  We introduce a new neural network model, together with a tractable and
monotone online learning algorithm. Our model describes feed-forward networks
for classification, with one output node for each class. The only nonlinear
operation is rectification using a ReLU function with a bias. However, there is
a rectifier on every edge rather than at the nodes of the network. There are
also weights, but these are positive, static, and associated with the nodes.
Our "rectified wire" networks are able to represent arbitrary Boolean
functions. Only the bias parameters, on the edges of the network, are learned.
Another departure in our approach, from standard neural networks, is that the
loss function is replaced by a constraint. This constraint is simply that the
value of the output node associated with the correct class should be zero. Our
model has the property that the exact norm-minimizing parameter update,
required to correctly classify a training item, is the solution to a quadratic
program that can be computed with a few passes through the network. We
demonstrate a training algorithm using this update, called sequential
deactivation (SDA), on MNIST and some synthetic datasets. Upon adopting a
natural choice for the nodal weights, SDA has no hyperparameters other than
those describing the network structure. Our experiments explore behavior with
respect to network size and depth in a family of sparse expander networks.
