Towards an Optimal Control Perspective of ResNet Training

We propose a training formulation for ResNets reflecting an optimal control problem that is applicable for standard architectures and general loss functions. We suggest bridging both worlds via penalizing intermediate outputs of hidden states corresponding to stage cost terms in optimal control. For standard ResNets, we obtain intermediate outputs by propagating the state through the subsequent skip connections and the output layer. We demonstrate that our training dynamic biases the weights of the unnecessary deeper residual layers to vanish. This indicates the potential for a theory-grounded layer pruning strategy.

On Dissipativity of Cross-Entropy Loss in Training ResNets

The training of ResNets and neural ODEs can be formulated and analyzed from the perspective of optimal control. This paper proposes a dissipative formulation of the training of ResNets and neural ODEs for classification problems by including a variant of the cross-entropy as a regularization in the stage cost. Based on the dissipative formulation of the training, we prove that the trained ResNet exhibit the turnpike phenomenon. We then illustrate that the training exhibits the turnpike phenomenon by training on the two spirals and MNIST datasets. This can be used to find very shallow networks suitable for a given classification task.