyzing the mean field limit. Recent works have successfully applied such analysis to two-layer networks and provided global convergence guarantees. The extension to multilayer ones however has been a highly challenging puzzle, and little is known about the optimization efficiency in the mean field regime when there are more than two layers. In this work, we prove a global convergence result for unregularized feedforward three-layer networks in the mean field regime. We first develop a rigorous framework to establish the mean field limit of three-layer networks under stochastic gradient descent training. To that end, we propose the idea of a \textit{neuronal embedding}, which comprises of a fixed probability space that encapsulates neural networks of arbitrary sizes. The identified mean field limit is then used to prove a global convergence guarantee under suitable regularity and convergence mode assumptions, which -- unlike previous works on two-layer networks -- does not rely critically on convexity. Underlying the result is a universal approximation property, natural of neural networks, which importantly is shown to hold at \textit{any} finite training time (not necessarily at convergence) via an algebraic topology argument.

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