Convergence of the gradient algorithm for linear regression models in the continuous and discrete time cases

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1 Convergence of he gradien algorihm for linear regreion model in he coninuou and dicree ime cae Lauren Praly To cie hi verion: Lauren Praly. Convergence of he gradien algorihm for linear regreion model in he coninuou and dicree ime cae. [Reearch Repor] PSL Reearch Univeriy; Mine PariTech. 07. <hal v3> HAL Id: hal hp://hal.archive-ouvere.fr/hal v3 Submied on 5 Feb 07 v3, la revied 8 May 08 v4 HAL i a muli-diciplinary open acce archive for he depoi and dieminaion of cienific reearch documen, wheher hey are publihed or no. The documen may come from eaching and reearch iniuion in France or abroad, or from public or privae reearch cener. L archive ouvere pluridiciplinaire HAL, e deinée au dépô e à la diffuion de documen cienifique de niveau recherche, publié ou non, émanan de éabliemen d eneignemen e de recherche françai ou éranger, de laboraoire public ou privé.

2 Convergence of he gradien algorihm for linear regreion model in he coninuou and dicree ime cae Lauren Praly Sared December 6, 06, Revied January 0, 067 verion : 3 Abrac We eablih convergence o zero of he oluion of θ = φφ θ or θ = θ φφ θ under a poibly vanihing perien exciaion condiion. Coninuou ime cae Given a coninuou vecor funcion R φ R n, we le T, be he raniion marix aociaed wih he non auonomou differenial equaion i.e. aifying θ = φφ θ T, = φφ T,, T, = I where denoe he parial derivaive wih repec o he fir argumen, here. We are inereed in ufficien condiion implying lim T, = lim + up T, x + x x = 0. MINES PariTech, PSL Reearch Univeriy, CAS - Cenre auomaique e yème, 35 rue S Honoré Fonainebleau, France Abou he moohne of φ we need only he differenial equaion o have oluion; o, ince we have boundedne, meaurabiliy i ufficien. We need alo o be able o change he order of inegraion. The funcion funcion φ i ofen obained from a pre-proceing from a raw funcion ay Φ. I may be Φ φ = γ + Φ or φ = γ Φ r, ṙ = λr + Φ.

3 Le x be an arbirary conan vecor. We have T, x = φ T, x By defining he funcion ˆf a ˆf = φ T, x, ˆf = φ x and inegraing, we ge T, x = x On anoher hand, by inegraing, we ge ˆf r dr 3 T, = I φrφr Tr, dr Incorporaing hi in he definiion of ˆf yield where we have le I follow ha we have 3 ˆf = φ T, x = φ I = φ x = f fu ˆf u = φrφr Tr, dr x φ φr φr Tr, x dr φ φr ˆf rdr 4 f = φ x. φu u φr ˆf rdr u φu φr ˆf rdr u u φu φr dr fu ˆf u u u du φu φr dr fu du φu du ˆf r dr ˆf u du + ˆf r dr fu ˆf u du ˆf r dr du 3 Thee inequaliie would be in par equaliie if here exi funcion µ : R R and ν : R R n aifying, φu = µu u φr ˆf rdr = νu ˆf u u [, ].

4 [ ˆf u du + φu du ] + φu du ˆf r dr ˆf r dr 5 So we ge finally T, x = x x x x In oher word we have imply I ˆf r dr fu du + φu du [φu x] du + φu du φuφu du x + φu du T, T, I φuφu du 6 + φu du Remark. Up o 4, we have ideniie. The conervaivene we may have in hi la inequaliy i only in he majoraion obained in 5. See foonoe 3.. An upperbound for T, T, can be obained in a very imilar way bu aring from T, r = T, rφrφr, inegraing in r backward from o and uing inead of ˆf r. ˆf r = x T, rφr, ˆf = x φ 3. The ep ued up o 4 for he differenial equaion can alo be ued for : θ = ψφ θ a we have for example in he lea quare algorihm. 3

5 4. Wih adaping he argumen ued in he proof of he claim p.369 of Appendix B. of [3], i hould be poible o exend he above reul o he yem η = Aη + Bφ θ, θ = φcη where he riple B, A, C i ricly poiive real. Dicree ime cae Given a equence of vecor φ R n bounded 4 in norm by φ, we le T, be he raniion marix aociaed wih he non auonomou dicree ime yem θ = I φφ θ i.e. aifying T, = I φφ T,, T, = I Our problem i o find ufficien condiion implying lim T, = lim + up T, x + x x = 0. Le x be an arbirary uni vecor. We have T, x = T, x φφ T, x Wih denoing, for +, ˆf = φ T, x, ˆf + = φ + x we ge T, x = T, x φ ˆf T, x φ ˆf = = T, x x T, φ ˆf φ φ ˆf T, x φ φ ˆf. T +, x = x φ + φ + ˆf + So ummaion give T, x = x r=+ φr φr ˆf r 7 4 See foonoe 4

6 On anoher hand, we have T, = T, φφ T, T, = T, φ φ T,. T +, = I φ + φ + T, So again ummaion give T, = I r=+ φrφr Tr, Incorporaing hi in he expreion of ˆf yield, for +, ˆf = φ T, x = φ I = φ x = f r=+ r=+ r=+ φrφr Tr, φ φr φr Tr, x φ φr ˆf r +, x where we have le We have alo f = φ x 8 ˆf + = f + Wih he Cauchy-Schwarz inequaliy we obain u fu + [φu φr] r=+ f + = ˆf + u + φu φv v=+ u r=+ u r=+ ˆf r ˆf r u + and herefore 5

