COMP9444 Neural Networks and Deep Learning 3. Backpropagation

Transcription

1 COMP9444 Neural Netwrks and Deep Learning 3. Backprpagatin Tetbk, Sectins 4.3, 5.2, 6.5.2

2 COMP s2 Backprpagatin 1 Outline Supervised Learning Ockham s Razr (5.2) Multi-Layer Netwrks Gradient Descent (4.3, 6.5.2)

3 COMP s2 Backprpagatin 2 Types f Learning Supervised Learning agent is presented with eamples f inputs and their target utputs Reinfrcement Learning agent is nt presented with target utputs, but is given a reward signal, which it aims t maimize Unsupervised Learning agent is nly presented with the inputs themselves, and aims t find structure in these inputs

4 COMP s2 Backprpagatin 3 Supervised Learning we have a training set and a test set, each cnsisting f a set f items; fr each item, a number f input attributes and a target value are specified. the aim is t predict the target value, based n the input attributes. agent is presented with the input and target utput fr each item in the training set; it must then predict the utput fr each item in the test set varius learning paradigms are available: Neural Netwrk Decisin Tree Supprt Vectr Machine, etc.

5 COMP s2 Backprpagatin 4 Supervised Learning Issues framewrk (decisin tree, neural netwrk, SVM, etc.) representatin (f inputs and utputs) pre-prcessing / pst-prcessing training methd (perceptrn learning, backprpagatin, etc.) generalizatin (avid ver-fitting) evaluatin (separate training and testing sets)

6 COMP s2 Backprpagatin 5 Curve Fitting Which curve gives the best fit t these data? f()

7 COMP s2 Backprpagatin 6 Curve Fitting Which curve gives the best fit t these data? f() straight line?

8 COMP s2 Backprpagatin 7 Curve Fitting Which curve gives the best fit t these data? f() parabla?

9 COMP s2 Backprpagatin 8 Curve Fitting Which curve gives the best fit t these data? f() 4th rder plynmial?

10 COMP s2 Backprpagatin 9 Curve Fitting Which curve gives the best fit t these data? f() Smething else?

11 COMP s2 Backprpagatin 10 Ockham s Razr The mst likely hypthesis is the simplest ne cnsistent with the data. inadequate gd cmprmise ver-fitting Since there can be nise in the measurements, in practice need t make a tradeff between simplicity f the hypthesis and hw well it fits the data.

12 COMP s2 Backprpagatin 11 Outliers Predicted Buchanan Vtes by Cunty [faculty.washingtn.edu/mtbrett]

13 COMP s2 Backprpagatin 12 Butterfly Ballt

14 COMP s2 Backprpagatin 13 Recall: Limitatins f Perceptrns Prblem: many useful functins are nt linearly separable (e.g. XOR) I 1 I 1 I ? I 2 I I 2 I 1 I 2 I 1 I 2 (a) and (b) r (c) r I 1 I 2 Pssible slutin: 1 XOR 2 can be written as: ( 1 AND 2 ) NOR ( 1 NOR 2 ) Recall that AND, OR and NOR can be implemented by perceptrns.

15 COMP s2 Backprpagatin 14 Multi-Layer Neural Netwrks XOR NOR AND NOR Prblem: Hw can we train it t learn a new functin? (credit assignment)

16 COMP s2 Backprpagatin 15 Tw-Layer Neural Netwrk Output units a i W j,i Hidden units a j W k,j Input units a k Nrmally, the numbers f input and utput units are fied, but we can chse the number f hidden units.

17 COMP s2 Backprpagatin 16 The XOR Prblem 1 2 target fr this ty prblem, there is nly a training set; there is n validatin r test set, s we dn t wrry abut verfitting the XOR data cannt be learned with a perceptrn, but can be achieved using a 2-layer netwrk with tw hidden units

18 COMP s2 Backprpagatin 17 Neural Netwrk Equatins b 1 u 1 w 11 c z s v 1 v 2 y y1 2 u 2 b 2 w 22 u 1 = b 1 + w w 12 2 y 1 = g(u 1 ) s = c+v 1 y 1 + v 2 y 2 z = g(s) E = 1 2 (z t) 2 w 21 w We smetimes use w as a shrthand fr any f the trainable weights {c,v 1,v 2,b 1,b 2,w 11,w 21,w 12,w 22 }.

19 COMP s2 Backprpagatin 18 NN Training as Cst Minimizatin We define an errr functin E t be (half) the sum ver all input patterns f the square f the difference between actual utput and desired utput E = 1 2 (z t) 2 If we think f E as height, it defines an errr landscape n the weight space. The aim is t find a set f weights fr which E is very lw.

