A Mean-field Analysis of Deep ResNet and Beyond: Towards Provable Optimization Via Overparameterization From Depth

Abstract

Training deep neural networks with stochastic gradient descent (SGD) can often achieve zero training loss on real-world tasks although the optimization landscape is known to be highly non-convex. To understand the success of SGD for training deep neural networks, this work presents a mean-field analysis of deep residual networks, based on a line of works that interpret the continuum limit of the deep residual network as an ordinary differential equation when the network capacity tends to infinity. Specifically, we propose a new continuum limit of deep residual networks, which enjoys a good landscape in the sense that every local minimizer is global. This characterization enables us to derive the first global convergence result for multilayer neural networks in the mean-field regime. Furthermore, without assuming the convexity of the loss landscape, our proof relies on a zero-loss assumption at the global minimizer that can be achieved when the model shares a universal approximation property. Key to our result is the observation that a deep residual network resembles a shallow network ensemble [59], i.e. a two-layer network. We bound the difference between the shallow network and our ResNet model via the adjoint sensitivity method, which enables us to apply existing mean-field analyses of two-layer networks to deep networks. Furthermore, we propose several novel training schemes based on the new continuous model, including one training procedure that switches the order of the residual blocks and results in strong empirical performance on the benchmark datasets.

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

“…As a remark, our approach can also be applied to prove a similar global convergence guarantee for two-layer networks, removing the convex loss assumption in previous works (Nguyen & Pham (2020)). The recent work Lu et al (2020) on a MF resnet model (a composition of many two-layer MF networks) and a recent update of Sirignano & Spiliopoulos (2019) essentially establish conditions of stationary points to be global optima. They however require strong assumptions on the support of the limit point.…”

Learning Poisson Systems and Trajectories of Autonomous Systems via Poisson Neural Networks

“…We then generate 500 training points followed by 100 test points with time step h = 0.01. That means the solution to this equation during the time interval [0, 5] is treated as the training set, and then we learn the data using a PNN and predict the solution between [5,6]. The result is shown in Fig.…”

“…T HE connection between dynamical systems and neural network models has been widely studied in the literature, see, for example, [1]- [5]. In general, neural networks can be considered as discrete dynamical systems with the basic dynamics at each step being a linear transformation followed by a component-wise nonlinear (activation) function.…”

Game on Random Environment, Mean-field Langevin System and Neural Networks

“…There is a rising interest in modeling the forward propagation of the deep neural networks using a controlled dynamics and in connecting the deep learning to the optimal control problems, see e.g. [5,6,9,17,20,21]. For the controlled processes in the particular form (4.1), we refer to Section 4 in [15] for the connection between the optimal control problem and the deep neural networks.…”