Minimum Squared Error


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1 Minimum Squred Error
2 LDF: Minimum SquredError Procedures Ide: conver o esier nd eer undersood prolem Percepron y i > 0 for ll smples y i solve sysem of liner inequliies MSE procedure y i i for ll smples y i solve sysem of liner equions Choose posiive consns,,, n ry o find weigh vecor s.. y i i for ll smples y i If we cn find weigh vecor such h y i i for ll smples y i, hen is soluion ecuse i s re posiive consider ll he smples (no jus he misclssified ones)
3 LDF: MSE Mrgins g(y) 0 y i y k Since we wn y i i, we expec smple y i o e disnce i from he sepring hyperplne (normlized y ) Thus,,, n give relive expeced disnces or mrgins of smples from he hyperplne Should mke i smll if smple i is expeced o e ner sepring hyperplne, nd mke i lrger oherwise In he sence of ny ddiionl informion, here re good resons o se n
4 LDF: MSE Mrix Noion Need o solve n equions Inroduce mrix noion: y y n M n ( 0) ( ) ( d) y y L y ( ) ( ) ( d) 0 0 y y L y ( ) ( ) ( ) M M M M M M d M 0 d y n n y n L y n Y Thus need o solve liner sysem Y
5 LDF: Exc Soluion is Rre Thus need o solve liner sysem Y Y is n n y (d +) mrix Exc soluion cn e found only if Y is nonsingulr nd squre, in which cse he inverse Y  exiss Y  (numer of smples) (numer of feures + ) lmos never hppens in prcice in his cse, gurneed o find he sepring hyperplne y y
6 LDF: Approxime Soluion Typiclly Y is overdeermined, h is i hs more rows (exmples) hn columns (feures) If i hs more feures hn exmples, should reduce dimensionliy Y Need Y, u no exc soluion exiss for n overdeermined sysem of equion More equions hn unknowns Find n pproxime soluion, h is Y Noe h pproxime soluion does no necessrily give he sepring hyperplne in he seprle cse Bu hyperplne corresponding o my sill e good soluion, especilly if here is no sepring hyperplne
7 LDF: MSE Crierion Funcion Minimum squred error pproch: find which minimizes he lengh of he error vecor e e Y e Thus minimize he minimum squred error crierion funcion: n ( ) Y y J s ( ) i i Unlike he percepron crierion funcion, we cn opimize he minimum squred error crierion funcion nlyiclly y seing he grdien o 0 i Y
8 LDF: Opimizing J s () J s ( ) Le s compue he grdien: J s n Y ( ) y i i i J ( ) M 0 Y ( Y ) s J Seing he grdien o 0: Y s d ( ) Y 0 Y Y Y
9 LDF: Pseudo Inverse Soluion Mrix Y Y is squre (i hs d + rows nd columns) nd i is ofen nonsingulr If Y Y is nonsingulr, is inverse exiss nd we cn solve for uniquely: ( ) Y Y Y pseudo inverse of Y ( ) ) Y Y Y Y ( Y Y) ( Y Y) I
10 LDF: Minimum SquredError Procedures If n, MSE procedure is equivlen o finding hyperplne of es fi hrough he smples y,,y n J ( ) Y s n n M n Then we shif his line o he origin, if his line ws good fi, ll smples will e clssified correcly
11 LDF: Minimum SquredError Procedures Only gurneed he sepring hyperplne if Y > 0 y h is if ll elemens of vecor Y M re posiive y n We hve Y + ε n + ε n Th is Y M where ε my e negive If ε,, ε n re smll relive o,, n, hen ech elemen of Y is posiive, nd gives sepring hyperplne If pproximion is no good, ε i my e lrge nd negive, for some i, hus i + ε i will e negive nd is no sepring hyperplne Thus in linerly seprle cse, les squres soluion does no necessrily gives sepring hyperplne Bu i will give resonle hyperplne
