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Wiley-Interscience Publication, John Wiley&Sons, Inc vMR]~(WYzfOo`SNYtWS_NbRv^(u0 <$ZCX          @    0         vQ[c4&6g  6  T~f[`Neli ~f[`N f[`NǏ zvlSR /ecTϑ:g SVM[O{l ^(ut P"0P< eea%a             ~f[`Nt"0 x Qpe0O!jWb 6  \f[`N7h,gvQpe: NuhV (G) generates observations x (typically in Rn), independently drawn from some fixed distribution F(x) ~hVSupervisor (S) labels each input x with an output value y according to some fixed distribution F(y|x) f[`N:gLearning Machine (LM)  learns from an i.i.d. l-sample of (x,y)-pairs output from G and S, by choosing a function that best approximates S from a parameterised function class f(x,a), where a is in L the parameter set sQ.i_: F(x,y), an i.i.d. l-sample on F, functions f(x,a) and the equivalent representation of each f using its index a PpP vPccccccck4cccccccccc'cccccccc cccccccc0cc%ccccc     gccccccc cc cc ccccc  +ccc     ,  9  g    r ggΘi&6 ~Θi&6 ~Θi&6 VC~'6  Ǐf[`N&6 y Θig\S& ` b 6&   _c1YQpe loss functional (L, Q) the error of a given function on a given example ΘiQperisk functional (R) the expected loss of a given function on an example drawn from F(x,y) the (usual concept of) generalisation error of a given functionp1 d FG (@f b f b b b 1b f f b b b ?b b b b b b Cb 4 K   {f[`NvN,h:y  b 6   tf[`Nvh Given an i.i.d. l-sample z1,& ,zl drawn from a fixed distribution F(z) For a function class loss functionals Q(z,a), with a in L We wish to minimise the risk, finding a function a* ` ` ` ` ` h ` ` h !` ` (` `            4    zf[`NvN,h:y  b 6   v~Θig\SR_~SR (The Empirical Risk Minimization (ERM) Inductive Principle) Define the empirical risk (sample/training error): Define the empirical risk minimiser: ERM approximates Q(z,a*) with Q(z,al) the Remp minimiser & that is ERM approximates a* with al Least-squares and Maximum-likelihood are realisations of ERM =3`<%@<^@ =   ;    P                   = 4    m |f[`NtvV*NR  b 6   t1. f[`NǏ zvN'`t What are (necessary and sufficient) conditions for consistency (convergence of Remp to R) of a learning process based on the ERM Principle? 2.f[`NǏ z6e[e^v^nяt How fast is the rate of convergence of a learning process? 3. c6Rf[`NǏ zvlSRt How can one control the rate of convergence (the generalization ability) of a learning process? 4. g f[`N{lvt How can one construct algorithms that can control the generalization ability?L" b  f Pb b j b b 4b b f ;b b b f `b b b  f Ub    P  ;   >  d   U }Q[c4&6g  6  V~f[`Neli f[`Nvh:y f[`NǏ zvlSR /ecTϑ:g SVM[O{l ^(ut P#0P<e e a%a            ~"~gΘig\SR_~SR (SRM)& ` `    ERM is intended for relatively large samples (large l/h) Large l/h induces a small e which decreases the the upper bound on risk Small samples? Small empirical risk doesn t guarantee anything! & we need to minimise both terms of the RHS of the risk bounds The empirical risk of the chosen aL An expression depending on the VC dimension of L9V " d d ` d d ` ` ` ` ` ` ` ` ` `  -` d d d d q` ` d d d `   ` d   V "~gΘig\SR_~SR (SRM)& ` `    The Structural Risk Minimisation (SRM) Principle Let S = {Q(z,a),aL}. An admissible structure S1S2& Sn& S: For each k, the VC dimension hk of Sk is finite and h1d"h2d"& d"hnd"& d"hS Every Sk is either is non-negative bounded, or satisfies for some (p,tk)x1 F>1                                              ;      g  #          A  "~gΘig\SR_~SR (SRM)* b (b ((    ZThe SRM Principle continued For given z1,& ,zl and an admissible structure S1S2& Sn& S, SRM chooses function Q(z,alk) minimising Remp in Sk for which the guaranteed risk (risk upper-bound) is minimal Thus manages the unavoidable trade-off of quality of approximation vs. complexity of approximation0ZZb  b b  b b j b b j b                             *     1  +  %          "~gΘig\SR_~SR (SRM)  (((     Overfitting and underfitting Q[c4&6g  6  V~f[`Neli f[`Nvh:y f[`NǏ zvlSR /ecTϑ:g SVM[O{l ^(u P#0P<e eea%a            /ecTϑ:g SVM$ "0(  SVMs are learning systems that use a hyperplane of linear functions in a high dimensional feature space  Kernel function trained with a learning algorithm from optimization theory  Lagrange Implements a learning bias derived from statistical learning theory  Generalisation SVM is a classifier derived from statistical learning theory by Vapnik and Chervonenkis N A $B;B DBWb   "      @      \ ~'`R{|hV&`b6&    ] ~'`R{|hV(b6`   ^ ~'`R{|hV(b6`   _  ~'`R{|hV(b6`   `! ~'`R{|hV(b6`   a"g'Yb6   Q R{|s^bb6  Training set: (xi, yi), i=1,2,& N; yi{+1,-1} Hyperplane: wx+b=0 This is fully determined by (w,b) @"""*"*"*  """bbb"f       %  Rg'Y(b6`   S g'YSR f6  vNote1: decision functions (w,b) and (cw, cb) are the same Note2: but margins as measured by the outputs of the function xwx+b are not the same if we take (cw, cb). Definition: geometric margin: the margin given by the canonical decision function, which is when c=1/||w|| Strategy: 1) we need to maximise the geometric margin! (cf result from learning theory) 2) subject to the constraint that training examples are classified correctly Z"" ""Q""" """""""""&    M        n T According to Note1, we can demand the function output for the nearest points to be +1 and  1 on the two sides of the decision function. This removes the scaling freedom. Denoting a nearest positive example x+ and a nearest negative example x-, this is Computing the geometric margin (that has to be maximised): And here are the constraints: V["*!"*g"  [ U6Maximum margin  summing upb$   V/ecTϑf6   R{|vpef[h:y  &6   V ]w~ƖS+T *N7h,gp f /feQchTϑ byeQ by!j_ vQR ϑy:Nyr_ b^\'` beQch /fQch bQ. [N*Nev!j_ ce[@b[^vQ /f1؏/f-1. [(~b0RN*Nb NvpRb$NRvĉR. ZGGGGOGGGGG&     O              ^ 9hnc~[v~Ɩ vQ-N [~b NvN*N[ f6q^ O  g'Yv 0 9Z;& & & & E&  & & h  %     )    Bl㉟SY&6 8:NBl㉟SY 9hncgOSt bNlS:N[vPegBl  ~'`SR&6 яNon-linear SVMs`&    \Transform x f(x) The linear algorithm depends only on xxi, hence transformed algorithm depends only on f(x)f(xi) Use kernel function K(xi,xj) such that K(xi,xj)= f(x)f(xi)  """  '   0      ""*""*""*""*"      "@       ^~'`R{|  ^~'`R{|  ^~'`R{|  ^~'`R{|  ^~'`R{| [s^~'`R{|v`` &6  [s^~'`R{|v``   8hQpe(8hbck[8h)[IN B      8hQpev b X Yy_Q8h"6  {The kind of kernel represents the inner product of two vector(point) in a feature space of dimension. For example"px 0xF| | Q[c4&6g  6  V~f[`Neli f[`Nvh:y f[`NǏ zvlSR /ecTϑ:g SVM[O{l ^(u P#0P<eee%ea            SVM[O{lPe%e6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  Q[c4&6g  6  V~f[`Neli f[`Nvh:y f[`NǏ zvlSR /ecTϑ:g SVM[O{l ^(u| P#0P<ee%ee            SVM applications Pattern recognition Features: words counts DNA array expression data analysis Features: expr. levels in diff. conditions Protein classification Features: AA composition o#+ o o#+ 2X  L |. ` cHandwritten Digits Recognition`  rHandwritten digits recognition Of interest to the US Postal services 4% error was obtained about 4% of the training data were SVs only Text categorisation Face detection DNA analysis & VZhZ3Z"hb2""2~  8  SVM APPLICATIONSb(   Face Detection MRI OCR zZZZZb"  b0  Classifier Using RBF Kernel(   An Example ` ZZZ b b &   v, References `   Vladimir Vapnik. The Nature of Statistical Learning Theory, Springer, 1995 Andrew W. Moore. cmsc726: SVMs. http://www.cs.cmu.edu/~awm/tutorials C. Burges. A tutorial on support vector machines for pattern recognition. Data Mining and Knowledge Discovery, 2(2):955-974, 1998. http://citeseer.nj.nec.com/burges98tutorial.html Vladimir Vapnik. Statistical Learning Theory. Wiley-Interscience; 1998 Thorsten Joachims (joachims_01a): A Statistical Learning Model of Text Classification for Support Vector Machines Ben Rubinstein. Statistical Learning Theory. Dept. Computer Science & Software Engineering, University of Melbourne; and Division of Genetics & Bioinformatics, Walter & Eliza Hall Institute Pp P(Pfb)fc$"b"Hb#"P",b b b "    :        &     b    0m<www.lianyimf.com/shizz/,b0f0   0 Questions?!2  g6b6  /            !"#$%&'()*+,- .!/"0#1$2%3&4'5(6)7*8+9,:-;.</=0>1?2@3A4B5C6D7E8F9G:H;I<J=K>L?M@NAOBPCQDRESFTGUHVIW   0` 33PP` 13` 3333` Q_{` 333fpKNāvI` j@v۩ῑ΂H>?" dd@,?n<d@ `7 c`2@C`7`c`2` n?" dd@   @@``PR    @ ` ` p>> <`(  <  < <̌  #" `   0   < 08  " )}   0 cN NN}/kGr ,{Nd\ ,{ Nd\ ,{Vd\ ,{Nd\    < 6@  "Y y   j*0    < 6  "Y0y    l* 0    < 6  "Ycy   l* 0  `B < s *DjJ"  < NA" ??"X $ 0 0B < s *޽h ? 333380___PPT10.(г%  Blends   0 zr@(  @  @ Bhx7 ?#" ` p 7 P cN NN}/kGrjL#j_     @ 0z7 " `   7 R cN NN}/kGroRjL#j_  @ 67 "`p  7 n*"0   @ 6H7 "`p  7 p*$0   @ 6L7 "` 7 p*$0  B @ s *޽h ? 333380___PPT10.(г%& 0 `6(    06  P   A V*    0L       f*   d  c $ ?     