Learning for Cognitive Wireless Users

Transcription

1 Learnng for Cognve Wreless Users Shor Paper Y Su and Mhaela van der Schaar Dep. of Elecrcal Engneerng UCLA {ysu haela}@ee.ucla.edu Absrac Ths paper sudes he value of learnng for cognve ranscevers n dynac wreless neworks. We quanfy he uly proveen ha can be obaned by a wdeband cognve user whch learns he saonary usage paern of he specru occuped by narrowband users and based on hs nforaon adap s ranssson. Specfcally we nvesgae he rade-off beween he learnng duraon and he achevable perforance n saonary envronens. We apply opzaon and large devaons heory o analycally derve an upper bound of he nu requred learnng duraon gven he user s olerable perforance loss and ouage probably. Furherore nocng ha learnng echnques requre he nforaon feedback of he specru usage paern beween he ranscevers we nvesgae how he cognve user can furher prove s perforance by akng accoun of s feedback delay. The pac of naccurae delay esaon on he achevable perforance s quanfed. Keywords- learnng feedback delay cognve wreless users I. INTODUCTION A prosng way of provng rado specru ulzaon s o buld cognve wreless devces ha can benef fro he opporunsc deployen of unused specral opporunes fro varous frequency bands [1]. Whle concepually sple he realzaon of cognve wreless devces s hghly challengng. Several probles us be solved: sensng over wde frequency bands; denfyng avalable specru opporunes; explong he denfed ranssson opporunes ec. In parcular a cognve wreless devce should be able o learn fro he envronen and adap s nernal saes o sascal varaons n he ncong F sul by akng correspondng changes n ceran operang paraeers (e.g. rans-power carrerfrequency and odulaon sraegy) n real-e [1]. Learnng echnques have already been deployed o prove he perforance of a broad class of wred and wreless councaons syses. They enable he dynacally neracng councaons devces o acqure nforaon buld knowledge and ulaely prove her perforance [3]-[5]. As opposed o he prevous works whch focus on sudyng eher he long-er convergence behavor of ceran learnng algorhs [3][4] or deerne he operaonal shorer-er perforance whou provdng any perforance guaranees [5] hs paper as o characerze and analycally quanfy he achevable perforance ha can be obaned by cognve wreless users wh learnng capables. We sudy how uch a cognve devce wh no pror knowledge should learn abou s envronen e.g. e-varyng nerference o reach s perforance requreen. Parcularly f he envronen s saonary we explcly quanfy he benefs ha a user can derve n ers of s proved uly by learnng for a longer Ths work was sponsored by NSF CAEE award no and an ON gran. duraon.e. based on a larger nuber of observaons abou he envronen. We apply opzaon and large devaons heory o derve an upper bound of he nu observaon duraon gven he perforance guaranee desred by he user. Then nocng ha he nforaon requred for cognve devces o perfor learnng s gahered hrough he nforaon feedback fro he recever o he ranser and hs nforaon can be delayed durng hs process we sudy how a cognve devce can prove s perforance f he feedback delay s accuraely known. We also quanfy he pac of perfec delay easureens on he acheved perforance. The res of he paper s organzed as follows. Secon II presens he syse odel and forulaes he proble of learnng and adapng o he specru usage paern. Secon III analycally derves an upper bound of he nu requred learnng duraon. Secon IV shows he nuercal resuls Secon V quanfes he pac of specru usage nforaon feedback delay. Conclusons are drawn n Secon VI. A. Syse Descrpon II. SYSTEM MODEL Fg. 1. Invesgaed cognve wreless neworks. We assue a cognve wreless syse slar wh he one suded n [2] (see Fg. 1). The oal nuber of frequency channels n he syse s N and each has a bandwdh of B. The ajory of rado devces n hs syse are narrowband users. These devces can dynacally ulze he dle specru