CONTROLLED LYAPUNOV-EXPONENTS IN OPTIMIZATION AND FINANCE

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1 { r CONTROLLED LYAPUNOVEXPONENTS IN OPTIMIZATION AND FINANCE L. Gerencsér, M. Rásonyi, Zs. Vágó Computer and Automation Institute of the Hungarian Academy of Sciences H1111 Budapest, Kende 117, Hungary. gerencser@sztaki.hu, rasonyi@sztaki.hu PPCU Department of Information Technology H1052 Budapest Piarista kz 1. vago@sztaki.hu Keyords: Random products, Lyapunov exponents, SPSA, groth rate, recursive estimation Astract Let e a stationary process of realvalued matrices, depending on some vectorvalued parameter, satisfying "!$#%# '&"(##*),+ for all. The toplyapunov exponent of is defined as./0%12 4 %5#%# 76 98;:=% 6 & #%# TopLyapunov exponents play a prominent role in randomization procedures for optimization, such as SPSA, and in finance, giving the grothrate of a selffinancing currencyportfolio ith a fixed strategy. We develop a convergent iterative procedure for the optimization of.. In the case hen is a Markovprocess, the proposed procedure is formally ithin the class defined in [1], hoever the general case requires fundamentally different techniques. 1 Random matrixproducts Let A 4 CBA A % e a stationary process of realvalued matrices over some proaility space ËDFAGHAJIK, satisfying! ## '&L#%#9)M+ (1) here %!KN denotes the positive part of N. It is ellknon (see [2]) that under the aove condition O12 exists. Here RS+ 4 F#%# 6 'L8;: P6 '&"## (2) is alloed. The folloing result is fundamental in multiplicative ergodic theory (see [2]): Theorem 1 Assume that the process > descried aove satisfies (1) and in addition it is ergodic. Then T almost surely 0%12 4 F#%# 6 'L8;: 6 '&"## () The numer, the exponential groth rate of the product #%# U6 98V:W%X6 & ##, is called the top Lyapunovexponent of the process YZ. X for reasons that ill ecome clear later. We also recall a part of Oseledec s theorem (see [8] and [6]) hich descries hat happens if e apply the aove random matrix products to a fixed vector. Theorem 2 Under the conditions of Theorem 1 there exists a suset D\[;]MD of proaility such that for all ^/ D[ there is a proper suspace _^`5]a S of fixed dimension such that for all c95de_f^` 1%2 4 Fg ;f^`h 'L8;:=^` 6(6i6 '&^`jc;gk Assume no that the matrices lma 4?BA common parameter, say, here onp] e, and n % depend on a is an open domain. is considered as a controlparameter that e can set freely. Thus the top Lyapunovexponent. ill e a function of, and ill e called a controlled Lyapunovexponent. The prolem that e consider in this paper is: 2'1%q " (4) A theoretical expression for can e otained as follos (cf. [2]). Let s t % : and define the normalized products u vs ## s ##. Then it can e shon that the process ẍu A! : is asymptotically stationary. Let y denote the stationary distriution of lz Aou :. Then e have S## z u/:p## y (5) Oviously this expression is not very useful for practical computations. 2 Minimization of the toplyapunov exponent In developing an iterative procedure for solving the aove minimization prolem an alternative expression for. ill play a key role. Let us define a } matrixvalued process u~zu A 4 0BA A as follos: u 0 ' 6 98V: % 6 '& #%# 6 98V: %6 '&"## (6) assuming that the denominator is not zero. In the latter case e rite u ~B. Oviously, um ẍu can e defined recursively as follos: u! :0 '! :(u ##! :(u V## (7)