0q   @   8USQdkYkHre,g7h_ ,{N~ ,{ N~ ,{V~ ,{N~    6`  `P    d*     68  `    f*   H  0޽h ? ̙3380___PPT10.]V@jWK  0 L0 ZR0(  x  c $$kA <wj  A   c $@A < <X<$  0 A   07  io,$  0 @/ecTϑ:g Support Vector Machinesh!Zg,c,b,, b ,&    H  0޽h ? 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Vh*$0 c     `D  ??   dhn$0 c     `  ?? pO  Uh$0 c     `ج  ??0 P  VS1$0 c     `T  ??0  P  VS*$0 c     `  ??0 P P  dSn$0 c     B`CDEFԔ``` @  `p  `   HL   <"`   B  s *޽h ? f3f3y___PPT10Y+D=' A= @B +'  0 p  g(  ~  s *  <   |  0 P rProblem: how rich class of classifications q(x;) to use.x:0 2G%CCCC C:x  <A ??    x  <A ??  x  <A  ??      0(  P  b underfitting$ 0 2 C   0$    a overfitting$ 0 2 C    0Љ  0   ^good fit$ 0 2 C L   0Џ  `Problem of generalization: a small emprical risk Remp does not imply small true expected risk R.Za0 2GCK+CaH  0޽h ? ^WNff36  0  x6(  x~ x s *\  <   x x c $m  <d    H x 0޽h ? 333380___PPT10.؂$  0 $(  r  S y  <   r  S z  <M6   H  0޽h ? 333380___PPT10.܂wr"  0L0 !y!45!(  ~  s *$-  <` p     BD/  1?"Z _a.t(<2g       B3  1?"Mp ~yest<t <c k      BT7  1?" odenotes +1 denotes -1*t(<2c  2   TSO?"@f52   TAb?"0@bPB   TDԔ?"p``B  NDԔ?"  2  N?"b & L 2  N?" D 2  N?" 2  N?"  2  N?"d2  N?"0 N 2  N?"2  N?" @ 2  T\?" 2  T\?"2  T\?"/  . O 2  T\?"2  T\?" 2  T\?" p. 2  T\?" 2  TAb?" 2  TAb?" , R 2   TAb?" 8 2 ! TAb?" 2 " TAb?"3 3 X 2 # TAb?" * 2 $ TAb?"J h2 % TAb?"2 & TAb?"  .2 ' TAb?" 2 ( TSO?"2 ) TSO?" p  2 * TSO?"  2 + TSO?"0 U 2 , TSO?"" D 2 - TSO?" 2 . TSO?")O2 / TSO?"n 2 0 TSO?" 0 8 ` 5  TH 1?"  v f < t(<2c$ c  B  ZD1?"    BN  1?" ' [x*t(<2c  B  ZD1?"B  ZD1?"p 1 BU  1?" ` _f(x,w,b) = sign(w. x - b)t(<2ccccgg cgcgcgc   2 NPa 1?"`` R(t <2c  3 Bc  1?"@0  y!How would you classify this data?("t <2"c " H  0޽h ? 3333y___PPT10Y+D=' A= @B +"  0L0 !!45%!(  ~  s *j  <` p     T(t 1?"  v f < t(<2c$ c  B  ZD1?"   Bȃ  1?" ' [x*t(<2c  B  ZD1?"  B$  1?"@0m _a.t(<2g    B  ZD1?"p   B  1?"p} ~yest<t <c k      B  1?" odenotes +1 denotes -1*t(<2c  2   TSO?"@f52   TAb?"0@bPB   TDԔ?"p``B  NDԔ?"  2  N?"b & L 2  N?" D 2  N?" 2  N?"  2  N?"d2  N?"0 N 2  N?"2  N?" @ 2  T\?" 2  T\?"2  T\?"/  . O 2  T\?"2  T\?" 2  T\?" p. 2  T\?" 2  TAb?" 2  TAb?" , R 2   TAb?" 8 2 ! TAb?" 2 " TAb?"3 3 X 2 # TAb?" * 2 $ TAb?"J h2 % TAb?"2 & TAb?"  .2 ' TAb?" 2 ( TSO?"2 ) TSO?" p  2 * TSO?"  2 + TSO?"0 U 2 , TSO?"" D 2 - TSO?" 2 . TSO?")O2 / TSO?"n 2 0 TSO?" 0  1 BS  1?"  ` _f(x,w,b) = sign(w. x - b)t(<2ccccgg cgcgcgc  B 2 ND1?"p`  3 ND 1?"`` R(t <2c  4 BDG  1?"@0  y!How would you classify this data?("t <2"c " H  0޽h ? 3333y___PPT10Y+D=' A= @B +#  0L0 ""55"(  ~  s *4N  <` p     T@ 1?"  v f < t(<2c$ c  B  ZD1?"   BO  1?" ' [x*t(<2c  B  ZD1?"  B  1?"@0m _a.t(<2g    B  ZD1?"p   B  1?"p} ~yest<t <c k      B"  1?" odenotes +1 denotes -1*t(<2c  2   TSO?"@f52   TAb?"0@bPB   TDԔ?"p``B  NDԔ?"  2  N?"b & L 2  N?" D 2  N?" 2  N?"  2  N?"d2  N?"0 N 2  N?"2  N?" @ 2  T\?" 2  T\?"2  T\?"/  . O 2  T\?"2  T\?" 2  T\?" p. 2  T\?" 2  TAb?" 2  TAb?" , R 2   TAb?" 8 2 ! TAb?" 2 " TAb?"3 3 X 2 # TAb?" * 2 $ TAb?"J h2 % TAb?"2 & TAb?"  .2 ' TAb?" 2 ( TSO?"2 ) TSO?" p  2 * TSO?"  2 + TSO?"0 U 2 , TSO?"" D 2 - TSO?" 2 . TSO?")O2 / TSO?"n 2 0 TSO?" 0  1 B0  1?"  ` _f(x,w,b) = sign(w. x - b)t(<2ccccgg cgcgcgc  B 2 ND1?"0  3 NT 1?"`` R(t <2c  4 B  1?"@0  y!How would you classify this data?("t <2"c "  5 6@ "  }'Copyright 2001, 2003, Andrew W. Moore&(0(g  ( H  0޽h ? 3333y___PPT10Y+D=' A= @B +#  0L0 ""55"(  ~  s *@ <` p     T1?"  v f < t(<2c$ c  B  ZD1?"   B 1?" ' [x*t(<2c  B  ZD1?"  B& 1?"@0m _a.t(<2g    B  ZD1?"p   Bc 1?"p} ~yest<t <c k      B`h 1?" odenotes +1 denotes -1*t(<2c  2   TSO?"@f52   TAb?"0@bPB   TDԔ?"p``B  NDԔ?"  2  N?"b & L 2  N?" D 2  N?" 2  N?"  2  N?"d2  N?"0 N 2  N?"2  N?" @ 2  T\?" 2  T\?"2  T\?"/  . O 2  T\?"2  T\?" 2  T\?" p. 2  T\?" 2  TAb?" 2  TAb?" , R 2   TAb?" 8 2 ! TAb?" 2 " TAb?"3 3 X 2 # TAb?" * 2 $ TAb?"J h2 % TAb?"2 & TAb?"  .2 ' TAb?" 2 ( TSO?"2 ) TSO?" p  2 * TSO?"  2 + TSO?"0 U 2 , TSO?"" D 2 - TSO?" 2 . TSO?")O2 / TSO?"n 2 0 TSO?" 0  1 BF 1?"  ` _f(x,w,b) = sign(w. x - b)t(<2ccccgg cgcgcgc  B 2 ND1?" p  3 NV1?"`` R(t <2c  4 BTy 1?"@0  y!How would you classify this data?("t <2"c "  5 6 } "  }'Copyright 2001, 2003, Andrew W. Moore&(0(g  ( H  0޽h ? 3333y___PPT10Y+D=' A= @B +#   0L0 ""55"(  ~  s * <` p     T1?"  v f < t(<2c$ c  B  ZD1?"   B芉 1?" ' [x*t(<2c  B  ZD1?"  B\ 1?"@0m _a.t(<2g    B  ZD1?"p   B] 1?"p} ~yest<t <c k      Bq 1?" odenotes +1 denotes -1*t(<2c  2   TSO?"@f52   TAb?"0@bPB   TDԔ?"p``B  NDԔ?"  2  N?"b & L 2  N?" D 2  N?" 2  N?"  2  N?"d2  N?"0 N 2  N?"2  N?" @ 2  T\?" 2  T\?"2  T\?"/  . O 2  T\?"2  T\?" 2  T\?" p. 2  T\?" 2  TAb?" 2  TAb?" , R 2   TAb?" 8 2 ! TAb?" 2 " TAb?"3 3 X 2 # TAb?" * 2 $ TAb?"J h2 % TAb?"2 & TAb?"  .2 ' TAb?" 2 ( TSO?"2 ) TSO?" p  2 * TSO?"  2 + TSO?"0 U 2 , TSO?"" D 2 - TSO?" 2 . TSO?")O2 / TSO?"n 2 0 TSO?" 0  1 Bл 1?"  ` _f(x,w,b) = sign(w. x - b)t(<2ccccgg cgcgcgc  B 2 ND1?" p  3 NΉ1?"`` R(t <2c  4 Bщ 1?"@0  y!How would you classify this data?("t <2"c "  5 6|։ "  }'Copyright 2001, 2003, Andrew W. Moore&(0(g  ( H  0޽h ? 3333y___PPT10Y+D=' A= @B +- ! 0L0 '%%77$(  B  NY Dޅ?"` B  NY D1?"0 ~  s *P݉ <` p     Tމ1?"  v f < t(<2c$ c  B  ZD1?"   B 1?" ' [x*t(<2c  B  ZD1?"   B 1?"@0m _a.t(<2g    B   ZD1?"p   B 1?"p} ~yest<t <c k      B 1?" odenotes +1 denotes -1*t(<2c  2   TSO?"@f52  TAb?"0@bPB  TDԔ?"p``B  NDԔ?"  2  N?"b & L 2  N?" D 2  N?" 2  N?"  2  N?"d2  N?"0 N 2  N?"2  N?" @ 2  T\?" 2  T\?"2  T\?"/  . O 2  T\?"2  T\?" 2  T\?" p. 2  T\?" 2   TAb?" 2 ! TAb?" , R 2 " TAb?" 8 2 # TAb?" 2 $ TAb?"3 3 X 2 % TAb?" * 2 & TAb?"J h2 ' TAb?"2 ( TAb?"  .2 ) TAb?" 2 * TSO?"2 + TSO?" p  2 , TSO?"  2 - TSO?"0 U 2 . TSO?"" D 2 / TSO?" 2 0 TSO?")O2 1 TSO?"n 2 2 TSO?" 0  3 B1 1?"  ` _f(x,w,b) = sign(w. x - b)t(<2ccccgg cgcgcgc   4 N=1?"`` R(t <2c  5 BP? 1?"I,$ 0 The maximum margin linear classifier is the linear classifier with the maximum margin. This is the simplest kind of SVM (Called an LSVM)Pt <2c gec  * 6 TG`Hj1?" B,$D 0 d Linear SVM* t(<2 c   7 63 "  }'Copyright 2001, 2003, Andrew W. Moore&(0(g  ( H  0޽h ?6 3333=5___PPT10+SD' A= @B DD' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*5W%(D2' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*5W%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*6%(+p+0+50 ++0+60 +V " 0 0(  ~  s *  <   x  c $ <0   X  0AR?H  0޽h ? ̙33 # 0 4,@(  ~  s *ܓ[ <  [   s *[AS < 0 [ " j  0H[pP   According to a theorem from Learning Theory, from all possible linear decision functions the one that maximises the margin of the training set will minimise the generalisation error.$0 2c  H  0޽h ? ̙33[ $ 0  P (  ~  s *\[ <  [   6Ԙ[AS P  " RB  s *DP x  c $[ <   [   0[@`  Uw$0 2c    0[ @  wx+b=0>0 2gg    0`[  0  wx+b>0>0 2gg     0[ @  wx+b<0>0 2gg  H  0޽h ? ̙33 % 0 rj` (  ~  s *p[ < )} [ `  c $AT ??  `  c $AU ?? r  `  c $AV ??   X  0 `  c $AW  ?? 2  XR   0 p p   <#[ "` j g'YSR f6  H  0޽h ? ̙33E  & 0   p  (    s [0e0eAS  < >  [ "   0.[  wx+b=0>0 2gg    03[  wx+b=1>0 2g!g!    06[  wx+b=-1>0 2g!g!    0[ 0 @  wx+b>1>0 2gg    0j[ P  wx+b<1>0 2gg    s pi[0e0e  <  [    0m[ N$0 2c    0hZ[0  ZGiven a linearly separable training set (xi, yi), i=1,2,& N; yi{+1,-1} Minimise ||w||2 Subject to This is a quadratic programming problem with linear inequality constraints. There are well known procedures for solving it JL0 2)cckckck         ~ &-   `   c $AX !??   !H  0޽h ? ̙33m ' 0 (  ~  s *-[ <  [   0l[AY <  } [ " M  0,P[@P   The training points that are nearest to the separating function are called support vectors. What is the output of our decision function for these points?$0 2c  H  0޽h ? ̙33 ( 0L0   |d(  |x | c $|[ <r  [ x | c $V[ <  [ ` | c $A h?? h` | c $A i??p9V i` | c $A j??io j` | c $A k??yY  k | HA l? ? # I l  | HA m? ? 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Wiley-Interscience Publication, John Wiley&Sons, Inc vMR]~(WYzfOo`SNYtWS_NbRv^(u0 <$ZCX          @    0          ~f[`Nt"0 x Qpe0O!jWb 6  \f[`N7h,gvQpe: NuhV (G) generates observations x (typically in Rn), independently drawn from some fixed distribution F(x) ~hVSupervisor (S) labels each input x with an output value y according to some fixed distribution F(y|x) f[`N:gLearning Machine (LM)  learns from an i.i.d. l-sample of (x,y)-pairs output from G and S, by choosing a function that best approximates S from a parameterised function class f(x,a), where a is in L the parameter set sQ.i_: F(x,y), an i.i.d. l-sample on F, functions f(x,a) and the equivalent representation of each f using its index a PpP vPccccccck4cccccccccc'cccccccc cccccccc0cc%ccccc     gccccccc cc cc ccccc  +ccc     ,  9  g    r ggΘi&6 ~Θi&6 ~Θi&6 VC~'6  Ǐf[`N&6 >Ǐf[`NOverfitting and underfitting( &0( |f[`NtvV*NR  b 6   t1. f[`NǏ zvN'`t What are (necessary and sufficient) conditions for consistency (convergence of Remp to R) of a learning process based on the ERM Principle? 