You will be given a definition of a task first, then some input of the task.
In this task, you are given the abstract of a research paper. Your task is to generate a summary of this abstract. Your summary should not be very short, but it's better if it's not more than 30 words.

We provide a novel perspective on the forward pass through a block of layers in a deep network. In particular, we show that a forward pass through a standard dropout layer followed by a linear layer and a non-linear activation is equivalent to optimizing a convex objective with a single iteration of a $ au$-nice Proximal Stochastic Gradient method. We further show that replacing standard Bernoulli dropout with additive dropout is equivalent to optimizing the same convex objective with a variance-reduced proximal method. By expressing both fully-connected and convolutional layers as special cases of a high-order tensor product, we unify the underlying convex optimization problem in the tensor setting and derive a formula for the Lipschitz constant $L$ used to determine the optimal step size of the above proximal methods. We conduct experiments with standard convolutional networks applied to the CIFAR-10 and CIFAR-100 datasets and show that replacing a block of layers with multiple iterations of the corresponding solver, with step size set via $L$, consistently improves classification accuracy.
Output:
A framework that links deep network layers to stochastic optimization algorithms; can be used to improve model accuracy and inform network design.