  We describe a novel family of models of multi- layer feedforward neural
networks in which the activation functions are encoded via penalties in the
training problem. Our approach is based on representing a non-decreasing
activation function as the argmin of an appropriate convex optimiza- tion
problem. The new framework allows for algo- rithms such as block-coordinate
descent methods to be applied, in which each step is composed of a simple (no
hidden layer) supervised learning problem that is parallelizable across data
points and/or layers. Experiments indicate that the pro- posed models provide
excellent initial guesses for weights for standard neural networks. In addi-
tion, the model provides avenues for interesting extensions, such as robustness
against noisy in- puts and optimizing over parameters in activation functions.