2.f[`NǏ z6e[e^v^nяt How fast is the rate of convergence of a learning process? 3. c6Rf[`NǏ zvlSRt How can one control the rate of convergence (the generalization ability) of a learning process? 4. g f[`N{lvt How can one construct algorithms that can control the generalization ability?L" b  f Pb b j b b 4b b f ;b b b f `b b b  f Ub    P  ;   >  d   U ~"~gΘig\SR_~SR (SRM)& ` `    ERM is intended for relatively large samples (large l/h) Large l/h induces a small e which decreases the the upper bound on risk Small samples? Small empirical risk doesn t guarantee anything! & we need to minimise both terms of the RHS of the risk bounds The empirical risk of the chosen aL An expression depending on the VC dimension of L9V " d d ` d d ` ` ` ` ` ` ` ` ` `  -` d d d d q` ` d d d `   ` d   V "~gΘig\SR_~SR (SRM)& ` `    The Structural Risk Minimisation (SRM) Principle Let S = {Q(z,a),aL}. An admissible structure S1S2& Sn& S: For each k, the VC dimension hk of Sk is finite and h1d"h2d"& d"hnd"& d"hS Every Sk is either is non-negative bounded, or satisfies for some (p,tk)x1 F>1                                              ;      g  #          A  "~gΘig\SR_~SR (SRM)* b (b ((    ZThe SRM Principle continued For given z1,& ,zl and an admissible structure S1S2& Sn& S, SRM chooses function Q(z,alk) minimising Remp in Sk for which the guaranteed risk (risk upper-bound) is minimal Thus manages the unavoidable trade-off of quality of approximation vs. complexity of approximation0ZZb  b b  b b j b b j b                             *     1  +  %          "~gΘig\SR_~SR (SRM)  (((     /ecTϑ:g SVM$ "0(  SVMs are learning systems that use a hyperplane of linear functions in a high dimensional feature space  Kernel function trained with a learning algorithm from optimization theory  Lagrange Implements a learning bias derived from statistical learning theory  Generalisation SVM is a classifier derived from statistical learning theory by Vapnik and Chervonenkis N A $B;B DBWb   "      @      \ ~'`R{|hV&`b6&    ] ~'`R{|hV(b6`   ^ ~'`R{|hV(b6`   _  ~'`R{|hV(b6`   `! ~'`R{|hV(b6`   a"g'Yb6   Q R{|s^bb6  Training set: (xi, yi), i=1,2,& N; yi{+1,-1} Hyperplane: wx+b=0 This is fully determined by (w,b) @"""*"*"*  """bbb"f       %  Rg'Y(b6`   S g'YSR f6  vNote1: decision functions (w,b) and (cw, cb) are the same Note2: but margins as measured by the outputs of the function xwx+b are not the same if we take (cw, cb). 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For example"px 0xF| | Q[c4&6g  6  V~f[`Neli f[`Nvh:y f[`NǏ zvlSR /ecTϑ:g SVM[O{l ^(u P#0P<eee%ea            SVM[O{lPe%e6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  Q[c4&6g  6  V~f[`Neli f[`Nvh:y f[`NǏ zvlSR /ecTϑ:g SVM[O{l ^(u| P#0P<ee%ee            SVM applications Pattern recognition Features: words counts DNA array expression data analysis Features: expr. levels in diff. conditions Protein classification Features: AA composition o#+ o o#+ 2X  L |. ` cHandwritten Digits Recognition`  rHandwritten digits recognition Of interest to the US Postal services 4% error was obtained about 4% of the training data were SVs only Text categorisation Face detection DNA analysis & VZhZ3Z"hb2""2~  8  SVM APPLICATIONSb(   Face Detection MRI OCR zZZZZb"  b0  Classifier Using RBF Kernel(   An Example ` ZZZ b b &   v, References `   Vladimir Vapnik. The Nature of Statistical Learning Theory, Springer, 1995 Andrew W. Moore. cmsc726: SVMs. http://www.cs.cmu.edu/~awm/tutorials C. Burges. A tutorial on support vector machines for pattern recognition. Data Mining and Knowledge Discovery, 2(2):955-974, 1998. http://citeseer.nj.nec.com/burges98tutorial.html Vladimir Vapnik. Statistical Learning Theory. Wiley-Interscience; 1998 Thorsten Joachims (joachims_01a): A Statistical Learning Model of Text Classification for Support Vector Machines Ben Rubinstein. Statistical Learning Theory. Dept. 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Wiley-Interscience Publication, John Wiley&Sons, Inc vMR]~(WYzfOo`SNYtWS_NbRv^(u0 <$ZCX          @    0          ~f[`Nt"0 x Qpe0O!jWb 6  \f[`N7h,gvQpe: NuhV (G) generates observations x (typically in Rn), independently drawn from some fixed distribution F(x) ~hVSupervisor (S) labels each input x with an output value y according to some fixed distribution F(y|x) f[`N:gLearning Machine (LM)  learns from an i.i.d. l-sample of (x,y)-pairs output from G and S, by choosing a function that best approximates S from a parameterised function class f(x,a), where a is in L the parameter set sQ.i_: F(x,y), an i.i.d. l-sample on F, functions f(x,a) and the equivalent representation of each f using its index a PpP vPccccccck4cccccccccc'cccccccc cccccccc0cc%ccccc     gccccccc cc cc ccccc  +ccc     ,  9  g    r ggΘi&6 ~Θi&6 ~Θi&6 VC~'6  Ǐf[`N&6 >Ǐf[`NOverfitting and underfitting( &0( |f[`NtvV*NR  b 6   t1. f[`NǏ zvN'`t What are (necessary and sufficient) conditions for consistency (convergence of Remp to R) of a learning process based on the ERM Principle? 2.f[`NǏ z6e[e^v^nяt How fast is the rate of convergence of a learning process? 3. c6Rf[`NǏ zvlSRt How can one control the rate of convergence (the generalization ability) of a learning process? 4. g f[`N{lvt How can one construct algorithms that can control the generalization ability?L" b  f Pb b j b b 4b b f ;b b b f `b b b  f Ub    P  ;   >  d   U ~"~gΘig\SR_~SR (SRM)& ` `    ERM is intended for relatively large samples (large l/h) Large l/h  induces a small e which decreases the the upper bound on risk Small samples? Small empirical risk doesn t guarantee anything! & we need to minimise both terms of the RHS of the risk bounds The empirical risk of the chosen aL An expression depending on the VC dimension of L9V " d d ` d d ` ` ` ` ` ` ` ` ` `  -` d d d d q` ` d d d `   ` d   V "~gΘig\SR_~SR (SRM)& ` `    The Structural Risk Minimisation (SRM) Principle Let S = {Q(z,a),aL}. An admissible structure S1S2& Sn& S: For each k, the VC dimension hk of Sk is finite and h1d"h2d"& d"hnd"& d"hS Every Sk is either is non-negative bounded, or satisfies for some (p,tk)x1 F>1                                              ;      g  #          A  "~gΘig\SR_~SR (SRM)* b (b ((    ZThe SRM Principle continued For given z1,& ,zl and an admissible structure S1S2& Sn& S, SRM chooses function Q(z,alk) minimising Remp in Sk for which the guaranteed risk (risk upper-bound) is minimal Thus manages the unavoidable trade-off of quality of approximation vs. complexity of approximation0ZZb  b b  b b j b b j b                             *     1  +  %          "~gΘig\SR_~SR (SRM)  (((     /ecTϑ:g SVM$ "0(  SVMs are learning systems that use a hyperplane of linear functions in a high dimensional feature space  Kernel function trained with a learning algorithm from optimization theory  Lagrange Implements a learning bias derived from statistical learning theory  Generalisation SVM is a classifier derived from statistical learning theory by Vapnik and Chervonenkis N A $B;B DBWb   "      @      \ ~'`R{|hV&`b6&    ] ~'`R{|hV(b6`   ^ ~'`R{|hV(b6`   _  ~'`R{|hV(b6`   `! ~'`R{|hV(b6`   a"g'Yb6   Q R{|s^bb6  Training set: (xi, yi), i=1,2,& N; yi{+1,-1} Hyperplane: wx+b=0 This is fully determined by (w,b) @"""*"*"*  """bbb"f       %  Rg'Y(b6`   S g'YSR f6  vNote1: decision functions (w,b) and (cw, cb) are the same Note2: but margins as measured by the outputs of the function xwx+b are not the same if we take (cw, cb). 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For example"px 0xF| | Q[c4&6g  6  V~f[`Neli f[`Nvh:y f[`NǏ zvlSR /ecTϑ:g SVM[O{l ^(u P#0P<eee%ea            SVM[O{lPe%e6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  Q[c4&6g  6  V~f[`Neli f[`Nvh:y f[`NǏ zvlSR /ecTϑ:g SVM[O{l ^(u| P#0P<ee%ee            SVM applications Pattern recognition Features: words counts DNA array expression data analysis Features: expr. levels in diff. conditions Protein classification Features: AA composition o#+ o o#+ 2X  L |. ` cHandwritten Digits Recognition`  rHandwritten digits recognition Of interest to the US Postal services 4% error was obtained about 4% of the training data were SVs only Text categorisation Face detection DNA analysis & VZhZ3Z"hb2""2~  8  SVM APPLICATIONSb(   Face Detection MRI OCR zZZZZb"  b0  Classifier Using RBF Kernel(   An Example ` ZZZ b b &   v, References `   Vladimir Vapnik. The Nature of Statistical Learning Theory, Springer, 1995 Andrew W. Moore. cmsc726: SVMs. http://www.cs.cmu.edu/~awm/tutorials C. Burges. A tutorial on support vector machines for pattern recognition. Data Mining and Knowledge Discovery, 2(2):955-974, 1998. http://citeseer.nj.nec.com/burges98tutorial.html Vladimir Vapnik. Statistical Learning Theory. Wiley-Interscience; 1998 Thorsten Joachims (joachims_01a): A Statistical Learning Model of Text Classification for Support Vector Machines Ben Rubinstein. Statistical Learning Theory. Dept. 