The Back Propagation Learning Algorithm

The Back Propagation Learning Algorithm For networks with hidden units. Error Correcting algorithm. Solves the credit (blame) assignment problem. 1

What is supervised learning? Can we teach a network to learn to associate a pattern of inputs with corresponding outputs? i.e. given initial set of weights, how can they be adapted to produce the desired output? Use a training set: y a f? d payment b e? c w p workload person workload pay P(happy) a 0.1 0.9 0.95 b 0.3 0.7 0.8 c 0.07 0.2 0.2 d 0.9 0.9 0.3 e 0.7 0.5 ?? f 0.4 0.8 ?? After training, how does network generalise to patterns unseen during learning? 2

Learning by Error Correction In the perceptron there was a binary valued output Ý and a target Ø. x1 x2 xN w1 w2 wN output y target t y Æ 1 Ý step ÛÜ ¼ 0 Σwi xi i Deﬁne this error measure: ½ ´Ø Ý µ¾ ¾ It counts the number of incorrect outputs. We want to design a weight changing procedure that minimises . 3

Learning by Error Correction How do we change the weights Û¼ Û½ ÛÆ so that error decreases? E Suppose error slope slope varies with weight -ve +ve Û like this. wi If we could measure the slope Û then changing weights by the negative of the slope will minimise . slope +ve ¡Û -ve move towards minimum of slope -ve ¡Û +ve 4

More Perceptron Problems For the perceptron, can’t be differentiated with respect to weights Û¼ Û½ ÛÆ because involves output Ý which is not differentiable. ½ ´Ø Ý µ¾ Ý step Æ ÛÜ ¾ ¼ Threshold Unit: y ´ ÈÆ Û Ü 1 ½ if ¼ Ý ¼ if ÈÆ ¼ Û Ü ¼ ¼ 0 Σwi xi i Sigmoid Unit: y ½ 1 Ý ÈÆ ¡ ½ · ÜÔ ÛÜ 0 Σwi xi i 5

Gradient Descent E The error is now slope slope a differentiable -ve +ve function. wi Change weights using negative slope ¡Û Û Û +ve ¡Û -ve move towards minimum of Û -ve ¡Û +ve This approach is called Gradient Descent 6

Derivation of Back Propagation x1 v1 y1 x2 v2 y2 xk vj yi uj k wi j xN vN yN inputs hidden outputs xk vj yi È ¡ output Ý sig Û Ú È ¡ hidden Ú sig Ù Ü error ½È È Ø Ý ¡¾ ¾ We need to ﬁnd the derivatives of with respect to weights Û and Ù . 7

Preliminaries xk ujk vj wij yi On a single pattern (drop ) ½ ¡¾ ¾ Ø Ý and ½ Ý ÈÆ ¡ ½ · ÜÔ Û Ú Note that: Ý ¡ Ú Ý ½ Ý Û Ý ¡ Û Ý ½ Ý Ú since if Ý ½ ½ · ÜÔ´ Üµ Ý then Ý ´½ Ý µ Ü 8

Between Hidden and Output Û xk ujk vj wij yi For weights between hidden units and output units. ½ ¡¾ ¾ Ø Ý Ý Û Ý Û ¡ Ý Ý Ø Ý Û Ý ´½ ÝµÚ ¡ Û Ý Ø ßÞ ´½ Ý µ Ú Ý call this Æ 9

Between Input and Hidden Ù xk ujk vj wij yi For weights between input units and hidden units. ½ ¡¾ ¾ Ø Ý Ý Ú Ù Ý Ú Ù ¡ Ý Ý Ø Ý Ú Ý ´½ ÝµÛ Ú Ù Ú ´½ Ú µ Ü ¡ Ù Ý Ø Ý ´½ Ý µ Û Ú ´½ Ú µ Ü Ù ÆÛ Ú ´½ Ú µ Ü 10

Between Hidden and Output ¡Û xk ujk vj wij yi Modifying weights between hidden units and output units using gradient descent. ¡Û Û ¡ Ý ßÞ Ø Ý ´½ ßÞ Ý µ Ú close to ¼ ½ small for Ý Learning constant “input” error ßÞ Æ 11

Between Input and Hidden ¡Ù xk ujk vj wij yi Modifying weights between input units and hidden units using gradient descent. ¡Ù Ù Æ Û Ú ´½ Ú µÜ back propagation of error The same procedure is applicable to a net with many hidden layers. 12

An Example x1 u x2 =0 2.0 21 .8 = u 11 =2.0 u 12 u 22 =0.8 Ü½ Ü¾ target Ø u 10 = -1.0 u 20 = -1.0 0 0 0 v1 v2 1 1 0 1 1 1 0 1 w1 =2.0 w2 = -1.0 1 1 0 1 y w0 = -1.0 ¡ hidden Ú½ sig Ù½½Ü½ · Ù½¾Ü¾ · Ù½¼ 0.9526 ¡ Ú¾ sig Ù¾½Ü½ · Ù¾¾Ü¾ · Ù¾¼ 0.6457 ¡ output Ý sig Û½Ú½ · Û¾Ú¾ · Û¼ 0.5645 error ½ Ø Ý ¡¾ ¾ 0.1593 13

An Example: updating the weights Learning constant ½¼ output Æ ´Ý Øµ Ý´½ Ýµ 0.1388 ¡Û¼ Æ½ ¼ -0.1388 ¡Û½ ÆÚ½ -0.1322 ¡Û¾ ÆÚ¾ -0.0896 hidden (to Ú½) hidden (to Ú¾) ¡Ù½¼ ÆÛ½ Ú½´½ Ú½µ½ ¼ ¡Ù¾¼ ÆÛ¾ Ú¾´½ Ú¾µ½ ¼ -0.0125 0.0318 ¡Ù½½ ÆÛ½ Ú½´½ Ú½µÜ½ ¡Ù¾½ ÆÛ¾ Ú¾´½ Ú¾µÜ½ -0.0125 0.0318 ¡Ù½¾ ÆÛ½ Ú½´½ Ú½µÜ¾ ¡Ù¾¾ ÆÛ¾ Ú¾´½ Ú¾µÜ¾ -0.0125 0.0318 14

An Example: a New Error x1 u x2 8 =0 1.9 21 .83 = u 11 =1.98 u 12 u 22 =0.83 Ü½ Ü¾ target Ø u 10 = -1.01 u 20 = -0.96 0 0 0 v1 v2 1 1 0 1 1 1 0 1 w1 =1.86 w2 = -1.08 1 1 0 1 y w0 = -1.13 ¡ hidden Ú½ sig Ù½½Ü½ · Ù½¾Ü¾ · Ù½¼ 0.9509 ¡ Ú¾ sig Ù¾½Ü½ · Ù¾¾Ü¾ · Ù¾¼ 0.6672 ¡ output Ý sig Û½Ú½ · Û¾Ú¾ · Û¼ 0.4776 error ½ Ø Ý ¡¾ ¾ 0.1140 The error has reduced for this pattern. 15

Summary Credit-assignment problem solved for hidden units: Input Output Æ½ Û½ Û¾ Æ Æ¾ Û¿ Æ ¼ ´ µÈ Û Æ Æ¿ Errors total input to unit ; 1st derivative of acti- ¼ vation function (sigmoid) Outstanding issues: 1. Number of layers; number and type of units in layer 2. Learning rates 3. Local or distributed representations 16

The Back Propagation Learning Algorithm

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