Abstract

Residual networks (ResNets) have displayed impressive results in pattern recognition and, recently, have garnered considerable theoretical interest due to a perceived link with neural ordinary differential equations (neural ODEs). This link relies on the convergence of network weights to a smooth function as the number of layers increases. We investigate the properties of weights trained by stochastic gradient descent and their scaling with network depth through detailed numerical experiments. We observe the existence of scaling regimes markedly different from those assumed in neural ODE literature: one may obtain an alternative ODE limit, a stochastic differential equation or neither of these. The scaling regime one ends up with depends on certain features of the network architecture, such as the smoothness of the activation function. These findings cast doubts on the validity of the neural ODE model as an adequate asymptotic description of deep ResNets and point to an alternative class of differential equations as a better description of the deep network limit.  In the case where the scaling limit is a stochastic differential equation, the deep network limit is shown to be described by a system of forward-backward stochastic differential equations. Joint work with: Alain-Sam Cohen (InstaDeep Ltd), Alain Rossier (Oxford), RenYuan Xu (University of Southern California).