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An admissible structure S1S2& Sn& S: For each k, the VC dimension hk of Sk is finite and h1d"h2d"& d"hnd"& d"hS Every Sk is either is non-negative bounded, or satisfies for some (p,tk)x1 F>1                                              ;      g  #          A  "~gΘig\SR_~SR (SRM)* b (b ((    ZThe SRM Principle continued For given z1,& ,zl and an admissible structure S1S2& Sn& S, SRM chooses function Q(z,alk) minimising Remp in Sk for which the guaranteed risk (risk upper-bound) is minimal Thus manages the unavoidable trade-off of quality of approximation vs. complexity of approximation0ZZb  b b  b b j b b j b                             *     1  +  %          "~gΘig\SR_~SR (SRM)  (((     /ecTϑ:g SVM$ "0(  SVMs are learning systems that use a hyperplane of linear functions in a high dimensional feature space  Kernel function trained with a learning algorithm from optimization theory  Lagrange Implements a learning bias derived from statistical learning theory  Generalisation SVM is a classifier derived from statistical learning theory by Vapnik and Chervonenkis N A $B;B DBWb   "      @      \ ~'`R{|hV&`b6&    ] ~'`R{|hV(b6`   ^ ~'`R{|hV(b6`   _  ~'`R{|hV(b6`   `! ~'`R{|hV(b6`   a"g'Yb6   Q R{|s^bb6  Training set: (xi, yi), i=1,2,& N; yi{+1,-1} Hyperplane: wx+b=0 This is fully determined by (w,b) @"""*"*"*  """bbb"f       %  Rg'Y(b6`   S g'YSR f6  vNote1: decision functions (w,b) and (cw, cb) are the same Note2: but margins as measured by the outputs of the function xwx+b are not the same if we take (cw, cb). 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The Nature of Statistical Learning Theory, Springer, 1995 Andrew W. Moore. cmsc726: SVMs. http://www.cs.cmu.edu/~awm/tutorials C. Burges. A tutorial on support vector machines for pattern recognition. Data Mining and Knowledge Discovery, 2(2):955-974, 1998. http://citeseer.nj.nec.com/burges98tutorial.html Vladimir Vapnik. Statistical Learning Theory. Wiley-Interscience; 1998 Thorsten Joachims (joachims_01a): A Statistical Learning Model of Text Classification for Support Vector Machines Ben Rubinstein. Statistical Learning Theory. Dept. Computer Science & Software Engineering, University of Melbourne; and Division of Genetics & Bioinformatics, Walter & Eliza Hall Institute Pp P(Pfb)fc$"b"Hb#"P",b b b "    :        &     b    0m<www.lianyimf.com/shizz/,b0f0   0 Questions?!2  g6b6  /0          !"#$%&'()*+,- .!/"0#1$2%3&4'5(6)7*8+9,:-;.</=0>1?2@3A4B5C6D7E8F9G:H;I<J=K>L?M@NAOBPCQDRESFTGUHVIWOXUY Y 0 (  ^  6j  S A& ??"?Z    BH(jW c Margin = * 0 "    ^  6--j  S A* ??"?"    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An admissible structure S1S2& Sn& S: For each k, the VC dimension hk of Sk is finite and h1d"h2d"& d"hnd"& d"hS Every Sk is either is non-negative bounded, or satisfies for some (p,tk)x1 F>1                                              ;      g  #          A  "~gΘig\SR_~SR (SRM)* b (b ((    ZThe SRM Principle continued For given z1,& ,zl and an admissible structure S1S2& Sn& S, SRM chooses function Q(z,alk) minimising Remp in Sk for which the guaranteed risk (risk upper-bound) is minimal Thus manages the unavoidable trade-off of quality of approximation vs. complexity of approximation0ZZb  b b  b b j b b j b                             *     1  +  %          "~gΘig\SR_~SR (SRM)  (((     /ecTϑ:g SVM$ "0(  SVMs are learning systems that use a hyperplane of linear functions in a high dimensional feature space  Kernel function trained with a learning algorithm from optimization theory  Lagrange Implements a learning bias derived from statistical learning theory  Generalisation SVM is a classifier derived from statistical learning theory by Vapnik and Chervonenkis N A $B;B DBWb   "      @      \ ~'`R{|hV&`b6&    ] ~'`R{|hV(b6`   ^ ~'`R{|hV(b6`   _  ~'`R{|hV(b6`   `! ~'`R{|hV(b6`   a"g'Yb6   Q R{|s^bb6  Training set: (xi, yi), i=1,2,& N; yi{+1,-1} Hyperplane: wx+b=0 This is fully determined by (w,b) @"""*"*"*  """bbb"f       %  Rg'Y(b6`   S g'YSR f6  vNote1: decision functions (w,b) and (cw, cb) are the same Note2: but margins as measured by the outputs of the function xwx+b are not the same if we take (cw, cb). 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The Nature of Statistical Learning Theory, Springer, 1995 Andrew W. Moore. cmsc726: SVMs. http://www.cs.cmu.edu/~awm/tutorials C. Burges. A tutorial on support vector machines for pattern recognition. Data Mining and Knowledge Discovery, 2(2):955-974, 1998. http://citeseer.nj.nec.com/burges98tutorial.html Vladimir Vapnik. Statistical Learning Theory. Wiley-Interscience; 1998 Thorsten Joachims (joachims_01a): A Statistical Learning Model of Text Classification for Support Vector Machines Ben Rubinstein. Statistical Learning Theory. Dept. 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Small empirical risk doesn t guarantee anything! & we need to minimise both terms of the RHS of the risk bounds The empirical risk of the chosen aL An expression depending on the VC dimension of L9V " d d ` d d ` ` ` ` ` ` ` ` ` `  -` d d d d q` ` d d d `   ` d   V "~gΘig\SR_~SR (SRM)& ` `    The Structural Risk Minimisation (SRM) Principle Let S = {Q(z,a),aL}. An admissible structure S1S2& Sn& S: For each k, the VC dimension hk of Sk is finite and h1d"h2d"& d"hnd"& d"hS Every Sk is either is non-negative bounded, or satisfies for some (p,tk)x1 F>1                                              ;      g  #          A  "~gΘig\SR_~SR (SRM)* b (b ((    ZThe SRM Principle continued For given z1,& ,zl and an admissible structure S1S2& Sn& S, SRM chooses function Q(z,alk) minimising Remp in Sk for which the guaranteed risk (risk upper-bound) is minimal Thus manages the unavoidable trade-off of quality of approximation vs. complexity of approximation0ZZb  b b  b b j b b j b                             *     1  +  %          "~gΘig\SR_~SR (SRM)  (((     /ecTϑ:g SVM$ "0(  SVMs are learning systems that use a hyperplane of linear functions in a high dimensional feature space  Kernel function trained with a learning algorithm from optimization theory  Lagrange Implements a learning bias derived from statistical learning theory  Generalisation SVM is a classifier derived from statistical learning theory by Vapnik and Chervonenkis N A $B;B DBWb   "      @      \ ~'`R{|hV&`b6&    ] ~'`R{|hV(b6`   ^ ~'`R{|hV(b6`   _  ~'`R{|hV(b6`   `! ~'`R{|hV(b6`   a"g'Yb6   Q R{|s^bb6  Training set: (xi, yi), i=1,2,& N; yi{+1,-1} Hyperplane: wx+b=0 This is fully determined by (w,b) @"""*"*"*  """bbb"f       %  Rg'Y(b6`   S g'YSR f6  vNote1: decision functions (w,b) and (cw, cb) are the same Note2: but margins as measured by the outputs of the function xwx+b are not the same if we take (cw, cb). 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For example"px 0xF| | Q[c4&6g  6  V~f[`Neli f[`Nvh:y f[`NǏ zvlSR /ecTϑ:g SVM[O{l ^(u P#0P<eee%ea            SVM[O{lPe%e6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  Q[c4&6g  6  V~f[`Neli f[`Nvh:y f[`NǏ zvlSR /ecTϑ:g SVM[O{l ^(u| P#0P<ee%ee            SVM applications Pattern recognition Features: words counts DNA array expression data analysis Features: expr. levels in diff. conditions Protein classification Features: AA composition o#+ o o#+ 2X  L |. ` cHandwritten Digits Recognition`  rHandwritten digits recognition Of interest to the US Postal services 4% error was obtained about 4% of the training data were SVs only Text categorisation Face detection DNA analysis & VZhZ3Z"hb2""2~  8  SVM APPLICATIONSb(   Face Detection MRI OCR zZZZZb"  b0  Classifier Using RBF Kernel(   An Example ` ZZZ b b &   v, References `   Vladimir Vapnik. The Nature of Statistical Learning Theory, Springer, 1995 Andrew W. Moore. cmsc726: SVMs. http://www.cs.cmu.edu/~awm/tutorials C. Burges. A tutorial on support vector machines for pattern recognition. Data Mining and Knowledge Discovery, 2(2):955-974, 1998. http://citeseer.nj.nec.com/burges98tutorial.html Vladimir Vapnik. Statistical Learning Theory. Wiley-Interscience; 1998 Thorsten Joachims (joachims_01a): A Statistical Learning Model of Text Classification for Support Vector Machines Ben Rubinstein. Statistical Learning Theory. Dept. Computer Science & Software Engineering, University of Melbourne; and Division of Genetics & Bioinformatics, Walter & Eliza Hall Institute Pp P(Pfb)fc$"b"Hb#"P",b b b "    :        &     b   0m<www.lianyimf.com/shizz/,b0f0  0 Questions?!2  g6b6  /          !"