2 B ith initial condition u &SM '& Bƒ~B. It is easily seen that %S#%# 76 98;:=% 6 & #%# Thus Theorem 1 implies T almost surely. M% V: & L8;: & ## 7&9#%#, and the convention that S##! : u #%# ˆ S## & #%# F#%#! : u ## (8) To compute the gradient of ith respect to consider first the expression ith Š Œ fixed. Let ẍu.žjj AŽ ZB e a smooth curve in S ŒX ith u.b ukaf u.b" u such that u YB. Then at ŽUYB e have, using ## u.žj(##` ufu5 ; Š :J z, that Ž #%# u ŽJi#%# equals š ufu 8;: z 6 u5u u u Using the identities o o5œ and oœ, e get Ž ## u ŽJi#%# usu } (9) and u e interchanged: let u5 ŒX e fixed and let.žj e a smooth curve in F ŒX ith ŠB"`a No let the role of such that už MB. Proceed as aove, and note that, in analogy ith (9) e have Ž ##.ŽJuK#%# Thus e finally arrive at the folloing result: ƒ usu (10) Lemma 1 Let.ŽJ A u.žj AJŽU ŸB e smooth curves in 5Œ, ith.b"kz ŠAou.Bk uka' ËB5 ˆAF u.bs u, such that už MB. Then at Ž MB e have Ž ## ŽJ u.žj(## ## u ## u5u }ufu (11) Let us no consider the case here here Uˆ ';. is a is an smooth function of, as aove, i.e. on?]ž, and n open domain. Assume that lm is nonsingular for all 4 and all on. Thus e get a elldefined sequence ẍu// u V, and for all 4 u ;. is a smooth function of. Let P for some A % A. e a fixed coordinate direction and let us introduce the notations rhx ;. u rjx u ; Differentiating (8), and using Lemma 1 ith ª! :P., u«u e get, after formal derivation, the folloing expression for rj ž = : 98;: u rhx 12 4 u5 }! :! :/ rjx! :iu u5 Š! : & #%#! :(u ## z (12) Introduce the notations: _ A' ˆAoukA5 uk usu5 V Š V Ÿ }ufuk * Š * z _'. AJ rh A ua u rj _. AJ rh A ua u rj _ ŠAJ r A ua u r._l: i( A A _ (( J (1) It is assumed that the partial derivatives rjx! : are availale explicitly. On the other hand the partial derivatives u rhx ill e computed recursively, taking into account the recursive definition of u given in (7). For this purpose consider the mapping of ŒX H Œ into ŒX defined y. AouS O u ## u ## (14) assuming that u± ²B. To otain the derivative of ith respect to u let }kœ e fixed and let u ŽJ AJŽ «B e a smooth curve in ŒX ith u ËB' uaf u.b' u. Then at Ž ~B e have Ž A u.žjj u Taking into account (9) e get Ž. Aou ŽJ u No interchanging the role of Ž ŽJ A u5 }u ## uk## u u ## u ## ³ z Ž and u e get u ³ Thus e arrive at the folloing result: #%# u ŽJi#%# ` ufu (15) F ufu (16) Lemma 2 Let ŽJ A u.žj AŽµ B e smooth curves in S ŒX ith.b"s, A u ËBF«uA'.B"S ˆAS u ËBƒ už 0B. Then at Ž 0B e have Ž u u ³U Thus e can rite Ž = u ` ufu V u F }usu h¹$ u such that»º; A ua' ˆAF us (17) ŽJ A u.žjj Mºm. AoukA7 AF u5 (18) Applying the aove notations e can express the derivatives u r. in a recursive manner for any as follos: u rhx! :Oº;! :WAou ma rhx! :WAou rh (19)

3 š ¾ A The iterative scheme. Assume, that at time 4 e have at our disposal the latest estimator P and the matrices maj u ma u r(. Oserve l! :! :== and r! :\M r(! :W.=. Then set u! :p! :(u ##! : u V## u rj! :p ºm.! :WA u ¼AJ rjx! :=Aou rjx L! : AJ r(! : Aou A u r( =! :p = 4 _' (20) An important technical tool is enforced oundedness hich is achieved y resetting: if! : ould leave a fixed compact domain then e reset to its initial value. The algorithm formally falls ithin the class of recursive estimation methods descried in [1] if is a Markovprocess, ut the application of the results of [1] is not straightforard. In particular, [1] does not consider the effect of resetting. The convergence analysis requires completely different tools if is not Markov. The first step is relatively easy: the extension of the ODEmethod to recursive estimation processes ith resetting, hen the correction term is strictly stationary (asymptotically) for each fixed. The hard part is to estalish uniform las of large numers ith respect to for sums defined in terms of the process.! : A u. Noisefree SPSA We consider the folloing prolem: 2'1qK½5. A here ½5" is defined for \. A key assumption is that the computation of ½5 is expensive and the gradient of ½5 is not computale at all. Therefore, e need a numerical procedure to estimate the gradient of ½S denoted y ¾./~½ r. (21) Folloing [7] e consider random perturations of the components of. For this e first consider a sequence of independent, identically distriuted (i.i.d.) random variales JA Z A A A % A. defined over some proaility space ËDFAJGµAhIK satisfying certain eak technical conditions given in [7]. E.g. they may e chosen Bernoulli ith T Àa š T ÀÁ š No let Â Ã B e a fixed sequence of numers. For any k e evaluate ½5 at to randomly and symmetrically chosen points V Â and ÀKÂ, respectively. Define the random vector 8;: Ä 8V: : A (( A 8;: Å Then the estimator of the gradient is defined as _ËmA/ 8V: Â ƒæ ½SS Â Ç½5 Â hè The fixed gain SPSA (simultaneous perturation stochastic approximation) procedure is then defined y! :k Ê9_ƒ (22) ith Ê Ã B fixed. The peculiarity of the procedure is, that for a and Â 0Ë B the correction term _Ë;AJ vanishes asymptotically. Fixed gain SPSA methods have een first considered in [4] in connection ith discrete optimization. A main result is that fixed gain SPSA applied to noisefree optimization yields geometric rate of convergence almost surely, just like deterministic gradient methods under appropriate conditions, see [5]. The convergence properties of the proposed fixed gain SPSA method can e easily estalished for quadratic functions. We have the folloing result: Theorem Let ½ e a positive definite quadratic function, ½5. š and let Â Â e fixed. Then, for sufficiently small Ê there is a deterministic constant )ZB, depending on Ê, such that for any initial condition =& outside of a set of Leesguemeasure zero e have 12 Ì Í %S# } # ith proaility 1. Sketch the proof: first, it is easy to see that for quadratic functions ¾ Since.K 0 : _ ËmA.'! :kž.ï Ê 8V:., e get the folloing recursion for 8;: A (2) No the sequence is i.i.d., hence the matrixvalued process ËÏ Ê 8V: is stationary and ergodic. Applying Oseledec s multiplicative ergodic theorem (cf. [8, 6]) the claim of the theorem follos immediately ith some deterministic, not necessarily negative. To sho that )OB for small Ê e use the result of []. Simple adaptive procedures for noisefree SPSA have een considered in earlier orks. A simple procedure is to use to gains and choose the one in each step that gives smaller function value. To our knoledge the est sitching strategy, minimizing the toplyapunov exponent is not knon. The prolem is hard even for to fixed matrices, and has een solved only recently y V. Blondel (yet unpulished ).

