ABOUT ONE APPROACH TO APPROXIMATION OF CONTINUOUS FUNCTION BY THREE-LAYERED NEURAL NETWORK
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1 ABOUT ONE APPROACH TO APPROXIMATION OF CONTINUOUS FUNCTION BY THREE-LAYERED NEURAL NETWORK Ram Rzayev Cyberetc Isttute of the Natoal Scece Academy of Azerbaa Republc Aygu Alasgarova Khazar Uversty, Azerbaa Republc Abstract Accordg to the accepted opo a three-layered eural etwork wth feedforward structure s cosdered as uversal appromator for ay cotuous fucto. However ths s rather covetoal represetato sce traed o fte set of pots the gve eural etwork wll be more lkely terpolator of cotuous fucto, tha ts appromator. Therefore, that retra the eural etwork up to uversal appromator so that appromato Veershtrass s theorem s full carred out t s offered the combed trag of eural etwork by Remez s algorthm wth error backpropagato ad to costruct the eural appromato kerel. Key words: cotuous fucto, Veershtrass s theorem, feedforward three-layered eural etwork, eural appromato kerel, Remez s algorthm, error backpropagato algorthm.. Itroducto Sce 986 year, after "error backpropagato" algorthm has bee offered by Rummelhart, varous o-recurret eural-set costructos bega to be appled wdely the decso of may appled problems such as modellg ad detfcato of systems, geerato ad decomposto of sgals, recogto ad classfcato of patters, etc. Based o the dfferetal ature of eural etworks ths algorthm has allowed to reveal appromate propertes of the etworks usg olear actvato fuctos to euros from hdde layers. It has bee establshed [] that feedforward eural etworks eve wth oe hdde olear layer are capable to be a uversal appromator for ay cotuous fucto o the set cosstg of fte umber of pots. Moreover, kow theorems of Chebshev ad Valle- Pusse [2] have show that the problem of appromato of cotuous fucto f o the closed lmted set Ω s equvalet to a problem of appromato of the same fucto o a subset Ω Ω cosstg of fte umber of pots. The problem of appromato of the cotuous fuctos by eural etworks was cosdered by may authors. I partcular, t s theoretcally establshed that the eural etwork 627
2 wth oe olear hdde layer ca appromate ay cotuous fucto wth ecessary accuracy [3]. Thus the estmato order of the appromato versely depeds o umber of euros used to the hdde layer. I practce to obta the requred estmato of appromato t s ecessary to volve sgfcat amout olear euros to the hdde layer of eural etwork, ad t, as a rule, leads to crease tme spet for eural etwork trag. It does usg of eural etworks less effectve, especally the solvg of maagemet problems, where ofte t s ecessary to make operatvely the "correctreasoable" decso respose to possble eteral ad teral factors of fluece. I the gve artcle the approach to trag the eural etworks appromatg cotuous fuctos s cosdered somewhat dfferet from tradtoal. The essece of ths approach cossts the followg. For appromato ay cotuous fucto f defed o a segmet [ a, stead of tradtoal eural etwork wth ay parameters ω ξ, θ y et ω ϕ ξ θ, ad lmted sgmod actvato fucto ϕ ϕ ξ θ of -th euro from the hdde layer, t s selected a etwork of the smlar structure y et ω ϕ ξ θ, 2 where parameters ξ, θ are advace optmzed accordg to equalty θ < ε, ε > ϕ ξ 3 by co-applcato of error backpropagato ad Remez s algorthm for ay trag set + {,}. The eural etwork ducg the polyomal et ϕ ξ θ ad creatg thus favourable codtos for appromato of ay cotuous fucto we shall ame as eural-appromato kerel. Trag of the eural etworks costructed o the bass of such kerel wll be lmted by optmzato oly output coectos weghts ω,. It s obvous, that t cosderably wll reduce tme allocated o trag. 628