bands by enablng carrer frequency swchng and packng all he acve rados ghly n he specral doan [2]. An exaple s gven n Fg. 1. If one devce releases f 2 he devce occupyng f 4 wll swch o f 2. The syse sae s defned as he nuber of occuped frequency channels n. Narrowband rados ener and ex he syse ndependenly followng Posson dsrbuons. The specru usage paern can be capured as a connuous e Markov chan wh nfnesal generaor [2][6][7] λ1 λ1 µ 1 ( λ 2 µ 1) λ + 2. (1) Q = λ N µ N µ N The Markov chan odel and s correspondng nfnesal generaor Q can ake varous fors based on he confguraon /8/$ IEEE

2 of he consdered wreless nework. Denoe he seady sae probably vecor of he specru usage paern as π = [ π π1 πn ] n whch π represens he probably of havng acve narrowband devces n he syse. No aer wha for he nfnesal generaor Q akes we always have π Q =. As shown n Fg. 1 we also consder a wdeband devce n he syse whch can rans over all N frequency channels. The nose power a frequency band s N and s channel gan s h. Each acve narrowband devce causes an nerference power of I o he wdeband recever. The wdeband devce s subjeced o a oal power consran of P. Denoe he power vecor across all frequency bands P = [ P ] T 1 PN n whch P s he power allocaed n frequency band. Hence he achevable rae s gven by N N hp n < hp ( π P) = πnblog 1+ + πnblog 1+ N I + N.(2) = 1 n n= B. Learnng Duraon and Perforance Fg. 2 shows hs learnng process n whch he wdeband recever perodcally senses he specru and feeds back o s ranser he nuber of nerferng narrowband devces n a e. Specfcally he wdeband devce odels s envronen by sply counng he nuber of acve narrowband devces encounered n he pas and approxang he saonary specru usage paern π by he observed frequences of he syse saes. We defne an eprcal frequency funcon ( n ) c N ( n ) c ( n ) n= = (3) where c ( n ) s a counng funcon 1 sasfyng c ( n ) = n { 1 N} and c 1 c ( n) + 1 f n = n n = (4) 1 c ( n) oherwse. ( ) The wdeband user approxaes he seady sae π usng he eprcal frequency funcon and akes he bes P P = response acon P ( ) ha zes ( ).e. ( ) ( P) arg wh 1 = [ 1 1] T. Denoe he achevable P T 1 P rae when he wdeband user akes he bes response o he eprcal frequency funcon = π P. as a ( ) ( ( )) Throughou hs paper he learnng duraon refers o he nuber of avalable observed specru usage paerns over e for he wdeband user o updae ( n) and approxae he seady sae dsrbuon π. Ths paper as o deerne how any observaons are suffcen for a learnng user o reach a ceran desrable perforance guaranee. Specfcally gven he olerable perforance loss wh respec o perfecly knowng π and he ouage probably δ we wan 1 Noe ha here we noralze he feedback perod and Secon II and III plcly assue ha hs perod s suffcenly large such ha adjacen saples of specru usage paern s ndependen of each oher. Secon V dscusses he opal adapaon sraeges for varous feedback delays and saplng nervals. o deerne he nu requred learnng duraon: ( ( π) ( ) ) n s.. Prob δ. (5) a a Undersandng hs proble s poran fro boh heorecal and praccal perspecves because due o he real-e adapaon requreen of cognve neworks [1] only led observaons are usually avalable o cognve users and s also necessary for he o undersand he basc rade-off of perforance vs. learnng duraon. Fg. 2. Specru usage feedback of he wdeband devce. III. MINIMUM EQUIED LEANING DUATION Alhough slar bounds exs n sascal learnng heory e.g. Hoeffdng's nequaly [8] s sll dffcul o solve he proble n (5) because hese bounds do no drecly apply o our consdered proble. However we can fnd an upper bound for he soluon of he proble n (5). For hs we adop ools fro large devaons heory [9]. Accordng o he large devaons heory he eprcal frequency funcon ( n) of a rando saple of sze drawn fro π sasfes + ( D ( ) ) δ Prob π δ 2 δ > (6) N where D( p q ) s he Kullback-Lebler (KL) dsance beween px ( ) and q( x. ) The basc dea n deernng