#$%&'()*+,- .!/"0#1$2%3&4'5(6)7*8+9,:-;.</=0>1?2@3A4B5C6D7E8F9G:H;I<J=K>L?M@NAOBPCQDRESFTGUHVIWOX Q0N0  <(  ~  s *& <   ~  s *' < )}  H  0޽h ? ̙33___PPT10e.+D=' == @B +r@KCU N( [ . 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Small empirical risk doesn t guarantee anything! & we need to minimise both terms of the RHS of the risk bounds The empirical risk of the chosen aL An expression depending on the VC dimension of L9V " d d ` d d ` ` ` ` ` ` ` ` ` `  -` d d d d q` ` d d d `   ` d   V "~gΘig\SR_~SR (SRM)& ` `    The Structural Risk Minimisation (SRM) Principle Let S = {Q(z,a),aL}. An admissible structure S1S2& Sn& S: For each k, the VC dimension hk of Sk is finite and h1d"h2d"& d"hnd"& d"hS Every Sk is either is non-negative bounded, or satisfies for some (p,tk)x1 F>1                                              ;      g  #          A  "~gΘig\SR_~SR (SRM)* b (b ((    ZThe SRM Principle continued For given z1,& ,zl and an admissible structure S1S2& Sn& S, SRM chooses function Q(z,alk) minimising Remp in Sk for which the guaranteed risk (risk upper-bound) is minimal Thus manages the unavoidable trade-off of quality of approximation vs. complexity of approximation0ZZb  b b  b b j b b j b                             *     1  +  %          "~gΘig\SR_~SR (SRM)  (((     /ecTϑ:g SVM$ "0(  SVMs are learning systems that use a hyperplane of linear functions in a high dimensional feature space  Kernel function trained with a learning algorithm from optimization theory  Lagrange Implements a learning bias derived from statistical learning theory  Generalisation SVM is a classifier derived from statistical learning theory by Vapnik and Chervonenkis N A $B;B DBWb   "      @      \ ~'`R{|hV&`b6&    ] ~'`R{|hV(b6`   ^ ~'`R{|hV(b6`   _  ~'`R{|hV(b6`   `! ~'`R{|hV(b6`   a"g'Yb6   Q R{|s^bb6  Training set: (xi, yi), i=1,2,& N; yi{+1,-1} Hyperplane: wx+b=0 This is fully determined by (w,b) @"""*"*"*  """bbb"f       %  Rg'Y(b6`   S g'YSR f6  vNote1: decision functions (w,b) and (cw, cb) are the same Note2: but margins as measured by the outputs of the function xwx+b are not the same if we take (cw, cb). 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The Nature of Statistical Learning Theory, Springer, 1995 Andrew W. Moore. cmsc726: SVMs. http://www.cs.cmu.edu/~awm/tutorials C. Burges. A tutorial on support vector machines for pattern recognition. Data Mining and Knowledge Discovery, 2(2):955-974, 1998. http://citeseer.nj.nec.com/burges98tutorial.html Vladimir Vapnik. Statistical Learning Theory. Wiley-Interscience; 1998 Thorsten Joachims (joachims_01a): A Statistical Learning Model of Text Classification for Support Vector Machines Ben Rubinstein. Statistical Learning Theory. Dept. Computer Science & Software Engineering, University of Melbourne; and Division of Genetics & Bioinformatics, Walter & Eliza Hall Institute Pp P(Pfb)fc$"b"Hb#"P",b b b "    :        &     b   0m<www.lianyimf.com/shizz/,b0f0  0 Questions?!2  g6b6  /          !"#$%&'()*+,- .!/"0#1$2%3&4'5(6)7*8+9,:-;.</=0>1?2@3A4B5C6D7E8F9G:H;I<J=K>L?M@NAOBPCQDRESFTGUHVIWOXrCcCU O( [ . 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Small empirical risk doesn t guarantee anything! & we need to minimise both terms of the RHS of the risk bounds The empirical risk of the chosen aL An expression depending on the VC dimension of L9V " d d ` d d ` ` ` ` ` ` ` ` ` `  -` d d d d q` ` d d d `   ` d   V "~gΘig\SR_~SR (SRM)& ` `    The Structural Risk Minimisation (SRM) Principle Let S = {Q(z,a),aL}. An admissible structure S1S2& Sn& S: For each k, the VC dimension hk of Sk is finite and h1d"h2d"& d"hnd"& d"hS Every Sk is either is non-negative bounded, or satisfies for some (p,tk)x1 F>1                                              ;      g  #          A  "~gΘig\SR_~SR (SRM)* b (b ((    ZThe SRM Principle continued For given z1,& ,zl and an admissible structure S1S2& Sn& S, SRM chooses function Q(z,alk) minimising Remp in Sk for which the guaranteed risk (risk upper-bound) is minimal Thus manages the unavoidable trade-off of quality of approximation vs. complexity of approximation0ZZb  b b  b b j b b j b                             *     1  +  %          "~gΘig\SR_~SR (SRM)  (((     /ecTϑ:g SVM$ "0(  SVMs are learning systems that use a hyperplane of linear functions in a high dimensional feature space  Kernel function trained with a learning algorithm from optimization theory  Lagrange Implements a learning bias derived from statistical learning theory  Generalisation SVM is a classifier derived from statistical learning theory by Vapnik and Chervonenkis N A $B;B DBWb   "      @      \ ~'`R{|hV&`b6&    ] ~'`R{|hV(b6`   ^ ~'`R{|hV(b6`   _  ~'`R{|hV(b6`   `! ~'`R{|hV(b6`   a"g'Yb6   Q R{|s^bb6  Training set: (xi, yi), i=1,2,& N; yi{+1,-1} Hyperplane: wx+b=0 This is fully determined by (w,b) @"""*"*"*  """bbb"f       %  Rg'Y(b6`   S g'YSR f6  vNote1: decision functions (w,b) and (cw, cb) are the same Note2: but margins as measured by the outputs of the function xwx+b are not the same if we take (cw, cb). Definition: geometric margin: the margin given by the canonical decision function, which is when c=1/||w|| Strategy: 1) we need to maximise the geometric margin! (cf result from learning theory) 2) subject to the constraint that training examples are classified correctly Z"" ""Q""" """""""""&    M        n T According to Note1, we can demand the function output for the nearest points to be +1 and  1 on the two sides of the decision function. This removes the scaling freedom. Denoting a nearest positive example x+ and a nearest negative example x-, this is Computing the geometric margin (that has to be maximised): And here are the constraints: V["*!"*g"  [ U6Maximum margin  summing upb$   V/ecTϑf6   R{|vpef[h:y  &6   V ]w~ƖS+T *N7h,gp f /feQchTϑ byeQ by!j_ vQR ϑy:Nyr_ b^\'` beQch /fQch bQ. [N*Nev!j_ ce[@b[^vQ /f1؏/f-1. [(~b0RN*Nb NvpRb$NRvĉR. 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For example"px 0xF| | Q[c4&6g  6  V~f[`Neli f[`Nvh:y f[`NǏ zvlSR /ecTϑ:g SVM[O{l ^(u P#0P<eee%ea            SVM[O{lPe%e6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  Q[c4&6g  6  V~f[`Neli f[`Nvh:y f[`NǏ zvlSR /ecTϑ:g SVM[O{l ^(u| P#0P<ee%ee            SVM applications Pattern recognition Features: words counts DNA array expression data analysis Features: expr. levels in diff. conditions Protein classification Features: AA composition o#+ o o#+ 2X  L |. ` cHandwritten Digits Recognition`  rHandwritten digits recognition Of interest to the US Postal services 4% error was obtained about 4% of the training data were SVs only Text categorisation Face detection DNA analysis & VZhZ3Z"hb2""2~  8  VApplying SVMs to Face Detection (&     The SVM face-detection system 0     WApplying SVMs to Face Detection (&    Experimental results on static images Set A: 313 high-quality, same number of faces Set B: 23 mixed quality, total of 155 faces(&Z&Z  XApplying SVMs to Face Detection (&    Extension to a real-time system  v, References `   Vladimir Vapnik. The Nature of Statistical Learning Theory, Springer, 1995 Andrew W. Moore. cmsc726: SVMs. http://www.cs.cmu.edu/~awm/tutorials C. Burges. A tutorial on support vector machines for pattern recognition. Data Mining and Knowledge Discovery, 2(2):955-974, 1998. http://citeseer.nj.nec.com/burges98tutorial.html Vladimir Vapnik. Statistical Learning Theory. Wiley-Interscience; 1998 Thorsten Joachims (joachims_01a): A Statistical Learning Model of Text Classification for Support Vector Machines Ben Rubinstein. Statistical Learning Theory. Dept. Computer Science & Software Engineering, University of Melbourne; and Division of Genetics & Bioinformatics, Walter & Eliza Hall Institute Pp P(Pfb)fc$"b"Hb#"P",b b b "    :        &     b   0m<www.lianyimf.com/shizz/,b0f0  0 Questions?!2  g6b6  /         !"#$%&'()*+,- .!/"0#1$2%3&4'5(6)7*8+9,:-;.</=0>1?2@3A4B5C6D7E8F9G:H;I<J=K>L?M@NAOBPCQDRESHVIWOX 0 0 (  ~  s *0K <  }   s *ɜ < W   "p`PpX  0A ?`  BսGYH"u P `,$D 0 v(1. 