3 2. Trag of eural etworks by combed applcato error backpropagato ad Remez s algorthm It s well kow that for ay cotuous fucto t s practcally mpossble to costruct a polyomal of ts best appromato P [2]. Therefore, the decso of appromato problems usually reduce to the fdg of the geeralzed polyomal P c ϕ, 4 whch dffers from P o ftesmal. For the fdg such appromate polyomal there are may appled methods. Amog these Remez s algorthm may respects s the most successful. Havg obtaed the geeral recogto, t s everywhere used practce for the appromate represetato of cotuous fuctos by polyomals. Sce all multlayered eural etworks wth feedforward structure fally duce the geeralzed polyomal type of 4 that, obvously, durg etwork trag t s meagful to use Remez s algorthm, whch cotrast to tradtoal methods allows to optmze the eural etwork up to uversal appromator for ay cotuous fucto o all feld of ts defto. Let's cosder the prcple of ths algorthm the combato to algorthm "error backpropagato" o a eample of cotuous fucto appromato of oe varable by the o-recurret eural etwork wth oe hdde olear layer. pots by formula Let f C[ a, ad o the segmet [ a, let s select the system of + 2 varous a a + + < <... < < b 5 2 π b a cos, +, + + Thkg y f let s create the trag set {, } y. 6 o the base of system 5. The o ts bass havg appled to a eural etwork the algorthm "error backpropagato" oe ca costruct the eural etwork as the best appromator of the fucto f the form of where,, N ω ϕ ξ θ, 7 ω ξ θ are et parameters optmzed o the pots of system 5. Takg respose that f N cost,, + let s suppose that E, f N r, r r ma f N E. a b 8 629
4 As s kow, sze E f of the best appromato of the fucto f o all segmet [ a, always ot less the best appromato E of ths fucto o the system,.e. E E f. O the other had, f t has appeared that eural etwork would be the best fucto appromator, ad ths process by that would be fshed. Therefore, let's beleve that f E E E <. 9 Further t s ecessary replace system of pots 5 by system a < <... < < + b 5 so that followg codtos were satsfed: sg r sg r ; r E ; ma r E. 2 2 O the bass of pots system 5 let s create trag set {, } y 2 f 2 +, where y, ad applyg algorthm «error backpropagato» for fucto r f N oe ca costruct the eural etwork π o system 5. The we ca assume that N N + π ; f N r π r ; , r E r E. 8 As well as the prevous case let s assume that 2 f E 2 f E E < for case of 2 E correspodg output N would be the best substtute for fucto f, so 2 Smlarly let s replace system 5 by system a 2 f E E E < < <... < < b 5 so that at all,, 2,..., + followg codtos are satsfed sg r sg r ; r E ; ma r 2 E. 2 After that t s ecessary create the eural etwork ducg output π as the best 2 2 appromator of the fucto r f N o pots system 5 ad let s assume N N + π ; f N r π r ; , r E r E 8 etc. whle we shall ot acheve for ay k realzato of the codto k f E E. All ths techology of trag of the chose eural etwork wth oe hdde layer for appromato 63
5 of cotuous fucto f ca be descrbed by teratve rato, whch essece rather easly reveals o the scheme preseted fg.. Fg. The scheme of the combed trag of eural etwork Thus, for ay [ a, the optmal sgal N k + beg the best appromator for correspodg value of fucto f s formed the output of the adder 6 fg. ad s defed by N k+ N k + π k k wth tal codto N. The output π s duced by the three-layer eural etwork 2 optmzed by algorthm "error backpropagato" o the bass of the trag set costructed o pots of system where k+ k+ k+ k+ a < <... < < b. 2 For each pot of the gve system should be satsfed followg three codtos: k k+ k k+ k k k sg r sg r ; r + k E ; k+ ma r + + k E, k k k f N r, r k k k r ma f N E a b k. E k, 3 I more detal let s cosder the scheme of the eural etwork combed trag. Sgal r k s formed the output of the block 3 ad after tradtoal trag through the comparso block 4 ad the trag block 5 the eural etwork 2 appromates t o pots 4 k system { } + +. After checkg the comparso block 8 codtos k E E f < E k 63
