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A local minimum for the 2-3-1 XOR network
IEEE Transactions on Neural Networks
|February 7, 2008
Summary
A previous assumption stated that two-layer feedforward neural networks cannot have suboptimal local minima. This paper presents a counterexample using the XOR problem, demonstrating local minima exist under standard definitions.
Area of Science:
- Artificial Intelligence
- Machine Learning
- Neural Networks
Background:
- Two-layer feedforward neural networks are widely used in machine learning.
- A common assumption in neural network training is the absence of suboptimal local minima under certain conditions (t-1 hidden nodes, t input patterns).
Discussion:
- This paper challenges the established assumption regarding the absence of local minima in two-layer feedforward neural networks.
- A specific counterexample is provided, involving a neural network with three hidden nodes trained on the XOR problem (four patterns).
- The existence of a region of local minima with non-zero error is demonstrated for this configuration.
Key Insights:
- The assumption that two-layer feedforward neural networks with t-1 hidden nodes and t input patterns are free from suboptimal local minima is disproven.
- The XOR problem serves as a concrete counterexample, illustrating the presence of local minima with non-zero error.
- The validity of the original proof is questioned due to its reliance on an unconventional definition of local minimum.
Outlook:
- Further research is needed to understand the prevalence and impact of local minima in neural network training.
- Investigating alternative training algorithms or network architectures may be necessary to overcome local minima.
- Clarifying the definition of local minima in the context of neural networks is crucial for theoretical advancements.
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