Multilayer perceptrons
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1 Multilayer perceptros If traiig set is ot liearly separable, a etwork of McCulloch-Pitts uits ca give a solutio If o loop exists i etwork, called a feedforward etwork (else, recurret etwork) A two-layer etwork solves a eve-parity problem I O I O= ( I I ) ( I I ) (Wag) 1
2 Multilayer etwork architecture For the restaurat (Willwait) problem 730(Wag) 2
3 Learig as error miimizatio Basic idea: defie a error fuctio that ca be miimized by adjustig weight values Move the weight vector i the directio of reducig error Error fuctio ca be defied over the etire set of traiig vectors or for a sigle example 730(Wag) 3
4 Error surface I terms of weights 730(Wag) 4
5 Gradiet descet search Defie error fuctio E that ca be miimized by adjustig weights For a fixed set of iput vectors, E depeds oly o weight values Fid miimum of E by gradiet descet Calculate gradiet of E i weight space Move weight vector W dowhill i gradiet directio 730(Wag) 5
6 Gradiet descet E E W Update rule: W W α E W W 730(Wag) 6
7 Backpropagatio algorithm Error fuctio measure square error 1 E = ( Ti Oi ) 2 i Key questio: how to determie appropriate error for hidde layers (uits)? Temporal credit assigmet problem Error ca be back-propagated layer by layer, hece the ame 2 730(Wag) 7
8 Two-layer perceptro Error fuctio 1 1 E( W ) = ( Ti Oi) = ( Ti g( Wji, aj)) 2 i 2 i j = ( Ti g( Wji, g( Wkj, Ik))) 2 i j k The gradiet for weights of the output layer E W ji, = a ( T O) g ( W a ) j i i j, i j j = a ( T O ) g ( i ) = a j i i i j i 2 730(Wag) 8
9 Two-layer perceptro, cot. For the weights of the hidde layer, derivatio is more complicated (ivolvig the chai rule of differetiatio), ad leads to: E = Ig k ( ij) Wj, i i = Ik j W i kj, Update rule has a learig rate Update rule is local Ivolvig oly presyaptic ad postsyaptic uits For sigmoid activatio: g = g( 1 g) 730(Wag) 9
10 The error term The error term is computed directly for the output layer Everythig else acts o weighted sum of backpropagated errors from the subsequet layer The two-layer versio exteds to a arbitrary umber of layers 730(Wag) 10
11 The backprop algorithm 1. Iitializatio. Set all weights to small radom values 2. Exteral presetatio. Preset a pair of a iput vector I ad a desired output vector T. Assig I to the iput layer. 3. Forward computatio. For each successive layer: Compute weighted sum Apply sigmoid fuctio to geerate output 730(Wag) 11
12 Backprop algorithm, cot. 4. Compute the errors for output layer by comparig actual output with T. Apply gradiet descet to adjust weights to output layer 5. Backpropagate the error. Compute similarly the errors for the precedig layer based o errors from ext layer. Adjust weights accordig to gradiet descet. 6. Repeat by goig to step 2 with aother pair util overall error is reduced to acceptable level. 730(Wag) 12
13 Backpropagatig the s First layer ( k 1)-th layer k-th layer (Wag) 13
14 WillWait problem, agai A compariso with decisio-tree learig 730(Wag) 14
15 Geeralizatio i eural etworks I most applicatios, dimesioality of iput vector is fairly large (100 or more) Impossible to sample more tha a tiy fractio of iput (e.g. the coectedess problem) IF classificatio of iputs is some fuctio close to a member of a set of fuctios implemetable by etwork, AND if the fit by traied data is good, THEN the etwork has likely captured a uderlyig fuctioal relatioship betwee iputs ad classificatio 730(Wag) 15
16 Other issues Expressiveess: attribute-based represetatio Ca represet ay iput-output fuctio (mappig) Havig trouble represetig relatios Computatioal efficiecy Traiig is geerally slow Sesitivity to oise: very tolerat Prior kowledge 730(Wag) 16
17 Example feed-forward et ALVINN: Autoomous Lad Vehicle i a Neural Net Oe hidde layer, oe output layer Five hidde odes, 32 output odes (steer left-steer right) 960 iputs (30 x 32 image itesity array) 5000 traiable weights 730(Wag) 17
18 Potetial problems with traiig ALVINN Huma drives the va, ad actual steerig agles take as correct output for correspodig iputs The etwork traied by backpropagatio But, 2 potetial problems Driver is usually drivig well, so etwork does ot get experiece with far-from-ceter positios O log stretches of road, etwork would be traied for log time o straight-ahead steerig, could swamp out learig o curves 730(Wag) 18
19 Solutio: Augmet traiig iputs Supplied with a set of huma-supplied traiig samples Rotates ad shifts each image to create 14 additioal images i which vehicle appears situated slightly differetly relative to road Uses model that tells system what steerig agle should be for each shifted image, give the huma-supplied steerig agle for the origial image 730(Wag) 19
20 ALVINN is a success Successfully steers Chevy va o highways ad ordiary roads at speeds of 55mph Previous top systems oly maaged 3 or 4mph Most recet versio drove across the Uited States ( No Hads Across America ) 98% of the time the system was steerig itself Huma still hadled brakig 730(Wag) 20
21 Summary of Backpropagatio Gradiet descet learig is hill-climbig search Covergece - ot guarateed Local miima - possible but rare Oe hidde layer is sufficiet Large volume of applicatios 730(Wag) 21
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