4 Ò Ò 4 Groth rate of ealthprocesses Let us consider a currency portfolio ÐŸÐ consisting of currencies. Thus Ð,Ð A A % A, here Ð denotes the asolute size of the portfolio held in the th currency at time 4. At any time 4 the exchange rates are collected in a `k matrix Ñ. Oviously, Ñ is random. Ñ ill e assumed to e a strictly stationary process. Based on past and present values of Ñ@ a realancing of the portfolio ill take place, so that a certain fixed percentage of dollar ill e converted into Euro or the other ay round. This realancing can e descried y a linear transformation: Ð@! :M Ð¼;A (24) here is a strictly stationary sequence of $Š random matrices, descriing the strategy of the investor. Let us focus on a parametric set of strategies 0. There is no reason to assume that l is a Markovprocess. The ealth or the value of the portfolio in say Euros ill e otained from a scalar product of the form l»ó Ð¼;A here ÓL is an appropriate ro of the random matrix Ñm. Then the groth rate of the ealth ill e Ô%12 Ì Í 4 hich, under reasonale conditions, is equal to the top Lyapunov exponent of lx. Its the maximization can e carried out y the procedure proposed in Section 1. 5 Simulation results Our first experiments sho the dependence of the top Lyapunov exponent on the stepsize Ê in the fixed gain SPSA method. As a enchmark example, e considered the prolem of minimizing a quadratic function of the form ½S ž The minimizing point as generated uniformly ithin the unit cue. The matrix as also generated randomly in the XÕ folloing ay: first e generated the eigenvalues of,, according to exponential distriution ith parameter y žb Ö, and considered the matrix Ø 0ÙL1ÛÚP¼ LÕ. Then e applied randomly chosen rotations, and considered «ÜˆØ Ü 8;:, here Ü is the product of random rotations. In Figure 1 e plotted the estimation of the toplyapunov exponent as the function of the stepsize Ê. In Figure 2 e plotted the estimation of the gradient of the top Lyapunov exponent, as it is computed in Section 2. It is seen, that the gradient vanishes around the minimizing point, Ê7Ý0B B Ö. In Figure e plotted the result of the proposed iterative scheme to find the optimal control Lyapunov exponents in 000 iterations. Acknoledgement The author expresses his thanks to James C. Spall and John L. Maryak of the Applied Physics Laoratory of Johns Hopkins University for cooperating in this research. This research has een supported y the National Research Foundation (OTKA) of Hungary under grant no T References [1] A. Benveniste, M. Métivier, and P. Priouret. Adaptive algorithms and stochastic approximations. SpringerVerlag, Berlin, [2] H. Furstenerg and H.Kesten. Products of random matrices. Ann. Math. Statist., 1: , [] L. Gerencsér. Almost sure exponential staility of random linear differential equations. Stochastics, 6: , [4] L. Gerencsér, S. D. Hill, and Zs. Vágó. Optimization over discrete sets via SPSA. In Proceedings of the 8th Conference on Decision and Control, CDC 99, pages IEEE, [5] L. Gerencsér and Zs. Vágó. A general frameork for noisefree SPSA. In Proceedings of the 40th IEEE Conference on Decision and Control, CDC 01, page sumitted., [6] M. S. Ragunathan. A proof of oseledec s multiplicative ergodic theorem. Israel Journal of Mathematics, 42:56 62, [7] J.C. Spall. Multivariate stochastic approximation using a simultaneous perturation gradient approximation. IEEE Trans. Automat. Contr., 7:2 41, [8] V.I.Oseledec. A multiplicative ergodic theorem. Lyapunov charasteristic numers for dynamical systems. Trans. Mosco Math. Soc., 19:197 21, 1968.

5 0.005 top Lyapunov exponent stepsize of the SPSA Figure 1: Top Ljapunov exponent as the function of the stepsize Estimation of the est stepsize Gradient of the Lyapunov exponent stepsize of the SPSA Figure : The result of the iterative scheme. Figure 2: The gradient of the toplyapunov exponent.

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/01/$ IEEE THE MATHEMATICS OF NOISE-FREE SPSA FrM08-4 László Gerencsér Computer and Automation Institute of the Hungarian Academy of Sciences, 13-17 Kende u., Budapest, 1111, Hungary gerencser@sztaki.hu Zsuzsanna