6 k through the block 9 replacemet of pots system { } k 2 by system { } + + s carred out + + so that codtos 3 were satsfed. To satsfy to these codtos, t s eough to replace k system { } oe pot by pot [ a, whch + + k+ k+ r E presece of such pot s caused by kow Veyershtrass s theorem, ad all the others to leave ot chaged k 2 ad obtaed system to accept as system { }. + + Process of replacemet ca be carred out as follows. If the pot s betwee two pots ad k + k + k+ + + of system { }, the by meas of t s ecessary to replace that from ts whch the dfferece r k + has the same sg as the pot. If the pot k+ + at the left of all pots from system { } k+ k+ k + ad sg r sg r k+ k+ k+ k+ ecessary to replace the pot by; f thus sg r sg r s, the t s, the as k 2 system { } t s ecessary to select the pots system + + k+ k+ k+,,,...,. I case of the pot k+ + s located at the rght of all pots of system { }, t s ecessary to act smlarly. Notce that practce t s desrable to replace o more pots of k system { } by ew pots oe of whch s + + so that thus, frst, all codtos 3 were k+ k+ 2 satsfed ad, secodly, that values r are wheever possble greater. 3. Appromato of cotuous fuctos wth usg of eural appromato kerel Earler we have establshed the cocept of eural appromato kerel whch uder the characterstcs creates "favourable" appromato evromet for the best appromate of cotuous fuctos by feedforward eural etworks wth sgle olear hdde layer. Ratoalty of gve approach let s cosder o a eample of appromato of the fucto depedg o oe varable. For costructo the eural appromato kerel let s choose a eural etwork wth oe olear hdde layer Fg. 2, where all output coectos weghts are equal to, ad other parameters ξ, put coectos weghts, θ, thresholds olear euros from the hdde layer, are select as ay real umbers. As s kow, all polyomal appromato kerels by meas of whch, as a rule, are proved drect appromato theorems 632
7 of the fuctos theory, at ay argumet are detcally equal. As a eample of such kerels oe ca to pot to the bomal seres C k k k, [,] 5 o the bass of whch for cotuous fucto s formulated Berste's appromato polyomal [4] B k k k C f. 6 Fg. 2 Neural Network appromato kerel Uderstadg that for ay put sgal t s practcally mpossble to obta the sgal the eural etwork output let s act as follows. Start from the pots system 5 ad costructed o ts bass the trag set {,} let s tra the gve eural etwork by the scheme offered fg.. As a result of trag oe ca determe such optmum values ξ,, for whch there s a ftesmal ε θ such as for ay [ a, wll be fulfl the equalty > + where et < ε, 7 et ϕ ξ θ 8 s the polyomal duced by eural etwork wth sgmod actvatos fuctos to the olear euros from the hdde layer type of ϕ ξ θ e. 9 Obtaed eural etwork wth parameters ξ,, we shall cosder as a eural appromato kerel. O ts bass ad replacemet the uty weghts of output coectos by ay real umbers ω, s possble to costruct the eural etwork capable to appromate ay cotuous fucto offered a tabular kd. Advatage of the gve approach cossts that trag ths etwork s lmted to trag oly weghts of output θ 633
8 coectos ω, at presece of advace optmzed other parameters creatg "favourable appromato evromet". Really, t s vsble from followg reasog. Let cotuous fucto f ad eural appromato kerel 8 s set o the segmet [ a,. The for the eural etwork type of ω ϕ ξ θ, 2 usg a equalty 3, lmtato of sgmod fuctos from above by ad belevg M ma f, we have: [, a θ f f ϕ ξ f ω ϕ ξ θ + + f ϕ ξ θ ω ϕ ξ θ θ + f ω ϕ ξ θ M ε + f ϕ ξ f ω. Thus, to obta performace of Veyershtrass s appromato theorem for the eural etwork 2 t s eough to optmze weghts ω, so that [ a, t s carred out f ω. 2 I the cocluso let s ote that optmzato of weghts ω, ca be made both tradtoal ways wth applcato "error backpropagato" ad usg the scheme of trag as dcated above. Refereces [] AHIEZER N.I. Lektse po teor appromats. Nauka, 965 Russa. [2] CHEN, H., CHEN, T. Appromatos of Cotuous Fuctoals by Neural Networks wth Applcato to Dyamc Systems. IEEE Trasactos o Neural Networks, vol.4, No.6, November 993, pp [3] DZJADYK V.K. Vvedee v teoru ravomeroqo prblea fuktsy polomam,. Nauka, 977, 52 str. Russa. [4] FUNAHASHI, K. O the appromate realzato of cotuous mappgs by eural etworks. Neural Networks, vol.2, pp ,
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