12 LDF: Minimum SquredError Procedures We re free o choose. My e emped o mke lrge s wy o insure Y > 0 Does no work Le β e sclr, le s ry β insed of if * is les squres soluion o Y, hen for ny sclr β, les squres soluion o Y β is β* rg min Y β rg minβ Y( / β) ( / ) rg min Y β β * hus if for some i h elemen of Y is less hn 0, h is y i < 0, hen y i (β) < 0, Relive difference eween componens of mers, u no he size of ech individul componen
13 LDF: Exmple Clss : (6 9), (5 7) Clss : (5 9), (0 4) Se vecors y, y, y 3, y 4 y dding exr feure nd normlizing 6 y 9 5 y 7 5 y 3 9 y Mrix Y is hen Y
14 LDF: Exmple Choose In ml, Y\ solves he les squres prolem Noe is n pproximion o Y, since no exc soluion exiss Y This soluion does give sepring hyperplne since Y > 0
15 LDF: Exmple Clss : (6 9), (5 7) Clss : (5 9), (0 0) The ls smple is very fr compred o ohers from he sepring hyperplne y Mrix y 7 5 y 3 9 y Y
16 LDF: Exmple Choose In ml, Y\ solves he les squres prolem Noe is n pproximion o Y, since no exc soluion exiss Y This soluion does no give sepring hyperplne since y 3 < 0
17 LDF: Exmple MSE pys o much enion o isoled noisy exmples (such exmples re clled ouliers) oulier MSE soluion desired soluion No prolems wih convergence hough, nd soluion i gives rnges from resonle o good
18 LDF: Exmple we know h 4 h poin is fr fr from sepring hyperplne In prcice we don know his Thus pproprie 0 In Ml, solve Y\ Noe is n pproximion o Y, This soluion does give he sepring hyperplne since Y > 0 Y 0
19 LDF: Grdien Descen for MSE soluion J s ( ) Y My wish o find MSE soluion y grdien descen:. Compuing he inverse of Y Y my e oo cosly. Y Y my e close o singulr if smples re highly correled (rows of Y re lmos liner cominions of ech oher) compuing he inverse of Y Y is no numericlly sle In he eginning of he lecure, compued he grdien: ( ) J Y ( Y ) s
20 LDF: WidrowHoff Procedure Thus he upde rule for grdien descen: ( ) ( ) ( ) ( ) J Y ( Y ) ( k+ ) ( k) ( k) ( k) η Y k If η η / k weigh vecor (k) converges o he MSE Y soluion, h is Y (Y)0 s WidrowHoff procedure reduces sorge requiremens y considering single smples sequenilly: ( y ) ( k+ ) ( k) ( k) ( k) η y i i i
21 LDF: HoKshyp Procedure In he MSE procedure, if is chosen rirrily, finding sepring hyperplne is no gurneed Suppose rining smples re linerly seprle. Then here is s nd posiive s s.. Y s s > 0 If we knew s could pply MSE procedure o find he sepring hyperplne Ide: find oh s nd s Minimize he following crierion funcion, resricing o posiive : ( ), Y J HK
22 LDF: HoKshyp Procedure J HK ( ), Y As usul, ke pril derivives w.r.. nd J HK J HK ( Y ) 0 Y ( Y ) 0 Use modified grdien descen procedure o find minimum of J HK (,) Alerne he wo seps elow unil convergence: ) Fix nd minimize J HK (,) wih respec o ) Fix nd minimize J HK (,) wih respec o
23 LDF: HoKshyp Procedure J HK ( Y ) 0 Y J HK ( Y ) 0 Alerne he wo seps elow unil convergence: ) Fix nd minimize J HK (,) wih respec o ) Fix nd minimize J HK (,) wih respec o Sep () cn e performed wih pseudoinverse For fixed minimum of J HK (,) wih respec o is found y solving Thus ( Y ) 0 Y ( ) Y Y Y
24 LDF: HoKshyp Procedure Sep : fix nd minimize J HK (,) wih respec o We cn use Y ecuse hs o e posiive Soluion: use modified grdien descen sr wih posiive, follow negive grdien u refuse o decrese ny componens of This cn e chieved y seing ll he posiive componens of J o 0 No doing seepes descen nymore, u we re sill doing descen nd ensure h is posiive
