Neural Nets via Forward State Transformation and Backward Loss Transformation

This article studies (multilayer perceptron) neural networks with an emphasis on the transformations involved — both forward and backward — in order to develop a semantic/logical perspective that is in line with standard program semantics. The common two-pass neural network training algorithms make this viewpoint particularly fitting. In the forward direction, neural networks act as state transformers, using Kleisli composition for the multiset monad — for the linear parts of network layers. In the reverse direction, however, neural networks change losses of outputs to losses of inputs, thereby acting like a (real-valued) predicate transformer. In this way, backpropagation is functorial by construction, as shown in other works recently. We illustrate this perspective by training a simple instance of a neural network.