Neural networks. Nuno Vasconcelos ECE Department, UCSD
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1 Neural networs Nuno Vasconcelos ECE Department, UCSD
2 Classfcaton a classfcaton problem has two types of varables e.g. X - vector of observatons (features) n the world Y - state (class) of the world x X R 2 = (fever, blood pressure) y Y = {dsease, no dsease} X, Y related by a (unnown) functon x f (.) y = f (x ) goal: desgn a classfer h: X Y such hthat th( h(x) = f(x) x 2
3 Perceptron classfer mplements the lnear decson rule h ( x ) = sgn [ g(x) () ] wth g ( x ) = w T x + b learnng s formulated as an optmzaton problem defne set of errors w { T x y ( w x + ) < 0} E = b defne the cost p ( ) ( T w b y w x b ) J, and mnmze + = x E J T p cannot be negatve snce, n E, all y (w x +b) are negatve at zero we now we have the best possble soluton (E empty) b w x g ( x ) w 3
4 Gradent descent s smplest possble mnmzaton technque pc ntal estmate x (0) follow the negatve gradent x ( ( ) x ) ( n + 1) ( n ) n f(x) (n ) η f = x η f ( x ) (n ) x usually the gradent s a functon of entre tranng set D more effcent alternatve s stochastc gradent descent tae the step mmedately after each pont no guarantee ths s a descent step but, on average, you follow the same drecton after processng entre D very popular n learnng, where D s usually large 4
5 Perceptron learnng for the Perceptron ths leads to: set = 0, w = 0, b = 0 set R = max x do { } for = 1:n { f y (w T x + b ) <= 0 then { } w +1 = w + η y x b +1 = b + η y R 2 =+1 +1 } untl y (w T x + b ) 0, (no errors) 5
6 Perceptron learnng the nterestng part s that ths s guaranteed to converge n fnte tme Theorem: Let D = {(x 1,y 1 ),..., (x n,y n )} and R = max x If there s (w*,b*) such that w* = 1 and ( ) T y w * x + b * > γ, then the Perceptron wll fnd an error free hyper-plane plane n at most 2 2R teratons γ the margn γ appears as a measure of the dffculty of the learnng problem 6
7 Some hstory Mnsy and Papert dentfed serous problems there are very smply logc problems that the Perceptron cannot solve later realzed that these can be elmnated by relyng on a mult-layered l Perceptron (MLP) or neural networ ths s a cascade of Perceptrons where x are the nput unts layer 1: h (x) = sgn[w T x] layer 2: u(x) = sgn[w T h(x)] ( 7
8 Graphcal representaton the Perceptron s usually represented as nput unts: coordnates of x weghts: coordnates of w homogeneous coordnates: x = (x,1) T bas term h ( x ) = sgn w x + w 0 = sgn ( T w x ) 8
9 Sgmods the sgn[.] functon s problematc n two ways: no dervatve at 0 s(x) non-smooth approxmatons t can be approxmated n varous ways for example by the hyperbolc tangent s (x) s ( x ) = tanh( σx ) = e e e + e σ x σ x σx σ controls the approxmaton error, but σx has dervatve everywhere smooth s (x) neural networs are mplemented wth these functons 9
10 Neural networ the MLP as functon approxmaton 10
11 Two modes of operaton normal mode, after tranng: feedforward collect z here enter x here feedforward 11
12 Two modes of operaton tranng mode: bacpropagaton compare z to the target t, e = z-t 2 bacpropagaton: update wegths to mnmze error enter x here feedforward 12
13 Bacpropagaton s ust gradent descent at the end of the day, the output z s ust a bg functon of nput vector x weghts, whch we can represent by a bg vector w e.g. z s obectve: J J v s w x = = = 1 = 1 = z( x; W ) wth hww ( v, w ) W* = arg mn W J ( W ) wth J ( W ) n = 1 = [ t z( x ; W )] 2 13
14 Bacpropagaton ths s conceptually trval, but computng the gradent of J loos qute messy e.g. for z what s dz/dw 13? J J s v s w x = 1 = 1 = t turns out that t s possble to do ths easly by dong a certan amount of boo-eepng the soluton s the bacpropagaton algorthm, whch s based on local updates the ey to understandng d t s to mae the rght defntons t 14
15 In detal notaton: nput : x weght to hdden unt : w hdden unt : g = w x y = s [ g ] weght to output unt : w output unt : u = w y z = s [ u ] before sgmod d after sgmod w w z u y g x 15
16 Computng the gradent of J the ey s the chan rule the output layer s easy J w = J u u w = δ y (*) where J J z δ = = (**) u z u = t z s' u ( ) ( ) s the senstvty of unt w w z u y g x u = w y s[ u ] 1 n z = J = [ ] 2 = 1 t z 2 16
