Multilayer neural networks
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1 Lecture Multlayer neural networks Mlos Hauskrecht 5329 Sennott Square Mdterm exam Mdterm Monday, March 2, 205 In-class (75 mnutes) closed book materal covered by February 25, 205
2 Multlayer neural networks Or another way of modelng nonlneartes for regresson and classfcaton problems Classfcaton wth the lnear model. Logstc regresson model defnes a lnear decson boundary Example: 2 classes (blue and red ponts) 2 Decson boundary
3 Lnear decson boundary logstc regresson model s not optmal, but not that bad When logstc regresson fals? Example n whch the logstc regresson model fals
4 Lmtatons of lnear unts. Logstc regresson does not work for party functons - no lnear decson boundary exsts Soluton: a model of a non-lnear decson boundary x Extensons of smple lnear unts use feature (bass) functons to model nonlneartes Lnear regresson m w0 w ( x) (x) ( x ) 2 ( x ) - an arbtrary functon of x w 0 w w 2 Logstc regresson g ( w0 w ( x)) m x d m ( x ) w m 4
5 Learnng wth extended lnear unts Feature (bass) functons model nonlneartes Lnear regresson m w0 w ( x) x ( x ) 2 ( x ) w 0 w w 2 Logstc regresson m g ( w0 w ( x)) x d m ( x ) w m Important property: The same problem as learnng of the weghts for lnear unts, the nput has changed but the weghts are lnear n the new nput Problem: too many weghts to learn Mult-layered neural networks An alternatve way to ntroduce nonlneartes to regresson/classfcaton models Key dea: Cascade several smple neural models wth logstc unts. Much lke neuron connectons. 5
6 Multlayer neural network Also called a multlayer perceptron (MLP) x x 2 x d Cascades multple logstc regresson unts Example: (2 layer) classfer wth non-lnear decson boundares w 0, () w 0,2 () w k, () w k,2 () z () z 2 () w 0, w, w 2, z p ( y x) Input layer Hdden layer Output layer Multlayer neural network Models non-lnearty through logstc regresson unts Can be appled to both regresson and bnary classfcaton problems Input layer x x 2 x d w 0, () w 0,2 () w k, () w k,2 () Hdden layer w 0, z () w, w 2, z 2 () Output layer regresson f ( x, z classfcaton p( y x, opton 6
7 Multlayer neural network Non-lneartes are modeled usng multple hdden logstc regresson unts (organzed n layers) The output layer determnes whether t s a regresson or a bnary classfcaton problem Input layer x Hdden layers Output layer regresson f ( x, x 2 classfcaton x d opton p( y x, Learnng wth MLP How to learn the parameters of the neural network? Gradent descent algorthm Weght updates based on the error: J ( D, w ) w w w J ( D, w ) We need to compute gradents for weghts n all unts Can be computed n one backward sweep through the net!!! The process s called back-propagaton 7
8 Backpropagaton (k-)-th level k-th level (k+)-th level x ( k ) x (k) w, z (k) ( k ) ( k ) w l, z l x l ( k ) (k) x (k) z w, - output of the unt on level k - nput to the sgmod functon on level k w x ( k z k) w ) (,0, x g( z ) - weght between unts and on levels (k-) and k Backpropagaton Update weght w, usng a data pont D { x, y } w, w, J ( D, w, Let J ( D, z J ( D, z Then: J ( D, x ( k ) w k) z w S.t. (k) s computed from (k) and the next layer ( k ), (, x l ( k ) wl, ( k ) x ( x ) l Last unt (s the same as for the regular lnear unts): ( K) ( y u f ( x u, ) It s the same for the classfcaton wth the log-lkelhood measure of ft and lnear regresson wth least-squares error!!! l 8
9 Learnng wth MLP Onlne gradent descent algorthm Weght update: w, ( k ) w, ( k ) J onlne ( D u, w ) w ( k ), J onlne ( Du, z J onlne ( Du, w, z w, x ( k ) w, ( k ) w, ( k ) ( k ) x ( k ) x ( k ) - -th output of the (k-) layer (k ) - dervatve computed va backpropagaton - a learnng rate Onlne gradent descent algorthm for MLP Onlne-gradent-descent (D, number of teratons) Intalze all weghts w, for =:: number of teratons do select a data pont D u =<x,y> from D set learnng rate compute outputs x (k ) for each unt compute dervatves (k ) va backpropagaton update all weghts (n parallel) w, ( k ) w, ( k ) ( k ) x ( k ) end for return weghts w 9
10 Xor Example. lnear decson boundary does not exst Xor example. Lnear unt 0
11 Xor example. Neural network wth 2 hdden unts Xor example. Neural network wth 0 hdden unts
12 MLP n practce Optcal character recognton dgts 20x20 Automatc sortng of mals 5 layer network wth multple output functons 0 outputs (0,, 9) layer Neurons Weghts x20 = 400 nputs
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