Supervised Learning NNs

Transcription

1 EE788 Robot Cognton and Plannng, Prof. J.-H. Km Lecture 6 Supervsed Learnng NNs Robot Intellgence Technolog Lab. From Jang, Sun, Mzutan, Ch.9, Neuro-Fuzz and Soft Computng, Prentce Hall

2 Contents. Introducton. Perceptrons 3. Adalne 4. Backpropagaton MLPs 5. Radal Bass Functon Netorks 6. Modular Netorks

3 . Introducton Artfcal neural netorks or smpl NNs Perceptron: McCulloch and Ptts 43 Sngle-laer perceptrons (Rosenblatt 6) Appled to pattern classfcaton learnng Interest n NNs dndled n the 97s. Lmtaton of sngle-laer sstems (Mnsk and Papert 69) The recent resurgence of nterest n the feld of NNs (snce 98s) Ne NN learnng algorthms Analog VLSI crcuts Parallel processng technques 3

4 . Introducton Classfcaton of NN models Learnng methods: supervsed vs. unsupervsed Archtectures: feedforard vs. recurrent Output tpes: bnar vs. contnuous Node tpes: unform vs. hbrd Implementatons: softare vs. hardare Connecton eghts: adustable vs. hardred Supervsed learnng or mappng netorks Desred nput-output data sets Adustable parameters Updated b a supervsed learnng rule 4

5 . Perceptrons Archtecture: a sngle-laer perceptron for pattern recognton g : maps all or a part of the nput pattern nto A bnar value {, } A bpolar value {-, } The term : : actve or ectator : nactve -: nhbtor 5

6 . Perceptrons Output: o f f n n f n th and here : a modfable eght, : the bas term Actvaton functon f( ): sgnum or step functon sgn( ) f, otherse step( ) f, otherse 6

7 . Perceptrons Learnng algorthm. Select an nput vector from the tranng data set.. If the perceptron gves an ncorrect response, modf all connecton eghts accordng to Δ = η t here t : a target output, η: learnng rate 3. Repeat and. Learnng rate A constant throughout tranng Proportonal to the error Faster convergence Ma lead to unstable learnng 7

8 8 Eclusve-OR problem: Not lnearl separable usng a sngle-laer perceptron because The to-laer perceptron: Multlaer perceptrons Solve nonlnearl separable problems.,,,. Perceptrons o: class, : class (Mnsk and Papert, 69)

9 9 3. Adalne Adalne (adaptve lnear element), Wdro and Hoff 6 Delta rule for adustng the eghts on the pth I/O pattern: n o p p p p p p p p p o t o t E o t E ) ( ) ( ) (

10 3. Adalne Delta rule Wdro-Hoff learnng rule Least mean square learnng procedure mnmzng squared errors Features: Smplct Dstrbuted learnng (performed locall at each node level) On-lne learnng (updated after presentaton of each pattern) Applcatons Adaptve nose cancellaton Interference cancelng n electrocardograms Echo elmnaton from long-dstance telephone transmsson lnes Antenna sdelobe nterference cancelng Adaptve nverse control

11 4. Backpropagaton MLPs MLP (Mult-Laer Perceptron) Feedforard netork that emplos the delta rule for tranng Feedforard netorks Full connected multlaer netork All neurons n a partcular laer are full connected to all neurons n the subsequent laer E) Three-laer feedforard NN Three neuron nput laer To neuron output laer Four neuron hdden laer (Laer ) (Laer ) (Laer )

12 4. Backpropagaton MLPs Basc model of a sngle artfcal neuron,, : nputs b f,, : eghts : a bas : the actvaton functon : the output Let s be a eghted sum, N s () t () t b s (t) = W +b

13 4. Backpropagaton MLPs Actvaton functon f(s) Bas b ll move the curve along the s-as. Sgmod actvaton functon: f ( s ) e Dfferentable Monotonc s Actvaton functons or, logstc 3

14 4. Backpropagaton MLPs Backpropagaton Supervsed learnng method to tran NNs Uses a gradent-descent optmzaton method, also referred to as the delta rule, hen appled to feedforard netorks Performance nde or cost functon J: M J ( d here d : desred netork output : actual netork output Usng gradent-descent, the eght ncrement s J here μ: a constant ) 4

15 5 Usng the chan rule, M d M J ) (, ) ( M d M J M d M J ) ( If the actvaton functon s the sgmod functon, then ts dervatve s ) ( s s s s e e e e s f or ) ( ) ( s f s f s f Snce f(s) s the neuron output, then above equaton can be rtten as ) ( s From (), agan usng the chan rule, s s () () (3) (4) 4. Backpropagaton MLPs

16 6 Bas b s called, thus N s ) ( N N s (5) (6) Substtutng (3) and (5) nto (4), Puttng (6) nto (), M M M d M J ) ( ) ( (7) here ) ( ) ( d (8) 4. Backpropagaton MLPs

17 4. Backpropagaton MLPs Substtutng (7) nto the eght ncrement here M M (9) Ths leads to a eght ncrement, called the delta rule, for a partcular neuron: ( kt ) here η s the learnng rate and s a value of beteen and. Hence the ne eght becomes () ( kt ) ( k ) T ( kt ) ( k ) T () 7