an upper bound s o fnd a value of δ such ha D ( π ) δ always leads o + a ( π) a ( ). By seng o sasfy δ 2 δ N we have Prob( D( π) δ) Prob ( a ( π) a ( ) ) and hs value provdes an upper bound for he proble n (5). The whole procedure s dvded no he followng hree seps. A. Exree Pons wh Perforance Loss Consrans Frs n he probably splex Ω = { 1 T = 1 } we consruc a convex se B ha conans he acual pf π. Le A = {{ k j} : k j { 1 N} and k < j} whch conans a + 1 oal nuber of M = cobnaons of any wo dfferen 2 negers n { 1 2 N }. Le ( S ) denoe he h eleen of se S. Based on he olerable perforance loss we choose 2M pfs and vew he as exree pons of he se B n whch we wll derve an upper bound of he nu requred learnng duraon. For = 1 2 2M he 2M pfs ha we are neresed n sasfy:

3 (P1) Ω; (P2) ( ) n = πn f n ( ). A (P2) ensures ha hese pfs have only wo eleens dfferen fro saonary dsrbuon π. The pfs sasfyng (P1) and (P2) can be rewren as ( n δ ) defned by ( ) πn δ f n = ( A) 1 ( n δ) = πn + δ f n = (( A ) ) = 12 2M. (7) 2 πn f n ( A) Denoe ( δ) = ( δ) ( N δ). We can choose he exree pons by seng he paraeer δ based on he olerable perforance loss. For = 1 2 M δ πl wh l = (( A ) ) f S 1 δ = = (8) n δ Sδ oherwse { } n whch Sδ = δ : a ( π) a ( ( δ) ) and δ and δ + M πl wh l = (( A ) ) f S 2 δ = = (9) n δ S δ oherwse { } n whch S δ = δ : a ( π) a ( ( δ) ) and δ. Due o he non-negave propery n (P1) when n ( A) f S δ = or S δ = we se ( n δ ) o be zero o ensure he perforance loss s as close o as possble. If S δ or S δ he exree pons are he pfs ha cause an exac perforance loss of. Usng he convex hull of he above 2M exree pons we consruc a convex se B whn whch o derve an upper bound of he nu requred learnng duraon n (5).e 2M 2M B = : = α ( δ) α and α = 1. (1) = 1 = 1 Proposon 1 (Sasfacon of Perforance Loss Consrans): Any B sasfes ( π) ( ). a a Proof: The proof s gven n [14]. Proposon 1 ensures ha any convex cobnaons of he exree pons sll sasfy he olerable perforance loss requreen whch enables us o apply opzaon heory o conver he erc of perforance loss no KL dsance δ n he followng sep. D n B. KL Dsance Mnzaon n Convex Se We apply large devaons heory o ranslae he perforance loss no anoher erc he KL dsance δ D. The basc dea s o solve an opzaon proble o fnd he nu KL dsance δ D such ha for any ha sasfes n D ( π ) δd we have ( ) ( ) n a π a. Parcularly he opzaon proble can be forulaed as n D( π) s.. S ( B ) (11) where S ( B ) s he surface of se B.e. S ( B ) = B \n( B ). Here we denoe he neror of se B as n( B ) [1]. Noe ha he KL dsance D ( π ) s convex n he par ( π ) and S ( B ) s a lnear consran. Therefore he proble n (11) essenally belongs o convex prograng and he opal soluon can be obaned effcenly by solvng he opzaon proble for each polyhedron on he boundary S ( B ) [11]. Because he convex cobnaons of he exree pons n B cover he adjacen regon of π δ D s suffcen n o ensure ( π) ( ). a a C. Mnu Learnng Duraon Calculaon The second sep shows ha D ( π ) δd leads o n a ( π) a ( ). Hence an upper bound of he soluon o he proble n (5) can be obaned by solvng ( ( π) D ) n s.. Prob D δ δ. (12) Applyng forula (6) fro large devaons heory we have he followng proposon: Proposon 2 (An Upper Bound): Suppose he wdeband devce updaes s eprcal frequency funcon and akes he bes-response acon wh respec o. An upper bound T of he soluon of proble (5) s ( δd δ) T = Mn_ N (13) y + n whch + Mn _ ( x y z) n : and 2 x = z Z y. Proof: I can be proved by cobnng (6) and (12). We consder a cognve syse wh N = 2 λ 1 = µ 2 = 2 λ 2 = µ 1 = 1 and he power consran of he wdeband devce s P = 1. Is channel condons and he power of nose and nerference are gven by h 1 = 2 h2 = 1 N1 = N2 = I = 1. I s easy o solve ha he saonary dsrbuon s π = We se he paraeers n he proble (5) o 2.5 be = 1 and 2 δ = 1. Fg. 3 llusraes he procedure of obanng he upper bound. Nong ha N = 2 we choose sx exree pons 1 6 n oal whch are deerned based on rae-pf curves ncludng () =.25 (1) =.5 and (2) =.25.e. () + (1) =.75. The convex hull of hese exree pons 1 6 s he exree pon se B. The dashed hexagon n Fg. 3 s he surface S ( B ) on whch we nze he KL dsance. Solvng he convex opzaon proble (11) leads o δ D = Usng (13) we oban n 2 ha T = Mn_ ( ) = 161. As shown n Fg. 3 f he learnng duraon s larger han T he KL dsance beween he acual saonary dsrbuon π and observed eprcal frequency funcon wll le whn he sold crcle wh an ouage probably less han δ. n n

