Linear regression II
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1 CS 75 Mache Learg Lecture 9 Lear regresso II Mlos Hauskrecht mlos@cs.ptt.eu 539 Seott Square Lear regresso Fucto f : X Y Y s a lear combato of put compoets f ( w w w w w w, w, w k - parameters (weghts w Bas term w w f (, w Iput vector w w
2 Lear regresso. Error. Data: D, Fucto: f ( We woul lke to have f ( for all,.., Error fucto measures how much our prectos evate from the esre aswers Mea-square error,.. ( y Learg: We wat to f the weghts mmzg the error! f ( Lear regresso. Eample mesoal put (
3 Lear regresso. Eample. mesoal put, ( Solvg lear regresso he optmal set of weghts satsfes: w( ( w ( w w Leas to a system of lear equatos (SLE wth + ukows of the form Soluto to SLE: Aw b,, w,, w,, w,, w A Assumg X s a ata matr wth rows correspog to eamples a colums to puts, a y s vector of outputs, the w ( X X X y b y, 3
4 Graet escet soluto Goal: the weght optmzato the lear regresso moel Error( w ( f (, w,.. A alteratve to SLE soluto: Graet escet Iea: Aust weghts the recto that mproves the Error he graet tells us what s the rght recto w w werror (w - a learg rate (scales the graet chages Batch vs Ole regresso algorthm he error fucto efe o the complete ataset D Error( w ( f (, w,.. We say we are learg the moel the batch moe: All eamples are avalable at the tme of learg Weghts are optmzes wth respect to all trag eamples A alteratve s to lear the moel the ole moe Eamples are arrvg sequetally Moel weghts are upate after every eample If eee eamples see ca be forgotte - 4
5 Etesos of smple lear moel Replace puts to lear uts wth m feature (bass fuctos to moel oleartes ( f ( w w ( ( w m - a arbtrary fucto of w f ( ( w m ( w m Orgal put New put Lear moel No-lear moel 5
6 Statstcal moel of regresso A statstcal moel of lear regresso: w y ~ N(, s a raom ose, represets thgs we caot capture wth w 3 5 E( y w 5 5 Gaussa ose ε y ~ N( w, Regularze lear regresso If the umber of parameters s large relatve to the umber of ata pots use to tra the moel, we face the threat of overfttg (geeralzato error of the moel goes up he precto accuracy ca be ofte mprove by settg some coeffcets to zero Icreases the bas, reuces the varace of estmates Solutos: Subset selecto Rge regresso Lasso regresso Prcpal compoet regresso 6
7 Regularzato: motvato If the moel s too comple a ca cause overfttg, ts precto accuracy ca be mprove by removg some puts from the moel = settg ther coeffcets to zero f ( w w w w w 3 3 w w, w, w k - parameters (weghts Iput vector w w w w f (, w f ( w w w w 3 3 w Rge regresso Questo: how to force the weghts to? Error fucto for the staar least squares estmates: ( w ( w Where w,.. We seek: * w arg m w Rge regresso: w ( w,.. ( y What oes the ew obectve fucto o? w ( w L,.. w w w L a 7
8 Staar regresso: Rge regresso: Rge regresso ( w ( w,.. w w w w L pealzes o-zero weghts wth the cost proportoal to (a shrkage coeffcet If a put attrbute has a small effect o mprovg the error fucto t s shut ow by the pealty term Icluso of a shrkage pealty s ofte referre to as regularzato. (rge regresso s relate to khoov regularzato ( w ( w w L,.. Regularze lear regresso How to solve the least squares problem f the error fucto s erche by the regularzato term w? Aswer: he soluto to the optmal set of weghts w s obtae aga by solvg a set of lear equato. Staar lear regresso: w( ( w ( w Soluto: w* ( X X X Regularze lear regresso: y where X s a matr wth rows correspog to eamples a colums to puts, y s vector of outputs w* ( I X X X y 8
9 Regularze lear regresso Rge regularzato s also relate to the Bayesa regresso wth the Gaussa pror Iea: Statstcal moel for lear regresso: w y ~ N(, p( y N( w, A a pror o w Posteror o w: p( w N(, I p( w X, Y, Gaussa he obectve fucto for the regularze least squares correspos to the problem of fg the moe of the posteror. p( w X, Y,, where Lasso regresso Staar regresso: w ( w (,.. Lasso regresso/regularzato: w ( w ( w L,.. w w pealzes o-zero weghts wth the cost L proportoal to. L s more aggressve pushg the weghts to compare to L he obectve fucto correspos to the moe of the posteror the Bayesa regresso whe the pror o w s moele usg a Laplace strbuto 9
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