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The error surface of the simplest XOR network has only global minima
I G Sprinkhuizen-Kuyper1, E J Boers
1Department of Computer Science, Leiden University, The Netherlands.
Neural Computation
|August 15, 1996
Summary
This study proves that artificial neural networks with a single hidden unit solving the XOR problem have zero-error minima. Training algorithms like backpropagation will find the correct solution with high probability.
Area of Science:
- Artificial Intelligence
- Machine Learning
- Computational Neuroscience
Background:
- The XOR problem is a classic benchmark for evaluating neural network capabilities.
- Understanding the error surface is crucial for effective neural network training.
Purpose of the Study:
- To analyze the error surface of a simple artificial neural network for the XOR problem.
- To determine the nature of stationary points in the network's weight space.
Main Methods:
- Mathematical analysis of the error surface for a single-hidden-unit network.
- Investigation of stationary points (minima and saddlepoints).
- Probabilistic analysis of training convergence.
Main Results:
- The error surface for the XOR problem exhibits zero-error minima.
- All other stationary points are identified as saddlepoints.
- The volume of regions with saddlepoints in weight space is zero.
Conclusions:
- Training algorithms like backpropagation with momentum are guaranteed to converge to the zero-error solution with probability one.
- This specific network architecture provides a theoretically sound foundation for solving the XOR problem.
- The findings contribute to the understanding of optimization landscapes in neural networks.