Roy S. Freedman

Roy S. Freedman

The exclusive or (xor) function is one of the simplest examples that illustrate why nonlinear feedforward networks are superior to linear regression for machine learning applications. We review the xor representation and approximation problems and discuss their solutions in terms of probabilistic logic and associative copula functions. After briefly reviewing the specification of feedforward networks, we compare the dynamics of learned error surfaces with different activation functions such as RELU and tanh through a set of colorful three-dimensional charts. The copula representations extend xor from Boolean to real values, thereby providing a convenient way to demonstrate the concept of cross-validation on in-sample and out-sample data sets. Our approach is pedagogical and is meant to be a machine learning prolegomenon.

Roy S. Freedman

We show how a declarative functional programming specification of backpropagation yields a visual and transparent implementation within spreadsheets. We call our method Visual Backpropagation. This backpropagation implementation exploits array worksheet formulas, manual calculation, and has a sequential order of computation similar to the processing of a systolic array. The implementation uses no hidden macros nor user-defined functions; there are no loops, assignment statements, or links to any procedural programs written in conventional languages. As an illustration, we compare a Visual Backpropagation solution to a Tensorflow (Python) solution on a standard regression problem.