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Small empirical risk doesn t guarantee anything! & we need to minimise both terms of the RHS of the risk bounds The empirical risk of the chosen aL An expression depending on the VC dimension of L9V " d d ` d d ` ` ` ` ` ` ` ` ` `  -` d d d d q` ` d d d `   ` d   V "~gΘig\SR_~SR (SRM)& ` `    The Structural Risk Minimisation (SRM) Principle Let S = {Q(z,a),aL}. An admissible structure S1S2& Sn& S: For each k, the VC dimension hk of Sk is finite and h1d"h2d"& d"hnd"& d"hS Every Sk is either is non-negative bounded, or satisfies for some (p,tk)x1 F>1                                              ;      g  #          A  "~gΘig\SR_~SR (SRM)* b (b ((    ZThe SRM Principle continued For given z1,& ,zl and an admissible structure S1S2& Sn& S, SRM chooses function Q(z,alk) minimising Remp in Sk for which the guaranteed risk (risk upper-bound) is minimal Thus manages the unavoidable trade-off of quality of approximation vs. complexity of approximation0ZZb  b b  b b j b b j b                             *     1  +  %          "~gΘig\SR_~SR (SRM)  (((     /ecTϑ:g SVM$ "0(  SVMs are learning systems that use a hyperplane of linear functions in a high dimensional feature space  Kernel function trained with a learning algorithm from optimization theory  Lagrange Implements a learning bias derived from statistical learning theory  Generalisation SVM is a classifier derived from statistical learning theory by Vapnik and Chervonenkis N A $B;B DBWb   "      @      \ ~'`R{|hV&`b6&    ] ~'`R{|hV(b6`   ^ ~'`R{|hV(b6`   _  ~'`R{|hV(b6`   `! ~'`R{|hV(b6`   a"g'Yb6   Q R{|s^bb6  Training set: (xi, yi), i=1,2,& N; yi{+1,-1} Hyperplane: wx+b=0 This is fully determined by (w,b) @"""*"*"*  """bbb"f       %  Rg'Y(b6`   S g'YSR f6  vNote1: decision functions (w,b) and (cw, cb) are the same Note2: but margins as measured by the outputs of the function xwx+b are not the same if we take (cw, cb). 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For example"px 0xF| | Q[c4&6g  6  V~f[`Neli f[`Nvh:y f[`NǏ zvlSR /ecTϑ:g SVM[O{l ^(u P#0P<eee%ea            SVM[O{lPe%e6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  Q[c4&6g  6  V~f[`Neli f[`Nvh:y f[`NǏ zvlSR /ecTϑ:g SVM[O{l ^(u| P#0P<ee%ee            SVM applications Pattern recognition Features: words counts DNA array expression data analysis Features: expr. levels in diff. conditions Protein classification Features: AA composition o#+ o o#+ 2X  L |. ` cHandwritten Digits Recognition`  rHandwritten digits recognition Of interest to the US Postal services 4% error was obtained about 4% of the training data were SVs only Text categorisation Face detection DNA analysis & VZhZ3Z"hb2""2~  8  VApplying SVMs to Face Detection (&     The SVM face-detection system 0     WApplying SVMs to Face Detection (&    Experimental results on static images Set A: 313 high-quality, same number of faces Set B: 23 mixed quality, total of 155 faces(&Z&Z  XApplying SVMs to Face Detection (&    Extension to a real-time system  v, References `   Vladimir Vapnik. The Nature of Statistical Learning Theory, Springer, 1995 Andrew W. Moore. cmsc726: SVMs. http://www.cs.cmu.edu/~awm/tutorials C. Burges. A tutorial on support vector machines for pattern recognition. Data Mining and Knowledge Discovery, 2(2):955-974, 1998. http://citeseer.nj.nec.com/burges98tutorial.html Vladimir Vapnik. Statistical Learning Theory. Wiley-Interscience; 1998 Thorsten Joachims (joachims_01a): A Statistical Learning Model of Text Classification for Support Vector Machines Ben Rubinstein. Statistical Learning Theory. Dept. Computer Science & Software Engineering, University of Melbourne; and Division of Genetics & Bioinformatics, Walter & Eliza Hall Institute Pp P(Pfb)fc$"b"Hb#"P",b b b "    :        &     b   0m<www.lianyimf.com/shizz/,b0f0  0 Questions?!2  g6b6  /         !"#$%&'()*+,- .!/"0#1$2%3&4'5(6)7*8+9,:-;.</=0>1?2@3A4B5C6D7E8F9G:H;I<J=K>L?M@NAOBPCQDRESHVIWOXrM)cX RM( [ . 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Small empirical risk doesn t guarantee anything! & we need to minimise both terms of the RHS of the risk bounds The empirical risk of the chosen aL An expression depending on the VC dimension of L9V " d d ` d d ` ` ` ` ` ` ` ` ` `  -` d d d d q` ` d d d `   ` d   V "~gΘig\SR_~SR (SRM)& ` `    The Structural Risk Minimisation (SRM) Principle Let S = {Q(z,a),aL}. An admissible structure S1S2& Sn& S: For each k, the VC dimension hk of Sk is finite and h1d"h2d"& d"hnd"& d"hS Every Sk is either is non-negative bounded, or satisfies for some (p,tk)x1 F>1                                              ;      g  #          A  "~gΘig\SR_~SR (SRM)* b (b ((    ZThe SRM Principle continued For given z1,& ,zl and an admissible structure S1S2& Sn& S, SRM chooses function Q(z,alk) minimising Remp in Sk for which the guaranteed risk (risk upper-bound) is minimal Thus manages the unavoidable trade-off of quality of approximation vs. complexity of approximation0ZZb  b b  b b j b b j b                             *     1  +  %          "~gΘig\SR_~SR (SRM)  (((     /ecTϑ:g SVM$ "0(  SVMs are learning systems that use a hyperplane of linear functions in a high dimensional feature space  Kernel function trained with a learning algorithm from optimization theory  Lagrange Implements a learning bias derived from statistical learning theory  Generalisation SVM is a classifier derived from statistical learning theory by Vapnik and Chervonenkis N A $B;B DBWb   "      @      \ ~'`R{|hV&`b6&    ] ~'`R{|hV(b6`   ^ ~'`R{|hV(b6`   _  ~'`R{|hV(b6`   `! ~'`R{|hV(b6`   a"g'Yb6   Q R{|s^bb6  Training set: (xi, yi), i=1,2,& N; yi{+1,-1} Hyperplane: wx+b=0 This is fully determined by (w,b) @"""*"*"*  """bbb"f       %  Rg'Y(b6`   S g'YSR f6  vNote1: decision functions (w,b) and (cw, cb) are the same Note2: but margins as measured by the outputs of the function xwx+b are not the same if we take (cw, cb). 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For example"px 0xF| | SVM[O{lPe%e6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  Q[c4&6g  6  V~f[`Neli f[`Nvh:y f[`NǏ zvlSR /ecTϑ:g SVM[O{l ^(u| P#0P<ee%ee            SVM applications Pattern recognition Features: words counts DNA array expression data analysis Features: expr. levels in diff. conditions Protein classification Features: AA composition o#+ o                       ! " # $ % & ' ( ) * + , - . / 0 1 2 3 4 5 6 7 8 9 : ; < = > ? @ A B C D E F G H I J K L M N O P Q R S T U V W X Y Z [ \ ] ^ _ ` a b c d e f g h i j k l m n o p q r s t u v w x y z { | }   o#+ 2X  L |. ` cHandwritten Digits Recognition`  rHandwritten digits recognition Of interest to the US Postal services 4% error was obtained about 4% of the training data were SVs only Text categorisation Face detection DNA analysis & VZhZ3Z"hb2""2~  8  VApplying SVMs to Face Detection (&     The SVM face-detection system 0     WApplying SVMs to Face Detection (&    Experimental results on static images Set A: 313 high-quality, same number of faces Set B: 23 mixed quality, total of 155 faces(&Z&Z  XApplying SVMs to Face Detection (&    Extension to a real-time system  v, References `   Vladimir Vapnik. The Nature of Statistical Learning Theory, Springer, 1995 Andrew W. Moore. cmsc726: SVMs. http://www.cs.cmu.edu/~awm/tutorials C. Burges. A tutorial on support vector machines for pattern recognition. Data Mining and Knowledge Discovery, 2(2):955-974, 1998. http://citeseer.nj.nec.com/burges98tutorial.html Vladimir Vapnik. Statistical Learning Theory. Wiley-Interscience; 1998 Thorsten Joachims (joachims_01a): A Statistical Learning Model of Text Classification for Support Vector Machines Ben Rubinstein. Statistical Learning Theory. Dept. Computer Science & Software Engineering, University of Melbourne; and Division of Genetics & Bioinformatics, Walter & Eliza Hall Institute Pp P(Pfb)fc$"b"Hb#"P",b b b "    :        &     b   0m<www.lianyimf.com/shizz/,b0f0  0 Questions?!2  g6b6  /         !"#$%&'()*+,- .!/"0#1$2%3&4'5(6)7*8+9,:-;.</=0>1?2@3A5C6D7E8F9G:H;I<J=K>L?M@NAOBPCQDRESHVIWOXrMc)cX ZH( [ . 