18 4. Backpropagaton MLPs Consder a three laered netork: Input laer (l=), hdden laer (l=), and output laer (l=) Back-propagaton commences th the output laer here d s knon and hence δ can be calculated usng (8), and the eghts adusted usng (). To adust the eghts on the hdden laer (l=), (8) s replaced b Three-laer feedforard NN: N [ ] l [ ( )] l () l 8

19 4. Backpropagaton MLPs Hence, the δ values for laer l are calculated usng the neuron outputs from laer l (hdden laer) together th the summaton of and δ products from laer l+ (output laer). The back-propagaton process contnues untl all eghts have been adusted. Then, usng a ne set of nputs, nformaton s fed forard through the netork (usng the ne eghts) and errors at the output laer computed. The process contnues untl () The performance nde J reaches an acceptable lo value () A mamum teraton count (number of epochs) has been eceeded () A tranng-tme perod has been eceeded. 9

20 4. Backpropagaton MLPs Equatons that govern the BPA can be summarzed as Sngle neuron summaton: Sgmod actvaton functon: s N ( t) ( t) b (3) e s (4) Delta rule: ( kt ) (5) Ne eght: ( kt ) ( k ) T ( kt ) (6) Output laer: Other laers: d ) ( ) (7) J ( M ( d ) l (8) N [ ] l [ ( )] l (9)

21 4. Backpropagaton MLPs Learnng th momentum Makng the current eght change equal to a proporton of the prevous eght change summed th the eght change calculated usng the delta rule Delta rule gven n (5) can be modfed as Δ ( kt ) ( α) ηδ αδ (( k ) T ) () here α s the momentum coeffcent, and has a value of beteen and. Used n BPA, the soluton stands less chance of becomng trapped n local mnma

22 4. Backpropagaton MLPs E) Tranng usng back-propagaton Calculate the output, and hence the ne values for the eghts and bases. Assume a learnng rate of.5. Current nputs =., =.6 Desred output d = Estng eghts and bases are: n the hdden laer, b 3. n the output laer, b 4.

23 4. Backpropagaton MLPs Sol.) Forard propagaton Hdden laer (l=): Sngle neuron summaton or () B sgmod actvaton functons (= to 3), () 3

24 4. Backpropagaton MLPs Insertng values nto () and (), Output laer (l=) (3) (4) Insertng values nto (3) and (4), 4

25 4. Backpropagaton MLPs Back propagaton Output laer (l=): From (8), Snce =, Delta rule: Ne eghts and bases for the output laer: 5

26 4. Backpropagaton MLPs Hdden laer (l=): From (9), To llustrate ths equaton, had there been to neurons n laer (l+),.e. the output laer, values for δ and δ for laer (l+) ould have been calculated. Thus, for laer l (the hdden laer), the [δ ] l values ould be Hoever, snce n ths eample there s onl a sngle neuron n laer (l+), δ =. Thus the δ values for laer l are 6

27 4. Backpropagaton MLPs Hence, usng the delta rule, the eght ncrements for the hdden laer are The ne eghts and bases for the hdden laer no become Intall, b..5.5 W b

28 5. Radal Bass Functon Netorks Locall tuned and overlappng receptve felds: Structures n the regons of the cerebral corte, the vsual corte, etc. RBFN: A netork structure emplong local receptve felds to perform functon mappngs The actvaton level of the th receptve feld unts (or hdden unt): R u /,,, H R..., (a) Weghted sum of the output values (b) Weghted average of the output values 8

29 5. Radal Bass Functon Netorks R ( ) s a radal bass functon: Gaussan functon: Logstc functon: R R u ep ep[ u / ] The actvaton level s mamum hen the nput vector s at the center u of that unt. Note that there are no connecton eghts beteen the nput laer and the hdden laer. 9

30 3 5. Radal Bass Functon Netorks Output of an RBFN Weghted sum of the output value assocated th each receptve feld: Weghted average of the output value assocated th each receptve feld: Dsadvantage: a hgher degree of computatonal complet Advantage: a ell-nterpolated overall output beteen the outputs of the overlappng receptve felds H H R c c d H H H H R R c c d

31 5. Radal Bass Functon Netorks RBFN s appromaton capact Further mproved th supervsed adustments of the center and shape of the receptve feld (or radal bass) functon Sequental tranng algorthm: F the receptve feld functons frst Then adust the eghts of the output laer Functonal equvalence to Fuzz Inference Sstem Same aggregaton method to derve ther overall outputs Weghted average or eghted sum Same number of receptve feld unts and fuzz f-then rules Each radal bass functon of the RBFN s equal to a multdmensonal composte MF. 3

32 6. Modular Netorks Modular netorks A herarchcal organzaton comprsng multple NNs To prncpal components: Local eperts (or epert netorks) Integratng unt (or gatng netork) Overall output usng estmated combnaton eghts (g ): Y K g O 3

33 7. Summar Learnng modes Characterstcs of avalable nformaton for learnng Supervsed: Instructve nformaton on desred responses, eplctl specfed b a teacher Renforcement: Partal nformaton about desred responses, or onl rght or rong, evaluatve nformaton Unsupervsed: No nformaton about desred response Recordng: A pror desgn nformaton for memor storng 33