203.5.14 Neural Network Appendix

Math- How to update the weights?

• We update the weights backwards by iteratively calculating the error.
• The formula for weights updating is done using gradient descent method or delta rule also known as Widrow-Hoff rule.
• First we calculate the weight corrections for the output layer then we take care of hidden layers.
• \(W_(jk) = W_(jk) + \Delta W_(jk)\)
  • where \(\Delta W_(jk) = \eta . y_j \delta_k\)
  • \(\eta\) is the learning parameter
  • \(\delta_k = y_k (1- y_k) * Err\) (for hidden layers \(\delta_k = y_k (1- y_k) * w_j * Err )\)
  • Err = Expected output-Actual output
• The weight corrections is calculated based on the error function.
• The new weights are chosen in such way that the final error in that network is minimized.

Math-How does the delta rule work?

• Let’s consider a simple example to understand the weight updating using the delta rule.

• If we are building a simple logistic regression line. We would like to find the weights using weight update rule.
• \(Y= \frac{1}{(1+e^(-wx))}\) is the equation.
• We are searching for the optimal w for our data

• Let w be 1
• \(Y=\frac{1}{(1+e^(-x))}\) is the initial equation
• The error in our initial step is 3.59
• To reduce the error we will add a delta to w and make it 1.5

• Now w is 1.5 (blue line)
• \(Y=\frac{1}{(1+e^(-1.5x))}\) the updated equation
• With the updated weight, the error is 1.57
• We can further reduce the error by increasing w by delta

• If we repeat the same process of adding delta and updating weights, we can finally end up with minimum error.
• The weight at that final step is the optimal weight.
• In this example the weight is 8, and the error is 0.
• \(Y=\frac{1}{(1+e^(-8x))}\) is the final equation.
• In this example, we manually changed the weights to reduce the error. This is just for intuition, manual updating is not feasible for complex optimization problems.
• In gradient descent is a scientific optimization method. We update the weights by calculating gradient of the function.

Does this method really work?

• We changed the weights did it reduce the overall error?
• Lets calculate the error with new weights and see the change