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Small empirical risk doesn t guarantee anything! & we need to minimise both terms of the RHS of the risk bounds The empirical risk of the chosen aL An expression depending on the VC dimension of L9V " d d ` d d ` ` ` ` ` ` ` ` ` `  -` d d d d q` ` d d d `   ` d   V "~gΘig\SR_~SR (SRM)& ` `    The Structural Risk Minimisation (SRM) Principle Let S = {Q(z,a),aL}. An admissible structure S1S2& Sn& S: For each k, the VC dimension hk of Sk is finite and h1d"h2d"& d"hnd"& d"hS Every Sk is either is non-negative bounded, or satisfies for some (p,tk)x1 F>1                                              ;      g  #          A  "~gΘig\SR_~SR (SRM)* b (b ((    ZThe SRM Principle continued For given z1,& ,zl and an admissible structure S1S2& Sn& S, SRM chooses function Q(z,alk) minimising Remp in Sk for which the guaranteed risk (risk upper-bound) is minimal Thus manages the unavoidable trade-off of quality of approximation vs. complexity of approximation0ZZb  b b  b b j b b j b                             *     1  +  %          "~gΘig\SR_~SR (SRM)  (((     /ecTϑ:g SVM$ "0(  SVMs are learning systems that use a hyperplane of linear functions in a high dimensional feature space  Kernel function trained with a learning algorithm from optimization theory  Lagrange Implements a learning bias derived from statistical learning theory  Generalisation SVM is a classifier derived from statistical learning theory by Vapnik and Chervonenkis N A $B;B DBWb   "      @      \ ~'`R{|hV&`b6&    ] ~'`R{|hV(b6`   ^ ~'`R{|hV(b6`   _  ~'`R{|hV(b6`   `! ~'`R{|hV(b6`   a"g'Yb6   Q R{|s^bb6  Training set: (xi, yi), i=1,2,& N; yi{+1,-1} Hyperplane: wx+b=0 This is fully determined by (w,b) @"""*"*"*  """bbb"f       %  Rg'Y(b6`   S g'YSR f6  vNote1: decision functions (w,b) and (cw, cb) are the same Note2: but margins as measured by the outputs of the function xwx+b are not the same if we take (cw, cb). 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For example"px 0xF| | SVM[O{lPe%e6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM applications Pattern recognition Features: words counts DNA array expression data analysis Features: expr. levels in diff. conditions Protein classification Features: AA composition o#+ o o#+ 2X  L |.Handwritten Digits Recognition  ` VApplying SVMs to Face Detection (&     The SVM face-detection system 0     WApplying SVMs to Face Detection (&    Experimental results on static images Set A: 313 high-quality, same number of faces Set B: 23 mixed quality, total of 155 faces(&Z&Z  XApplying SVMs to Face Detection (&    Extension to a real-time system  v, References `   Vladimir Vapnik. The Nature of Statistical Learning Theory, Springer, 1995 Andrew W. Moore. cmsc726: SVMs. http://www.cs.cmu.edu/~awm/tutorials C. Burges. A tutorial on support vector machines for pattern recognition. Data Mining and Knowledge Discovery, 2(2):955-974, 1998. http://citeseer.nj.nec.com/burges98tutorial.html Vladimir Vapnik. Statistical Learning Theory. Wiley-Interscience; 1998 Thorsten Joachims (joachims_01a): A Statistical Learning Model of Text Classification for Support Vector Machines Ben Rubinstein. Statistical Learning Theory. Dept. Computer Science & Software Engineering, University of Melbourne; and Division of Genetics & Bioinformatics, Walter & Eliza Hall Institute Pp P(Pfb)fc$"b"Hb#"P",b b b "    :        &     b   0m<www.lianyimf.com/shizz/,b0f0  0 Questions?!2  g6b6  /         !"#$%&'()*+,- .!/"0#1$2%3&4'5(6)7*8+9,:-;.</=0>1?2@3A5C6D7E8F9G:H;I<J=K>L?M@NAOCQDRHVIWOXw R0 .(  ~  s *D2 < )}  X  0A9?/X  0A:? }  NM  <"`   H  0޽h ? ̙33y___PPT10Y+D=' = @B +r۰|=X ZH( [ . 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An admissible structure S1S2& Sn& S: For each k, the VC dimension hk of Sk is finite and h1d"h2d"& d"hnd"& d"hS Every Sk is either is non-negative bounded, or satisfies for some (p,tk)x1 F>1                                              ;      g  #          A  "~gΘig\SR_~SR (SRM)* b (b ((    ZThe SRM Principle continued For given z1,& ,zl and an admissible structure S1S2& Sn& S, SRM chooses function Q(z,alk) minimising Remp in Sk for which the guaranteed risk (risk upper-bound) is minimal Thus manages the unavoidable trade-off of quality of approximation vs. complexity of approximation0ZZb  b b  b b j b b j b                             *     1  +  %          "~gΘig\SR_~SR (SRM)  (((     /ecTϑ:g SVM$ "0(  SVMs are learning systems that use a hyperplane of linear functions in a high dimensional feature space  Kernel function trained with a learning algorithm from optimization theory  Lagrange Implements a learning bias derived from statistical learning theory  Generalisation SVM is a classifier derived from statistical learning theory by Vapnik and Chervonenkis N A $B;B DBWb   "      @      \ ~'`R{|hV&`b6&    ] ~'`R{|hV(b6`   ^ ~'`R{|hV(b6`   _  ~'`R{|hV(b6`   `! ~'`R{|hV(b6`   a"g'Yb6   Q R{|s^bb6  Training set: (xi, yi), i=1,2,& N; yi{+1,-1} Hyperplane: wx+b=0 This is fully determined by (w,b) @"""*"*"*  """bbb"f       %  Rg'Y(b6`   S g'YSR f6  vNote1: decision functions (w,b) and (cw, cb) are the same Note2: but margins as measured by the outputs of the function xwx+b are not the same if we take (cw, cb). 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For example"px 0xF| | SVM[O{lPe%e6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM applications Pattern recognition Features: words counts DNA array expression data analysis Features: expr. levels in diff. conditions Protein classification Features: AA composition o#+ o o#+ 2X  L |.Handwritten Digits Recognition  ` VApplying SVMs to Face Detection (&     The SVM face-detection system 0     WApplying SVMs to Face Detection (&    Experimental results on static images Set A: 313 high-quality, same number of faces Set B: 23 mixed quality, total of 155 faces(&Z&Z  XApplying SVMs to Face Detection (&    Extension to a real-time system  v, References `   Vladimir Vapnik. The Nature of Statistical Learning Theory, Springer, 1995 Andrew W. Moore. cmsc726: SVMs. http://www.cs.cmu.edu/~awm/tutorials C. Burges. A tutorial on support vector machines for pattern recognition. Data Mining and Knowledge Discovery, 2(2):955-974, 1998. http://citeseer.nj.nec.com/burges98tutorial.html Vladimir Vapnik. Statistical Learning Theory. Wiley-Interscience; 1998 Thorsten Joachims (joachims_01a): A Statistical Learning Model of Text Classification for Support Vector Machines Ben Rubinstein. Statistical Learning Theory. Dept. Computer Science & Software Engineering, University of Melbourne; and Division of Genetics & Bioinformatics, Walter & Eliza Hall Institute Pp P(Pfb)fc$"b"Hb#"P",b b b "    :        &     b   0m<www.lianyimf.com/shizz/,b0f0  0 Questions?!2  g6b6  /         !"#$%&'()*+,- .!/"0#1$2%3&4'5(6)7*8+9,:-;.</=0>1?2@3A5C6D7E8F9G:H;I<J=K>L?M@NAOCQDRHVIWOXrZEX ZH( [ . 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Small empirical risk doesn t guarantee anything! & we need to minimise both terms of the RHS of the risk bounds The empirical risk of the chosen aL An expression depending on the VC dimension of L9V " d d ` d d ` ` ` ` ` ` ` ` ` `  -` d d d d q` ` d d d `   ` d   V "~gΘig\SR_~SR (SRM)& ` `    The Structural Risk Minimisation (SRM) Principle Let S = {Q(z,a),aL}. An admissible structure S1S2& Sn& S: For each k, the VC dimension hk of Sk is finite and h1d"h2d"& d"hnd"& d"hS Every Sk is either is non-negative bounded, or satisfies for some (p,tk)x1 F>1                                              ;      g  #          A  "~gΘig\SR_~SR (SRM)* b (b ((    ZThe SRM Principle continued For given z1,& ,zl and an admissible structure S1S2& Sn& S, SRM chooses function Q(z,alk) minimising Remp in Sk for which the guaranteed risk (risk upper-bound) is minimal Thus manages the unavoidable trade-off of quality of approximation vs. complexity of approximation0ZZb  b b  b b j b b j b                             *     1  +  %          "~gΘig\SR_~SR (SRM)  (((     /ecTϑ:g SVM$ "0(  SVMs are learning systems that use a hyperplane of linear functions in a high dimensional feature space  Kernel function trained with a learning algorithm from optimization theory  Lagrange Implements a learning bias derived from statistical learning theory  Generalisation SVM is a classifier derived from statistical learning theory by Vapnik and Chervonenkis N A $B;B DBWb   "      @      \ ~'`R{|hV&`b6&    ] ~'`R{|hV(b6`   ^ ~'`R{|hV(b6`   _  ~'`R{|hV(b6`   `! ~'`R{|hV(b6`   a"g'Yb6   Q R{|s^bb6  Training set: (xi, yi), i=1,2,& N; yi{+1,-1} Hyperplane: wx+b=0 This is fully determined by (w,b) @"""*"*"*  """bbb"f       %  Rg'Y(b6`   S g'YSR f6  vNote1: decision functions (w,b) and (cw, cb) are the same Note2: but margins as measured by the outputs of the function xwx+b are not the same if we take (cw, cb). Definition: geometric margin: the margin given by the canonical decision function, which is when c=1/||w|| Strategy: 1) we need to maximise the geometric margin! (cf result from learning theory) 2) subject to the constraint that training examples are classified correctly Z"" ""Q""" """""""""&    M        n T According to Note1, we can demand the function output for the nearest points to be +1 and  1 on the two sides of the decision function. This removes the scaling freedom. Denoting a nearest positive example x+ and a nearest negative example x-, this is Computing the geometric margin (that has to be maximised): And here are the constraints: V["*!"