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It was assumed proven that two-layer feedforward neural networks with t-1 hidden nodes, when presented with t input patterns, can not have any suboptimal local minima on the error surface. In this paper, however, we shall give a counterexample to this assumption. This counterexample consists of a region of local minima with nonzero error on the error surface of a neural network with three hidden nodes when presented with four patterns (the XOR problem). We will also show that the original proof is valid only when an unusual definition of local minimum is used.

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.

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