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Learning rate and attractor size of the single-layer perceptron
Martin S Singleton1, Alfred W Hübler
1Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA. martin@math.uiuc.edu
Summary
We analyzed single-layer perceptrons using the delta rule to find average convergence rates. Results show convergence depends on inputs, threshold, and initial distribution size, providing insights into machine learning model behavior.
Area of Science:
- Artificial Intelligence
- Machine Learning
- Computational Neuroscience
Background:
- Single-layer perceptrons are fundamental neural network models.
- Understanding their convergence properties is crucial for algorithm development.
- The delta rule is a common online learning algorithm for perceptrons.
Purpose of the Study:
- To derive closed-form expressions for the mean convergence rates of a single-layer perceptron.
- To investigate the convergence behavior in weight space for linearly separable rules.
- To establish relationships between convergence, learning rate, and attractor properties.
Main Methods:
- Analysis of the simplest order one single-layer perceptron with two inputs.
- Application of the delta rule with online learning.
- Derivation of closed-form expressions for mean convergence rates.
- Investigation of weight vector trajectories in weight space.
Main Results:
- Convergence rate is an Nth order polynomial in threshold (t) and initial weight distribution size (l).
- Exact average convergence values derived for N=2, covering five linearly separable classes.
- Learning rate is determined by the attractor size.
- Attractors partition the input space for N-input perceptrons.
Conclusions:
- The study provides analytical expressions for perceptron convergence rates.
- Convergence dynamics are predictable based on network parameters and initial conditions.
- Attractor properties significantly influence the learning process and model behavior.
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