Analysis of momentum term in back-propagation

The back-propagation algorithm has been applied to many fields, and has shown large capability of neural networks. Many people use the back-propagation algorithm together with a momentum term to accelerate its convergence. However, in spite of the importance for theoretical studies, theoretical background of a momentum term has been unknown so far. First, this paper explains clearly the theoretical origin of a momentum term in the back-propagation algorithm for both a batch mode learning and a pattern-by-pattern learning. We will prove that the back-propagation algorithm having a momentum term can be derived through the following two assumptions: 1) The cost function is Eⁿ = n/Σ/μ α^n-μ E_μ, where E_μ is the summation of squared error at the output layer at the μth learning time and a is the momentum coefficient. 2) The latest weights are assumed in calculating the cost function Eⁿ. Next, we derive a simple relationship between momentum, learning rate, and learning speed and then further discussion is made with computer simulation.