Backpropagation

Linear separability constraint

Input 1 Input 2 Output 1 1 2 3 w1 w2   1

What if we add an extra layer between input and output?

5   w5 w6   3 4     w2 w3 w1 w4 1 2 Same as a linear network without any hidden layer!

What if we use thresholded units?

5   w5 w6 If netj > thresh, aj = 1 Else aj = 0 3 4 w2 w3 w1 w4 1 2

  5 If netj > 9.9, aj = 1 Else aj = 0 10 -10 3 4 1 Unit 3 10 10 5 5 1 2 1 1 Unit 4

So with thresholded units and a hidden layer, solutions exist… …and solutions can be viewed as “re-representing” the inputs, so as to make the mapping to the output unit learnable. BUT, how can we learn the correct weights instead of just setting them by hand?

But what if:   Simple delta rule:       …What function should we use for aj?

Net input Change in activation Activation 1.00 0.90 0.80 0.70 0.60   1.00 0.90 0.80 0.70 Change in activation 0.60 0.50 0.40 Activation 0.30 0.20 0.10 0.00 -10 -5 5 10 Net input

  Simple delta rule:          

  5       w5 w6 3 4 w2 w3   w1 w4 1 2      

5 6 Targets For outputs delta computed directly based on error. Delta is stored at each unit and also used directly to adjust each incoming weight. 3 4 5 1 2 6 Output For hidden units, there are no targets; “error” signal is instead the sum of the output unit deltas. These are used to compute deltas for the hidden units, which are again stored with unit and used to directly change incoming weights. Hidden Deltas, and hence error signal at output, can propagate backward through network through many layers until it reaches the input. Input

Alternative error functions.

  Sum-squared error: 5 w5 w6 3 4 w2 w3 w1 w4 1 2   Cross-entropy error:

5 w5 w6 3 4 w2 w3 w1 w4 1 2

Input 1 Input 2 New input Output 1 1 3 w1 w2 1 2 2