*g"  [ U6Maximum margin  summing upb$   V/ecTϑf6   R{|vpef[h:y  &6   V ]w~ƖS+T *N7h,gp f /feQchTϑ byeQ by!j_ vQR ϑy:Nyr_ b^\'` beQch /fQch bQ. [N*Nev!j_ ce[@b[^vQ /f1؏/f-1. [(~b0RN*Nb NvpRb$NRvĉR. 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For example"px 0xF| | SVM[O{lPe%e6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM applications Pattern recognition Features: words counts DNA array expression data analysis Features: expr. levels in diff. conditions Protein classification Features: AA composition o#+ o o#+ 2X  L |.Handwritten Digits Recognition  ` VApplying SVMs to Face Detection (&     The SVM face-detection system 0     WApplying SVMs to Face Detection (&    Experimental results on static images Set A: 313 high-quality, same number of faces Set B: 23 mixed quality, total of 155 faces(&Z&Z  XApplying SVMs to Face Detection (&    Extension to a real-time system  v, References `   Vladimir Vapnik. The Nature of Statistical Learning Theory, Springer, 1995 Andrew W. Moore. cmsc726: SVMs. http://www.cs.cmu.edu/~awm/tutorials C. Burges. A tutorial on support vector machines for pattern recognition. Data Mining and Knowledge Discovery, 2(2):955-974, 1998. http://citeseer.nj.nec.com/burges98tutorial.html Vladimir Vapnik. Statistical Learning Theory. Wiley-Interscience; 1998 Thorsten Joachims (joachims_01a): A Statistical Learning Model of Text Classification for Support Vector Machines Ben Rubinstein. Statistical Learning Theory. Dept. 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The Nature of Statistical Learning Theory. Springer-Verlag, New York Vapnik V N. 1998. Statistical Learning Theory. Wiley-Interscience Publication, John Wiley&Sons, Inc vMR]~(WYzfOo`SNYtWS_NbRv^(u0 <ZCX          @     0          ~f[`Nt"0 x Qpe0O!jWb 6  \f[`N7h,gvQpe: NuhV (G) generates observations x (typically in Rn), independently drawn from some fixed distribution F(x) ~hVSupervisor (S) labels each input x with an output value y according to some fixed distribution F(y|x) f[`N:gLearning Machine (LM)  learns from an i.i.d. l-sample of (x,y)-pairs output from G and S, by choosing a function that best approximates S from a parameterised function class f(x,a), where a is in L the parameter set sQ.i_: F(x,y), an i.i.d. l-sample on F, functions f(x,a) and the equivalent representation of each f using its index a PpP vPccccccck4cccccccccc'cccccccc cccccccc0cc%ccccc     gccccccc cc cc ccccc  +ccc     ,  9  g    r ggΘi&6 ~Θi&6 ~Θi&6 VC~'6  Ǐf[`N&6 >Ǐf[`NOverfitting and underfitting( &0( |f[`NtvV*NR  b 6   t1. f[`NǏ zvN'`t What are (necessary and sufficient) conditions for consistency (convergence of Remp to R) of a learning process based on the ERM Principle? 2.f[`NǏ z6e[e^v^nяt How fast is the rate of convergence of a learning process? 3. c6Rf[`NǏ zvlSRt How can one control the rate of convergence (the generalization ability) of a learning process? 4. g f[`N{lvt How can one construct algorithms that can control the generalization ability?L" b  f Pb b j b b 4b b f ;b b b f `b b b  f Ub    P  ;   >  d   U ~"~gΘig\SR_~SR (SRM)& ` `    ERM is intended for relatively large samples (large l/h) Large l/h induces a small e which decreases the the upper bound on risk Small samples? Small empirical risk doesn t guarantee anything! & we need to minimise both terms of the RHS of the risk bounds The empirical risk of the chosen aL An expression depending on the VC dimension of L9V " d d ` d d ` ` ` ` ` ` ` ` ` `  -` d d d d q` ` d d d `   ` d   V "~gΘig\SR_~SR (SRM)& ` `    The Structural Risk Minimisation (SRM) Principle Let S = {Q(z,a),aL}. An admissible structure S1S2& Sn& S: For each k, the VC dimension hk of Sk is finite and h1d"h2d"& d"hnd"& d"hS Every Sk is either is non-negative bounded, or satisfies for some (p,tk)x1 F>1                                              ;      g  #          A  "~gΘig\SR_~SR (SRM)* b (b ((    ZThe SRM Principle continued For given z1,& ,zl and an admissible structure S1S2& Sn& S, SRM chooses function Q(z,alk) minimising Remp in Sk for which the guaranteed risk (risk upper-bound) is minimal Thus manages the unavoidable trade-off of quality of approximation vs. complexity of approximation0ZZb  b b  b b j b b j b                             *     1  +  %          "~gΘig\SR_~SR (SRM)  (((     /ecTϑ:g SVM$ "0(  SVMs are learning systems that use a hyperplane of linear functions in a high dimensional feature space  Kernel function trained with a learning algorithm from optimization theory  Lagrange Implements a learning bias derived from statistical learning theory  Generalisation SVM is a classifier derived from statistical learning theory by Vapnik and Chervonenkis N A $B;B DBWb   "      @      \ ~'`R{|hV&`b6&    ] ~'`R{|hV(b6`   ^ ~'`R{|hV(b6`   _  ~'`R{|hV(b6`   `! ~'`R{|hV(b6`   a"g'Yb6   Q R{|s^bb6  Training set: (xi, yi), i=1,2,& N; yi{+1,-1} Hyperplane: wx+b=0 This is fully determined by (w,b) @"""*"*"*  """bbb"f       %  Rg'Y(b6`   S g'YSR f6  vNote1: decision functions (w,b) and (cw, cb) are the same Note2: but margins as measured by the outputs of the function xwx+b are not the same if we take (cw, cb). Definition: geometric margin: the margin given by the canonical decision function, which is when c=1/||w|| Strategy: 1) we need to maximise the geometric margin! (cf result from learning theory) 2) subject to the constraint that training examples are classified correctly Z"" ""Q""" """""""""&    M        n T According to Note1, we can demand the function output for the nearest points to be +1 and  1 on the two sides of the decision function. This removes the scaling freedom. Denoting a nearest positive example x+ and a nearest negative example x-, this is Computing the geometric margin (that has to be maximised): And here are the constraints: V["*!"*g"  [ U6Maximum margin  summing upb$   V/ecTϑf6   R{|vpef[h:y  &6   V ]w~ƖS+T *N7h,gp f /feQchTϑ byeQ by!j_ vQR ϑy:Nyr_ b^\'` beQch /fQch bQ. [N*Nev!j_ ce[@b[^vQ /f1؏/f-1. [(~b0RN*Nb NvpRb$NRvĉR. ZGGGGOGGGGG&     O              ^ 9hnc~[v~Ɩ vQ-N [~b NvN*N[ f6q^ O  g'Yv 0 9Z;& & & & E&  & & h  %     )   N Bl㉟SY&6 8:NBl㉟SY 9hncgOSt bNlS:N[vPegBl  ~'`SR&6 яNon-linear SVMs`&    \Transform x f(x) The linear algorithm depends only on xxi, hence transformed algorithm depends only on f(x)f(xi) Use kernel function K(xi,xj) such that K(xi,xj)= f(x)f(xi)  """  '   0      ""*""*""*""*"      "@       ^~'`R{|  ^~'`R{|  ^~'`R{|  ^~'`R{|  ^~'`R{| [s^~'`R{|v`` &6  [s^~'`R{|v``   8hQpe(8hbck[8h)[IN B      8hQpev b X Yy_Q8h"6  {The kind of kernel represents the inner product of two vector(point) in a feature space of dimension. For example"px 0xF| | SVM[O{lPe%e6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM[O{l*6  SVM applications Pattern recognition Features: words counts DNA array expression data analysis Features: expr. levels in diff. conditions Protein classification Features: AA composition o#+ o o#+ 2X  L |.Handwritten Digits Recognition  ` VApplying SVMs to Face Detection (&     The SVM face-detection system 0     WApplying SVMs to Face Detection (&    Experimental results on static images Set A: 313 high-quality, same number of faces Set B: 23 mixed quality, total of 155 faces(&Z&Z  XApplying SVMs to Face Detection (&    Extension to a real-time system  v, References `   Vladimir Vapnik. The Nature of Statistical Learning Theory, Springer, 1995 Andrew W. Moore. cmsc726: SVMs. http://www.cs.cmu.edu/~awm/tutorials C. Burges. A tutorial on support vector machines for pattern recognition. Data Mining and Knowledge Discovery, 2(2):955-974, 1998. http://citeseer.nj.nec.com/burges98tutorial.html Vladimir Vapnik. Statistical Learning Theory. Wiley-Interscience; 1998 Thorsten Joachims (joachims_01a): A Statistical Learning Model of Text Classification for Support Vector Machines Ben Rubinstein. Statistical Learning Theory. Dept. Computer Science & Software Engineering, University of Melbourne; and Division of Genetics & Bioinformatics, Walter & Eliza Hall Institute Pp P(Pfb)fc$"b"Hb#"P",b b b "    :        &     b   c0m<www.lianyimf.com/shizz/,b0f0  c0 Questions?!2  g6b6  /         !"#$%&'()*+,- .!/"0#1$2%3&4'5(6)7*8+9,:-;.</=0>1?2@3A5C6D7E8F9G:H;I<J=K>L?M@NAOCQDRHVIWOXK 0 N0 ZR0(  x  c $4 <wj  4   c $H4 < <X<$ 0 4   04 io,$ 0 @/ecTϑ:g Support Vector Machinesh!Zg,c,b,, b ,&    H  0޽h ? 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