4 () (.23.52) 3 ( ) D( π)=δ Dn 4 (.5.5) π (.5.25) 6 ( ).1 1 (.5) 2 ( ) (1) Fg. 3. KL dsance nzaon n S ( B ). IV. IMPACT OF FEEDBACK DELAY d n d Fg. 4. Feedback delay of he specru usage. Ths secon dscusses he pac of he feedback delay of specru usage nforaon whch causes he receved nforaon ou of dae and degrades he perforance. The feedback delay exss due o several reasons e.g. wreless propagaon sgnal processng expense and proocol overhead. We denoe he feedback delay of he specru usage paern n fro he recever o he ranser as d. As shown n Fg. 4 he specru usage paern ha he ranser receves a e s he usage paern he recever experenced a e d. Sj Defne he ranson probably arx S ( ) n whch () s he probably ha a Markov process s n sae j a e gven ha s n sae a e. Based on he sochasc process heory [7] we know ha S ( ) s he soluon of he Kologorov equaon whch akes he for of N + 1 () S = ve ξ ω (14) = 1 n whch ξ 1 ξ 2 ξ N + 1 are he N + 1 dsnc egenvalues of arx Q and v 1 v 2 v N + 1 and ω1 ω 2 ω N + 1 are he correspondng rgh and lef egenvecors of arx Q. In parcular he arx Q for he consdered Markov process has an egenvalue ξ 1 = wh he correspondng rgh and lef T egenvecors v 1 = [11 1] and ω1 = π. All he oher egenvalues ξ2 ξ N + 1 of Q have srcly negave real pars. d n Gven he laes feedback he opzaon of power allocaon a he ranser s convered no P T 1 P d ( π n ) P (15) n whch π = π π1 π N s he probably vecor of he d specru usage paern n wh π n = S ( d ) d ( n n n ) = Pr =. Fro (14) we have l n d n n S() = v 1ω. (16) + 1 d n Therefore f d + regardless of we always have d π ω1 = π. As a resul ( π P n ) n (15) s reduced o ( π P) n equaon (2). We can conclude ha learnng he saonary dsrbuon π of frequency usage paern and opzng he power allocaon wh respec o hs dsrbuon s opal only when he feedback delay s large. On he oher hand we also consder he led feedback delay scenaros. Noe ha n hese cases he bes sraegy s no o learn he saonary dsrbuon and he ranser needs o d explore he elness of he feedback nforaon n because π n (15) s a funcon of he led feedback delay d d. In parcular ( π P n ) n he opal ranssson sraegy of (15) wll becoe: N N d ( π P n ) = ( S d ( d ) ) d ( ) : P = n S d n n = 1 n (17) n hp < hp Blog 1 S d ( d ) + + log 1 n n B + N I + N n= where S: ( d ) represens he h row of S ( d ). The proble s convered no how o accuraely esae S ( ) a = d. Due o he perodc naure of he feedback nforaon n he wdeband devce s able o saple he ranson probably arx S ( ) a = 2 by updang eprcal frequency funcons 2 and use nuercal algorhs such as curve fng o esae S ( ) for non-neger ulples of. As long as he envronen s saonary and he saplng daa s large enough he wdeband devce can esae S ( ) accuraely. Now we nvesgae he pac of perfec esaon of he feedback delay d. Praccal ehods of easurng he feedback can be found n [12]. Denoe d he esae ha he wdeband devce has abou he feedback delay d. The per- d of perfec esaon d s forance degradaon ( ) N ( ) ( ( ) ( ) ) ( ) π : ( ) ( : ( ) ) : P P ( : ).(18) d = S d S d S d S d = We derve an upper bound of hs perforance degradaon based on Markov chan heory and forally sae as follows. Theore 1: The perforance degradaon d ( ) depends on 2 Ths secon assues he saplng perod s uch saller han he xng e of he Markov chan such ha he adjacen saples are no ndependen of each oher. d