7 fu u=+ ˆf u= u + φu φv u=+ v=+ r=+ u=r u r=+ ˆf r u + φu φv ˆf + v=+ u + + φu φv u=+ u=+ u=+ r=+ u=r v=+ φu φv ˆf + v=+ + r + + φu φu v=+ φu φv r=+ ˆf r r=+ v=+ ˆf r φv ˆf r ˆf r 9 Wih he definiion 8 of fu hi inequaliy i x φuφu x + φu u=+ u=+ r=+ ˆf r Wih 7, i allow u o obain he following upperbound for T, x { T, x x min φr φr } ˆf r r {+,...,} r=+ { min φr φr } x r {+,...,} x φuφu x + φu u=+ u=+ In oher word we have T, T, I { min } φr r {+,...,} + φu u=+ u=+ φuφu. 0 6

8 Remark The ame final remark a for he coninuou ime cae can be done here. In paricular abou he majoraion 9, a le conervaive bound i obained in he proof of [, Theorem 4.5] or of [5, Theorem., column, p. 05], for he cae where he preproceing menioned a he beginning of hi ecion i φ = Φ r, r = r + Φ and he above analyi in carried ou exploiing he aumpion ha he equence r goe o infiniy in ome pecific way. 3 Convergence for he dicree and coninuou ime cae Le i be ricly poiive real number going o + wih 0 = 0. For any, here exi τ uch ha i beween τ and +τ. We have een ha boh in he dicree and coninuou ime cae, here exi a real number π i in [0, ] aifying T i, i π i Specifically, ince a a in he coninuou ime cae, 6 give i π i λ min φuφu du i i + φu du i Wih denoing φ i = eup [i, i ] φ, a more conervaive lowerbound for π i i π i λ min i i φuφu du + i i φ4 i Alo, in [], he following aumpion i inroduced: exi a funcion T uch ha we have, for all 0 There exi α and β uch ha here α I T φrφr dr β I. Since we have T T T φr dr = race φrφr dr nλ max φrφr du The above aumpion implie ha,by leing i = T i 0, 7

9 we have, for all i i α λ min φrφr dr i, i i φr dr n β and herefore π i = α + n β in he dicree ime cae, wih φ maller han, 0 give { min } φr r {+ i,..., i } π i i i i + φu u=+ i λ min i φuφu 3 + i or he more conervaive lower bound π i = φ i i + i i φ 4 λ i min φuφu + i So we have T, 0 = τ T, τ T i, i T, τ τ T i, i τ π i τ exp π i where o obain he la inequaliy we have ued he propery x exp x x R. We conclude ha T, 0 end o 0 if we can find T and he i uch ha we ge π i = +. Dicuion : To how he inere of hi reul, we compare i wih he perien exciaion panning condiion. We do hi here for he coninuou ime cae only, bu he ame hold for he dicree ime cae. 8

10 The vecor funcion φ i aid perienly exciing or panning if here exi wo ricly poiive real number ε and T uch ha, for any, he Gram marix ime window wih widh T i above he level ε, i.e. +T λ min φφ d ε 0. +T φφ d on a I i eablihed inn [4] ha hi condiion i neceary and ufficien o have he uniform aympoic abiliy of he origin for The condiion of non ummabiliy of π i above implie aracivene bu no uniform aracivene. I can be een weaker han he perien exciaion or panning condiion in wo way: he level ε may decreae wih, he widh T of he ime window may increae wih. Specifically, le T be fixed and le ε i be he level reached by he Gram marix on he ih ime window [i T, it ], i.e. ε i = λ min it i T φφ d hen, wih, π i i no ummable if ε i i no, i.e. it λ min i T φφ d = +. Now, le ε be fixed and, wih 0 = 0, le i be he malle ime uch ha he Gram marix on he ime window [ i, ] i larger han he level ε, i.e. hen, wih, π i i no ummable if Reference i = min : λ min i φφ d ε + i i = +. [] D. Aeyel, R. Sepulchre. On he convergence of a ime-varian linear differenial equaion ariing in idenificaion. Kyberneika 994 Vol. 30, N. 6, pp [] H.-F. Chen, L. Guo. Idenificaion and ochaic adapive conrol. Springer Science+Buine, LLC 99. [3] R. Marino, P. Tomei. Nonlinear conrol deign, geomeric, adapive and robu. Prenice Hall

11 [4] A. Morgan, K. Narendra. On he uniform aympoic abiliy of cerain linear nonauonomou differenial equaion. SIAM J. Conrol and Opimizaion Vol. 5, No., January 977 [5] W. Ren, P. Kumar. Sochaic adapive predicion and model reference conrol. IEEE Tranacion on Auomaic Conrol, Vol. 39. No. 0. ocober Hiory of he verion Modificaion on January 0, 07 Addiion of he reference [] uggeed by Romeo Orega and of he commen on how π i can be choen conan when he aumpion propoed in ha paper hold. Modificaion on January 5, 07 Addiion of he dicuion on he relaion beween non ummabiliy of π i and he perien exciaion panning condiion. Modificaion on January 3, 07 The lower bound and 3 have been changed o follow a uggeion of Romeo Orega of giving le conervaive lower bound for π i. Before hey were, wih i+ i T, for he coninuou ime cae, π i = + T φ 4 λ min i φuφu du i for he dicree ime cae π i = φ T + T φ 4 λ i min φuφu + i 0