25 LDF: HoKshyp Procedure The HoKshyp procedure: 0) Sr wih rirry () nd () > 0, le k repe seps () hrough (4) ( k) ( k) ( k) ) e Y ) Solve for (k+) using (k) nd (k) [ e e ] ( k+ ) ( k) ( k) ( k) + η + 3) Solve for (k+) using (k+) 4) k k + ( k+ ) ( ) ( k+ ) unil e (k) > 0 or k > k mx or (k+) (k) Y Y For convergence, lerning re should e fixed eween 0 < η < Y
26 LDF: HoKshyp Procedure In he linerly seprle cse, e (k) 0, found soluion, sop one of componens of e (k) is posiive, lgorihm coninues In non seprle cse, e (k) will hve only negive componens evenully, hus found proof of nonsepriliy No ound on how mny ierion need for he proof of nonsepriliy
27 LDF: HoKshyp Procedure Exmple Clss : (6 9), (5 7) Clss : (5 9), (0 0) Mrix Y ( ) Sr wih nd Use fixed lerning η ( ) A he sr Y ( )
28 LDF: HoKshyp Procedure Exmple Ierion : e ( ) ( ) ( ) Y solve for () using () nd () [ e + e ] ( ) ( ) ( ) ( ) solve for () using () ( ) ( ) ( ) Y Y Y *
29 LDF: HoKshyp Procedure Exmple Coninue ierions unil Y > 0 In prcice, coninue unil minimum componen of Y is less hen 0.0 Afer 04 ierions converged o soluion does gives sepring hyperplne Y
30 LDF: MSE for Muliple Clsses Suppose we hve m clsses Define m liner discriminn funcions g i ( x) w x+ w 0 i i i,...,m Given x, ssign clss c i if g i ( x ) g ( x ) j j i Such clssifier is clled liner mchine A liner mchine divides he feure spce ino c decision regions, wih g i (x) eing he lrges discriminn if x is in he region R i
31 LDF: MSE for Muliple Clsses For ech clss i, find weigh vecor i, s.. i i y y 0 y y clss clss Le Y i e mrix whose rows re smples from clss i, so i hs d + columns nd n i rows i i Le s pile ll smples in n y d + mrix Y: Y Y Y M Y m smple from smple from M smple from smple from clss clss clss m clss m
32 LDF: MSE for Muliple Clsses Le i e column vecor of lengh n which is 0 everywhere excep rows corresponding o smples from clss i, where i is : i 0 M M 0 M rows corresponding o smples from clss i
33 LDF: MSE for Muliple Clsses Le s pile ll i s columns in n y c mrix B B [ L ] n Le s pile ll i s columns in d + y m mrix A A [ L ] m m LSE prolems cn e represened in YA B: smple from clss smple from clss smple from clss smple from clss 3 smple from clss 3 smple from clss Y A B
34 LDF: MSE for Muliple Clsses Our ojecive funcion is: J m ( A) i Y i i J(A) is minimized wih he use of pseudoinverse A ( Y Y) YB
35 LDF: Summry Percepron procedures find sepring hyperplne in he linerly seprle cse, do no converge in he nonseprle cse cn force convergence y using decresing lerning re, u re no gurneed resonle sopping poin MSE procedures converge in seprle nd no seprle cse my no find sepring hyperplne if clsses re linerly seprle use pseudoinverse if Y Y is no singulr nd no oo lrge use grdien descen (WidrowHoff procedure) oherwise HoKshyp procedures lwys converge find sepring hyperplne in he linerly seprle cse more cosly
Minimum Squared Error
Minimum Squred Error LDF: Minimum SquredError Procedures Ide: conver o esier nd eer undersood prolem Percepron y i > for ll smples y i solve sysem of liner inequliies MSE procedure y i = i for ll smples
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