17 Computng the gradent of J for the hdden layer J w J = y J = y y g s ' g w ( g ) x w ths s a lttle more subtle snce J depends on y w through all the z x z u y g g = w x y = s[ g ] J = 1 n 2 = 1 [ ] t z 2 17
18 Computng the gradent of J J y = y = = = 1 ( t z ) 2 2 ( t z ) z y z u ( t z ) u y ( t z ) s' ( u ) w and, from (**), J y y overall J w = δ w = δ w s ' ( g ) x w w u = w y z = s[ u ] z u y g x J = 1 n 2 = 1 [ ] t z 2 18
19 Computng the gradent of J J w = δ w s' by analogy wth (*) J w wth δ = δ x = δ w ( g ) x w w z u y g x u = w y z = s[ u ] J = 1 n 2 = 1 [ ] t z 2 19
20 In summary for any par (,) J w wth δ = δ y = δ w f s hdden and δ = ( t z ) s' ( u ) w f s output. The weght updates are w ( n+ 1) = w ( n+ 1) J η w the error s bacpropagated by local message passng! 20
21 Feature transformaton MLP can be seen as: non-lnear feature transformaton + lnear dscrmnant Perceptron feature transformaton y = Φ x ( ) 21
22 Feature transformaton the feature transformaton searches for the space where the patterns become separable example: two class problem 2-1 networ non lnearly separable on the space of Xs made lnearly separable on the space of Ys the fgure shows evoluton of Ys and dthe tranng error 22
23 Feature transformaton Q: s separablty always possble? A: not really, depends on the number of unts example two class problem 2-1 networ s not enough but 3-1 networ s n practce art-form tral and error 23
24 Other problems the optmzaton surface can be qute nasty example: scalar problem 1-1 networ cost has many plateaus global optmal soluton has no error but gradent frequently close to zero slow progress n general: one plateau per tranng pont mproves wth more ponts, degrades wth more weghts (dms) 24
25 Other problems how do we set the learnng rate η? f too small or too bg, we wll need varous teratons could even dverge lne search: pc η (0) ( ) ( n ) (0) ( n ) compute x ' = x η f x and then f(x ) f not good, mae η (+1) = α η () (wth α <1) and repeat untl you get a mnmum of f(x) ( n + 1) ( n ) ( n ) x = x η f ( x ) 25
26 Structural rs mnmzaton what about complexty penaltes, overfttng and all that? SRM, n general: 1. start from nested collecton of famles of functonss1 L S 2. for each S,, fnd the set of parameters that mnmzes the emprcal rs * 3. select the functon class such that { R = mn R ( )} emp + Φ h where Φ(h) ( ) s a functon of the VC dmenson (complexty) of the famly S can be done by 1. S = {MLPs such that W 2 < λ } 2. bacpropagaton n ths famly 26
27 Structural rs mnmzaton nstead of W * = solve arg mn J ( W ) W n wth J ( W ) = [ t z ( x ; W ) ] = 1 T W * = arg mn J ( W ) subect to W W < λ W we wll see that ths s equvalent to 2ε T W * = arg mn J ( W ) + W W W η re-worng out bacpropagaton ths can be done by h shrnng after each weght update do w = w ( 1-ε ) ths s nown as weght decay and penalzes complex models new old 2 27
28 In summary ths wors, but requres tunnng ε the cost surface s nasty one needs to try dfferent archtectures hence, tranng can be panfully slow wees s qute common a good neural networ may tae years to tran however, when you are fnshed t tends to wor well examples: the Rowley and Kanade face detector the LeCunn dgt recognzer (see n com/e html) 28
29 Rowley and Kanade Neural networ-based face detecton Rowley, H.A.; Balua, S.; Kanade, T.; IEEE Transactons on PAMI,Volume: 20, Issue: 1, Jan the face detector: 29
30 Results 30
31 Trcs (good for any learnng algorthm) expand the tranng set to cover for most varaton a more exhaustve tranng set always produces better results than a less exhaustve one f you can create nterestng examples artfcally, then by all means... e.g. n vson rotate, scale, translate ndependently of what algorthm you are usng 31
32 Trcs (good for any learnng algorthm) where do I get negatve examples? fndng a good negatve example s dffcult use the classfer tself to do t: 1. put together tranng set D 1 2. tran classfer C wth tranng set D 3. run on a dataset that has no postve examples 4. mae D +1 = {examples classfed as postve} U D 5. goto 2. e.g. close non-face examples 32
33 33
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