15-381: Artificial Intelligence. Regression and neural networks (NN)

Transcription

1 5-38: Artfcal Itellece Reresso ad eural etorks NN)

2 Mmck the bra I the earl das of AI there as a lot of terest develop models that ca mmc huma thk. Whle o oe ke eactl ho the bra orks ad, eve thouh there as a lot of proress sce, there s stll lttle ko), some of the basc computatoal uts ere ko A ke compoet of these uts s the euro.

3 he Neuro A cell the bra Hhl coected to other euros houht to perform computatos b terat sals from other euros Outputs of these computato ma be trasmtted to oe or more euros

4 What ca e do th NN? Classfcato - We alread metoed ma useful applcatos Reresso Iput: Real valued varables Output: Oe or more real values Eamples: - Predct the prce of Goole s stock from Mcrosoft s stock prce - Predct dstace to obstacle from varous sesors

5 Lear reresso Gve a put e ould lke to compute a output I lear reresso e assume that ad are related th the follo equato: +ε here s a parameter ad ε represets measuremet or other ose Y X

6 Multvarate reresso: Least squares We ca re-rte multvarate reresso as X he soluto s: X X) - X he s a stace of a larer set of computatoal solutos hch are usuall referred to as eeralzed least squares X X s a k b k matr X s a vector th k etres

7 Multvarate reresso: Least squares We ca re-rte our model as X he soluto turs out to be: X X) - X We eed to vert a k b k matr hs takes Ok 3 ) Deped o k ths ca be rather slo

8 Where e are Lear reresso solved But - Soluto ma be slo - Does ot address eeral reresso problems of the form fx)

9 Back to NN: Preceptro he basc process ut of a eural et f ) k k

10 Lear reresso Lets start b sett f ) We are back to lear reresso Ulke our oral lear reresso soluto, for perceptros e ll use a dfferet strate Wh? - We ll dscuss ths later, for o lets focus o the soluto 2 k 0 2 k

11 Gradet descet zf)-) 2 Slope z/ Δz Δ Go the opposte drecto to the slope ll lead to a smaller z But ot too much, otherse e ould o beod the optmal

12 Gradet descet Go the opposte drecto to the slope ll lead to a smaller z But ot too much, otherse e ould o beod the optmal We thus update the ehts b sett: z # " $ here λ s small costat hch s teded to prevet us from pass the optmal

13 Eample he choos the rht λ We et a mootocall decreas error as e perform more updates

14 Gradet descet for lear reresso We compute the radet.r.t. to each Ad f e have measuremets the here, s the th value of the th put vector ) 2 2 " " # $ % & ' " ) ) k k k k k k " # #, 2 ) 2 ) - -

15 Gradet descet for lear reresso If e have measuremets the Set he our update rule ca be rtte as " # #, 2 ) 2 ) - - ) - + ", $2 #

16 Gradet descet alorthm for lear reresso.chose λ 2.Start th a uess for 3.Compute δ for all 4.For all set " 5.If o mprovemet for + $2 # stop. Otherse o to step 3, - 2 )

17 W 2 Eample

18 Gradet descet vs. matr verso Advataes of matr verso - No teratos - No eed to specf parameters - Closed form soluto a predctable tme Advataes of radet descet - Applcable reardless of the umber of parameters - Geeral, apples to other forms of reresso

19 Perceptros for classfcato So far e dscussed reresso Hoever, perceptros ca also be used for classfcato For eample, output s > 0 ad - otherse Problem?

20 Perceptros for classfcato So far e dscussed reresso Hoever, perceptros ca also be used for classfcato Outputs ether 0 or We predct f > /2 ad 0 otherse Problem? Best least squares ft Best classfer

21 he smod fucto o classf us a perceptro e replace the lear fucto th the smod fucto: h) h + e Us the smod e ould mmze " Where s ether 0 or deped o the class 2 ))

22 Gradet descet th smod Oce e defed our taret fucto, e ca mmze t us radet descet hs volves some math, ad reles o the follo dervato*: So, )) ) ) ' h h h *I have cluded a dervato of ths at the ed of the lecture otes, 2 )) ) )) 2 ) ' )) 2 )) )) 2 )) " " " " " " # # # #

23 Gradet descet th smod ) " ), 2 )) ) )) 2 )) " " # # Set, 2 ) 2 )) " " # # $, ) 2 + " # $ % So our update rule s:

24 Revsed alorthm for smod reresso.chose λ 2.Start th a uess for 3.Compute δ for all 4.For all set " + % 2 # $ ), 5.If o mprovemet for - 2 )) stop. Otherse o to step 3

25 Multlaer eural etorks So far e dscussed etorks th oe laer. But these etorks ca be eteded to combe several laers, creas the set of fuctos that ca be represeted us a NN Ofte called the hdde laer 0,2,2 0,, 2, v ) 2 v v) 2 2,2 v 2 )

26 Lear the parameters for multlaer etorks Gradet descet orks b coect the output to the puts. But ho do e use t for a multlaer etork? We eed to accout for both, the output ehts ad the hdde laer ehts 0,2,2 0,, 2, v ) 2 v v) 2 2,2 v 2 )

27 Lear the parameters for multlaer etorks Its eas to compute the update rule for the output ehts ad 2 : + % 2 $ ) v, " # here " v ) 0,2,2 0,, 2, v ) 2 v) 2 2,2 v 2 )

28 Lear the parameters for multlaer etorks Its eas to compute the update rule for the output ehts ad 2 : + % 2 $ ) v, " # here " v ) But hat s the error assocated th each of the 0, hdde laer states? 0,2,2, 2, v ) 2 v v) 2 2,2 v 2 )

29 Backpropaato A method for dstrbut the error amo hdde laer states Us the error for each of these states e ca emplo radet descet to update them Set ", # $ ) output error eht 0,2,2 0,, 2, v ) 2 v v) 2 2,2 v 2 )

30 Backpropaato A method for dstrbut the error amo hdde laer states Us the error for each of these states e ca emplo radet descet to update them Set ", # $ ) Our update rule chaes to: k, # k, + % 2 $ ",,, ), k

31 Backpropaato he correct error term for each hdde state ca be determed b tak the partal dervatve for each of the eht parameters of the hdde laer.r.t. the lobal error fucto*: k, # k, + 2 $ 2 Err % ",, )), ), k *See RN book for detals paes )

32 Revsed alorthm for multlaered eural etork.chose λ 2.Start th a uess for, 3.Compute values v, for all hdde laer states ad puts 4.Compute δ for all : " v ) 5.Compute Δ,I 6.For all set + % 2 $ ) v 7. For all k ad set, k, # k, + % 2 $ ",,, ), k " # If o mprovemet for # + ", stop. Otherse o to step 3 s

33 Eamples Fure : Feedforard ANN desed ad tested for predcto of tactcal ar combat maeuvers.

34 What ou should ko Lear reresso - Solv a lear reresso problem Gradet descet Perceptros - Smod fuctos for classfcato Multlaered eural etorks - Backpropaato

35 Derv ) Recall that ) s the smod fucto so ) + e he dervato of ) s belo