5 wo ers d d and n ( d d) and s bounded as n ( d d ) ( ) α( ) d d d e β (19) n whch α() s a non-negave funcon sasfyng α ( ) = and l + α() exss and β >. The proof can be found n [14]. Two key observaons can be ade fro he above heore. Frs s sraghforward o see ha he perforance loss s a funcon of d d and he perforance loss s zero f d = d. Ths can be nerpreed as he shor er behavor of he perfec esaon. More poranly he heore ndcaes ha he perforance loss decreases a leas exponenally wh n ( d d) whch conrols he long-er behavor of he perfec esaon. Ths resul quanfes he sgnfcance of he elness of he nforaon feedback. Besdes he exsence of l + α () ples ha nfne esaon error of he feedback delay causes bounded perforance loss. Wh he ncrease of n ( d d) he effec of naccurae esaon of he delay d over he perforance dnshes a leas exponenally. perforance loss (d' ) perforance loss log( (d' )) d = d' Fg. 5. Perforance loss of naccurae esae over d d -d' = d' Fg. 6. Perforance loss of naccurae esae for fxed d d. We nuercally show he proveen of easurng he feedback delay d. Consder an exaple wh he paraeers N = 2 λ 1 = µ 2 =.2 and λ 2 = µ 1 =.1. The feedback delay d s assued o be 2 and he perforance loss d ( ) s ploed n Fg. 5. We can see ha agrees wh he arguen ha α ( ) = and l + α() exss n Theore 1. Copared wh akng bes response o saonary dsrbuon π perfecly knowng he value of feedback delay can ncrease he achevable rae by 3.5%. We vary d whle fxng d d o be 1 and plo he correspondng d ( ) n Fg. 6. We can see ha he perforance loss d ( ) decrease exponenally wh d whch coples wh Theore 1. V. CONCLUSIONS Ths paper sudes he nu requred observaons a wdeband user should have n order o learn abou he saonary probably dsrbuon of s experenced envronen gven he requred perforance guaranee. The derved resuls provde several nsghs for undersandng he basc rade-off ha can be ade n councaon syses beween he learnng duraon and he achevable perforance. We also consder he pac of nforaon feedback delay and quanfy he perforance loss for perfec esaon of he delay. Such nsghs are poran for desgnng and evaluang fuure councaons proocols wh learnng capables such ha engneers can buld praccal syses wh desred coplexy and perforance rade-off. EFEENCES [1] S. Haykn "Cognve rado: bran-epowered wreless councaons" IEEE JSAC. vol. 23 pp Feb. 25. [2] Y. Xng. Chandraoul S. Mangold and S. Shankar Dynac Specru Access n Open Specru Wreless Neworks IEEE JSAC Specal ssue on 4G Wreless Syses vol. 24 pp Mar. 26. [3] E. Fredan and S. Shenker. "Learnng and Ipleenaon on he Inerne." Manuscrp. New Brunswck: ugers Unversy Deparen of Econocs hp://ceseer.s.psu.edu/erc98learnng.hl [4] C. Pandana and K.J.. Lu "Near Opal enforceen Learnng Fraework for Energy-Aware Wreless Sensor Councaons" IEEE JSAC vol. 23 no 4 pp Apr. 25. [5] F. Fu and M. van der Schaar "Dynac Specru Sharng Usng Learnng for Delay-Sensve Applcaons" Proc. ICC 28 o appear [6] X.. Zhu L.F. Shen and T. S. Yu "Analyss of Cognve ado Specru Access wh Opal Channel eservaon" IEEE Cou. Leer vol. 11 No. 4 pp. 1-3 Aprl 27 [7].G. Gallager Dscree Sochasc Processes Sprnger 1995 [8] O. Bousque S. Boucheron and G. Lugos Inroducon o Sascal Learnng Theory Advanced Lecures on Machne Learnng Lecure Noes n Arfcal Inellgence vol pp Sprnger 24 [9] I. Csszár and P. C. Shelds Inforaon heory and sascs: a uoral Cou. and Infor. Theory vol.1 Issue.4 pp [1] J. B. Conway A Course n Funconal Analyss 2nd edon Sprnger- Verlag [11] S. Boyd and L. Vandeerghe Convex Opzaon Cabrdge Unversy Press 24. [12] M. Kazanzds and M. Gerla "End-o-end versus Explc Feedback Measureen n Neworks" Proc. ISCC pp [13] J. S. osehal Markov chan convergence: Fro fne o nfne Sochasc Processes and her Applcaons vol. 62 pp [14] Y. Su and M. van der Schaar Mnu equred Learnng and Ipac of Inforaon Feedback Delay for Cognve Users UCLA Techncal epor 28