A mean-field limit for certain deep neural networks
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Understanding deep neural networks (DNNs) is a key challenge in the theory of machine learning, with potential applications to the many fields where DNNs have been successfully used. This article presents a scaling limit for a DNN being trained by stochastic gradient descent. Our networks have a fixed (but arbitrary) number L≥ 2 of inner layers; N≫ 1 neurons per layer; full connections between layers; and fixed weights (or "random features" that are not trained) near the input and output. Our results describe the evolution of the DNN during training in the limit when N→ +∞, which we relate to a mean field model of McKean-Vlasov type. Specifically, we show that network weights are approximated by certain "ideal particles" whose distribution and dependencies are described by the mean-field model. A key part of the proof is to show existence and uniqueness for our McKean-Vlasov problem, which does not seem to be amenable to existing theory. Our paper extends previous work on the L=1 case by Mei, Montanari and Nguyen; Rotskoff and Vanden-Eijnden; and Sirignano and Spiliopoulos. We also complement recent independent work on L>1 by Sirignano and Spiliopoulos (who consider a less natural scaling limit) and Nguyen (who nonrigorously derives similar results).
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