20 COMP s2 Backprpagatin 19 Lcal Search in Weight Space Prblem: because f the step functin, the landscape will nt be smth but will instead cnsist almst entirely f flat lcal regins and shulders, with ccasinal discntinuus jumps.

21 COMP s2 Backprpagatin 20 Key Idea a i a i a i t in i in i in i 1 (a) Step functin (b) Sign functin (c) Sigmid functin Replace the (discntinuus) step functin with a differentiable functin, such as the sigmid: g(s)= 1 1+e s r hyperblic tangent g(s)=tanh(s)= es e s e s + e s = 2( 1 1+e 2s ) 1

22 COMP s2 Backprpagatin 21 Gradient Descent (4.3) Recall that the errr functin E is (half) the sum ver all input patterns f the square f the difference between actual utput and desired utput E = 1 2 (z t) 2 The aim is t find a set f weights fr which E is very lw. If the functins invlved are smth, we can use multi-variable calculus t adjust the weights in such a way as t take us in the steepest dwnhill directin. w w η E w Parameter η is called the learning rate.

23 COMP s2 Backprpagatin 22 Chain Rule (6.5.2) If, say y=y(u) u=u() Then y = y u u This principle can be used t cmpute the partial derivatives in an efficient and lcalized manner. Nte that the transfer functin must be differentiable (usually sigmid, r tanh). Nte: if z(s)= 1 1+e s, z (s)=z(1 z). if z(s)=tanh(s), z (s)= 1 z 2.

24 COMP s2 Backprpagatin 23 Frward Pass b 1 u 1 w 11 c z s v 1 v 2 y1 2 u 2 w 21 w 12 y b 2 w 22 u 1 = b 1 + w w 12 2 y 1 = g(u 1 ) s = c+v 1 y 1 + v 2 y 2 z = g(s) E = 1 2 (z t) 2 1 2

25 COMP s2 Backprpagatin 24 Backprpagatin Partial Derivatives E z dz ds = z t = g (s)=z(1 z) s y 1 = v 1 dy 1 du 1 = y 1 (1 y 1 ) Useful ntatin δ ut = E δ 1 = E δ 2 = E s u 1 u 2 Then δ ut = (z t) z(1 z) E v 1 = δ ut y 1 δ 1 = δ ut v 1 y 1 (1 y 1 ) E w 11 = δ 1 1 Partial derivatives can be calculated efficiently by packprpagating deltas thrugh the netwrk.

26 COMP s2 Backprpagatin 25 Tw-Layer NN s Applicatins Medical Dignsis Autnmus Driving Game Playing Credit Card Fraud Detectin Handwriting Recgnitin Financial Predictin

27 COMP s2 Backprpagatin 26 Eample: Pima Indians Diabetes Dataset Attribute mean stdv 1. Number f times pregnant Plasma glucse cncentratin Diastlic bld pressure (mm Hg) Triceps skin fld thickness (mm) Hur serum insulin (mu U/ml) Bdy mass inde (weight in kg/(height in m) 2 ) Diabetes pedigree functin Age (years) Class variable (0 r 1)

28 COMP s2 Backprpagatin 27 Training Tips re-scale inputs and utputs t be in the range 0 t 1 r 1 t 1 replace missing values with mean value fr that attribute initialize weights t very small randm values n-line r batch learning three different ways t prevent verfitting: limit the number f hidden ndes r cnnectins limit the training time, using a validatin set weight decay adjust learning rate (and mmentum) t suit the particular task

29 COMP s2 Backprpagatin 28 Overfitting in Neural Netwrks Errr Errr versus weight updates (eample 1) Training set errr Validatin set errr Number f weight updates Nte: -ais culd als be number f hidden ndes r cnnectins

30 COMP s2 Backprpagatin 29 Overfitting in Neural Netwrks Errr Errr versus weight updates (eample 2) Training set errr Validatin set errr Number f weight updates

31 COMP s2 Backprpagatin 30 ALVINN (Pmerleau 1991, 1993)

32 COMP s2 Backprpagatin 31 ALVINN Sharp Left Straight Ahead Sharp Right 30 Output Units 4 Hidden Units 3032 Sensr Input Retina

33 COMP s2 Backprpagatin 32 ALVINN Autnmus Land Vehicle In a Neural Netwrk later versin included a snar range finder 8 32 range finder input retina 29 hidden units 45 utput units Supervised Learning, frm human actins (Behaviral Clning) additinal transfrmed training items t cver emergency situatins drve autnmusly frm cast t cast

34 COMP s2 Backprpagatin 33 Summary Neural netwrks are bilgically inspired Multi-layer neural netwrks can learn nn linearly separable functins Backprpagatin is effective and widely used