The Error Surface of the Simplest XOR Network Has Only Global Minima

Abstract

The artificial neural network with one hidden unit and the input units connected to the output unit is considered. It is proven that the error surface of this network for the patterns of the XOR problem has minimum values with zero error and that all other stationary points of the error surface are saddlepoints. Also, the volume of the regions in weight space with saddlepoints is zero, hence training this network on the four patterns of the XOR problem using, e.g., backpropagation with momentum, the correct solution with error zero will be reached in the limit with probability one.